CATEGORIFIED JONES-WENZL PROJECTORS FOR ODD KHOVANOV HOMOLOGY By Dean Demetri Spyropoulos A DISSERTATION Submitted to Michigan State University in partial fulfillment of the requirements for the degree of Mathematics—Doctor of Philosophy 2025 ABSTRACT The Jones-Wenzl projectors are particular elements of the Temperley-Lieb algebra essential to the construction of quantum 3-manifold invariants. As a first step toward categorifying quantum 3-manifold invariants, Cooper and Krushkal categorified these projectors. In another direction, Naisse and Putyra gave a categorification of the Temperley-Lieb algebra compatible with odd Khovanov homology, introducing new machinery called grading categories. The first goal of this thesis is to provide a generalization of Naisse and Putyra’s work in the spirit of Bar-Natan’s canopolies or Jones’s planar algebras, replacing grading categories with grading multicategories. From this updated viewpoint, we describe an invariant of diskular tangles from odd Khovanov homology, naturally extending Naisse and Putyra’s tangle theory. In this thesis, the main application of our theory for diskular tangles is a proof of the existence and uniqueness of categorified Jones-Wenzl projectors in odd Khovanov homology. These results have a nearly immediate award: the existence of a new, “odd” categorification of the colored Jones polynomial. Finally, a major motivation to develop a tangle theory for odd Khovanov homology is to ultimately determine the state of its functoriality. In forthcoming work by the author, we study this question by approximating Khovanov’s argument for the original theory. In doing so, we develop a theory of Hochschild homology for modules and algebras graded by categories, indicating that the new constructions offered by grading categories are also deserving of study. Copyright by DEAN DEMETRI SPYROPOULOS 2025 For my father, and in loving memory of Paul Spyropoulos. iv ACKNOWLEDGEMENTS It is a pleasure to thank my advisors, Efstratia Kalfagianni and Matthew Stoffregen, for their guidance and support. Both have been fervent advocates for me, remaining constantly accessible and eternally encouraging—whatever success I am blessed to garner as a research mathematician will be owed to their many hours of assistance. I would also like to thank the remainder of my guidance committee, Teena Gerhardt and Matthew Hedden, for their interest in my research as well as their teaching insight. I am extremely grateful to have been part of the excellent topology group at MSU, of which these four are the backbone. I am indebted to Adam Lowrance for his advice and friendship, and for setting the example of the mathematician I aspire to be. Thanks are also due to Grégoire Naisse, Krzysztof Putyra, David Rose, Pedro Vaz, Stephan Wehrli, and Michael Willis for helpful conversations and suggestions related this thesis. I wish to thank my other collaborators—Amey Joshi, Rithwik Vidyarthi, and Chen Zhang—for their friendship (and their patience while I completed this thesis). Thank you to Joe Melby, Tristan Wells, Zhonghui Sun, and all the friends I’ve made at MSU. Thank you to Laura, Taylor, Alyssa, and the rest of the Math Department’s staff for their kindness and help in diffusing administrative issues. Finally, thank you to Elena, Charlie, Steve, my friends, and my family. v TABLE OF CONTENTS CHAPTER 1 . INTRODUCTION . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Unified projectors . 1.2 A gluing theorem for diskular tangles 1.3 Other applications . . 1.4 Future goals . . . 1.5 Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 3 7 9 . 11 . 13 . . . . . . . . . CHAPTER 2 CLASSICAL CATEGORIFICATIONS OF 𝑇 𝐿𝑛 AND PROJECTORS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15 . 15 2.1 Temperley-Lieb algebras and Jones-Wenzl projectors 2.2 Categorifications of the Temperley-Lieb algebra . . . . . . . . . . . . . . . . . 17 . 26 2.3 Cooper-Krushkal projectors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 3 THE ODD SETTING: CHRONOLOGIES AND G-GRADED STRUCTURES . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29 3.1 Chronological cobordisms and changes of chronology . . . . . . . . . . . . . . 29 3.2 Unified arc algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33 . . . . . . . . . . . . . . . . . . . . . . 35 3.3 A brief outline of C-graded structures CHAPTER 4 GRADING MULTICATEGORIES AND PLANAR ARC DIAGRAMS . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 44 4.1 . 50 (Grading) multicategories . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4.2 G as a grading multicategory . . . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.3 Generalities on modules graded by grading multicategories . . . . . . . . . . . 59 . 64 4.4 G -graded arc modules . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 5 C -SHIFTING SYSTEMS AND COBORDISMS . . . . . . . . . . . . . 68 5.1 A system of grading shifting functors for G . . . . . . . . . . . . . . . . . . . 68 5.2 Generalities on shifting systems for grading multicategories . . . . . . . . . . . 78 5.3 Homogeneous maps . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80 5.4 Changes of chronology . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 94 . CHAPTER 6 TANGLES, DG-MULTIMODULES, AND MULTIGLUING . . . . . . . 98 6.1 C -graded dg-multimodules and related concepts . . . . . . . . . . . . . . . . . 98 6.2 Resolution of diskular tangles . . . . . . . . . . . . . . . . . . . . . . . . . . . 100 . 110 dg-C -graded multimodules . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 CHAPTER 7 AN INVARIANT OF DISKULAR TANGLES . . . . . . . . . . . . . . 120 7.1 Quick computations in unified Khovanov homology . . . . . . . . . . . . . . . 120 . 125 7.2 Tangle invariant . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . CHAPTER 8 UNIFIED AND ODD PROJECTORS . . . . . . . . . . . . . . . . . . 8.1 Operations defined via multigluing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8.2 Duals and mirrors . . . . 141 . 141 . 148 vi 8.3 Definition and properties of unified projectors . . . . . . . . . . . . . . . . . . 159 8.4 Explicit computations for the 2-stranded projector . . . . . . . . . . . . . . . . 162 8.5 Existence of unified projectors . . . . . . . . . . . . . . . . . . . . . . . . . . 168 8.6 A unified colored link homology . . . . . . . . . . . . . . . . . . . . . . . . . 174 CHAPTER 9 TOWARD A HOCHSHILD (CO)HOMOLOGY FOR C-GRADED ALGEBRAS . . 177 9.1 More on C-graded algebras and bimodules . . . . . . . . . . . . . . . . . . . . 180 9.2 A C-graded bar resolution . . . . . . . . . . . . . . . . . . . . . . . . . . . . 191 9.3 The universal trace and C-graded Hochschild homology . . . . . . . . . . . . . 198 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . BIBLIOGRAPHY . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 206 vii CHAPTER 1 INTRODUCTION The Temperley-Lieb algebras, 𝑇 𝐿𝑛, are diagrammatic algebras originating from operator alge- bra theory which entered low-dimensional quantum topology with the construction of the Jones polynomial via representations of the braid group [Jon87]. Elements of particular importance are special idempotents of the Temperley-Lieb algebra, 𝑝𝑛 ∈ 𝑇 𝐿𝑛, called Jones-Wenzl projec- tors. These projectors have been studied extensively, and they are vital to the construction of the colored Jones polynomials and the skein theoretic construction of the Witten-Reshetikhin-Turaev 3-manifold invariants (cf. [Lic97], Chapter 13). In [Kho00], Khovanov provided a homological invariant of links whose graded Euler charac- teristic 𝜒 was the Jones polynomial, initiating the study of categorification. Since then, a major motivating question has been whether Khovanov’s categorification can be extended to a categorifi- cation of quantum 3-manifold invariants. It would stand to reason that the first step in replicating the procedures of the decategorified setting would be to construct categorical lifts of the Jones-Wenzl projectors, living in some categorification of the Temperley-Lieb algebra. A categorification of the Jones-Wenzl projectors was achieved by Cooper and Krushkal in [CK12]. First, Bar-Natan [BN05] provided a categorification of the Temperley-Lieb algebra in the sense that he constructed a category Kom(𝑛) whose Grothendieck group 𝐾0 was isomorphic to 𝑇 𝐿𝑛. Cooper and Krushkal then prove the existence of objects 𝑃CK 𝑛 of Kom(𝑛) which satisfy [𝑃CK 𝑛 ] = 𝑝𝑛, for [𝑃CK 𝑛 ] the equivalence class of 𝑃CK 𝑛 in 𝐾0(Kom(𝑛)). Said another way, 𝜒(𝑃CK 𝑛 ) = 𝑝𝑛. Rozansky [Roz14] has also given a construction of categorified projectors using the Khovanov complex associated to an infinite torus braid. For recent progress toward the categorification of quantum 3-manifold invariants from Khovanov homology, see [HRW22]. In this thesis, we initiate an investigation of similar phenomena for a different categorification of the Jones polynomial, called odd Khovanov homology. Suppose 𝐿 is a link. In [OS05], Ozsváth and Szabó constructed a spectral sequence converging to the Heegaard Floer homology of the double branched cover of 𝐿, ”HF(Σ(−𝐿); Z/2Z), with 𝐸2 page the (reduced) Khovanov homology of 1 𝐿, (cid:102)Kh(𝐿; Z/2Z). In an attempt to lift the spectral sequence to Z coefficients, Ozsváth, Rasmussen, and Szabó realized that the 𝐸2 page could no longer be ordinary reduced Khovanov homology. Instead, they produced a new candidate, another homological link invariant categorifying the Jones polynomial, closely related to Khovanov’s construction (indeed, necessarily agreeing over Z/2Z coefficients). Ozsváth, Rasmussen, and Szabó’s new construction [ORS13] is called odd Khovanov homol- ogy, which we denote by Kh𝑜 in this introduction; to avoid confusion, the original theory of [Kho00] has been retroactively declared even Khovanov homology, denoted Kh𝑒. While agree- ing in Z/2Z coefficients, there exist pairs of links 𝐿1 ≠ 𝐿2 for which Kh𝑒(𝐿1; Z) (cid:27) Kh𝑒(𝐿2; Z), but Kh𝑜(𝐿1; Z) ≇ Kh𝑜(𝐿2; Z), and vice-versa; see [Shu11]. We remark that spectral sequences from odd Khovanov homology to flavors of Floer homology have been discovered: Daemi [Dae15] showed that there is a spectral sequence from odd Khovanov homology to the plane Floer homology of the double branched cover, and Scaduto [Sca15] showed that another spectral sequence starting at odd Khovanov homology converges to the framed instanton homology of the double branched cover. Recall that even Khovanov homology is built from a functor F𝑒 with source the category whose objects are closed 1-manifolds and whose morphisms are embedded cobordisms, and a target category of K-modules, for some ring K. In the literature, a functor of this form is called a (1 + 1)-dimensional TQFT. Likewise, the original definition of odd Khovanov homology is built from a (perhaps misleadingly named) “projective TQFT”—that is, a TQFT well-defined only up to sign—of embedded cobordisms. Indeed, the TQFT of [ORS13], which we will denote by F𝑜, depends on some additional information. Using notation which will be introduced later (§3.1), this is pictured as á 𝑎𝑖 ë 𝑎𝑖+1 á 𝑎𝑖 ë 𝑎𝑖+1 F𝑜 = −F𝑜 . (1.0.1) 𝑎 𝑎 Moreover, F𝑜 is known to be sensitive to the exchange of critical points in embedded cobordisms between 1-manifolds. 2 Putyra, first in his Master’s thesis [Put10] and then in [Put14], introduced a refinement of the source category so that F𝑜 may be improved to a genuine functor. By a chronological cobordism, we mean a cobordism endowed with a framed Morse function, called a chronology, separating critical points; see §3.1. The chronology induces an orientation on each unstable manifold of index 1 and 2 critical points, which we draw as an arrow (as shown in (1.0.1) for an index 1 case). Consequently, F𝑜 is upgraded to a genuine functor: the equality above is reinterpreted as a relation between the maps on modules associated with two distinct chronological cobordisms. Going forward, functors from a category of chronological cobordisms to the category of K-modules will be called chronological TQFTs. Also introduced in [Put14] is the notion of a unified Khovanov complex, which is a complex over the ground ring 𝑅 = Z[𝑋, 𝑌 , 𝑍 ±1](cid:14)(𝑋 2 = 𝑌 2 = 1). The homology of this complex is called unified (also called covering or generalized in the literature) Khovanov homology. The unified Khovanov complex has the incredibly desirable feature of specializing to the even theory if one sets 𝑋 = 𝑌 = 𝑍 = 1, and to the odd theory by setting 𝑋 = 𝑍 = 1 and 𝑌 = −1. We use F (see §3.2) to denote the chronological TQFT for unified Khovanov homology. 1.1 Unified projectors As in Cooper and Krushkal’s work, our projectors will live in a categorification of the Temperley- Lieb algebra, which we denote by Chom(𝑛)G 𝑅 is the category of G -graded 𝐻𝑛-modules all of whose entries come from flat diskular tangles (see Figure 1.2 for an example of 𝑅 . Specifically, Chom(𝑛)G a non-flat diskular tangle). The algebra 𝐻𝑛 is the 𝑛th unified arc algebra; we review Khovanov’s arc algebras in §2.2.2 and unified arc algebras in §3.2. The notation “Chom” is meant to impress that we think of this category like the category “Kom” of [BN05], but with chronological cobordisms present. The notation G refers to the new grading essential to this thesis; we defer an introduction to G momentarily. The G -grading determines an integral 𝑞-grading (see §7.2.1). We let 𝐾 𝑞 0 denote the Grothendieck group which remembers only the 𝑞-grading and not the whole G -grading 3 information. Then, Chom(𝑛)G 𝑅 categorifies 𝑇 𝐿𝑛 in the sense that 𝐾 𝑞 0 (Chom(𝑛)G 𝑅 ) (cid:27) 𝑇 𝐿𝑛 as Z[𝑞, 𝑞−1]-algebras; see Definition 8.1.2 of §8.1. Specializing the ground ring 𝑅 by 𝑋, 𝑌 , 𝑍 = 1 defines a forgetful functor from Chom(𝑛)G 𝑅 to the category Kom(𝐻𝑛PMod), another categorification of 𝑇 𝐿𝑛 compatible with even Khovanov homology. This is the categorification of Khovanov, provided in [Kho02], using projective 𝐻𝑛- modules. Indeed, we will see that the G -grading is not essential to the even case—the objects of Kom(𝐻𝑛PMod) are not G -graded. Likewise, specializing by 𝑋, 𝑍 = 1 and 𝑌 = −1 induces a forgetful functor from Chom(𝑛)G 𝑅 to what we’ll denote by Chom(𝑛)G 𝑜 , a categorification of 𝑇 𝐿𝑛 implicit in the work of Naisse and Putyra. We call these the even and odd forgetful functors, and denote them by 𝜋𝑒 and 𝜋𝑜 respectively. Notice that the Z/2Z-reductions of both Kom(𝐻𝑛PMod) and Chom(𝑛)G 𝑜 agree; we denote by Kom(𝐻𝑛PMod)Z/2Z the corresponding category. The G -grading is also nonessential to the Z/2Z-reduction. We’ll denote the corresponding forgetful functors by 𝔣. Then 𝔣 ◦ 𝜋𝑒 = 𝔣 ◦ 𝜋𝑜; i.e., the diagram Chom(𝑛)G 𝑅 𝜋𝑒 𝜋𝑜 Kom(𝐻𝑛PMod) Chom(𝑛)G 𝑜 commutes. The following is proven in Chapter 8 as a combination of Proposition 8.3.5 and Theorem Kom(𝐻𝑛PModZ/2Z) 8.5.3. Theorem A. There exist categorifications of the Jones-Wenzl projectors, called unified projectors, 𝑃𝑛 in Chom(𝑛)G mean that [𝑃𝑛] ∈ 𝐾 𝑞 8.3.3). On one hand, 𝜋𝑒(𝑃𝑛) is a categorified projector in Kom(𝐻𝑛PMod), and 𝜋𝑒(𝑃𝑛) = 𝑃CK 𝑅 , which are unique up to chain-homotopy equivalence. By a categorification, we 𝑅 ) is equal to 𝑝𝑛 ∈ 𝑇 𝐿𝑛 (for a complete description, see Definition 0 (Chom(𝑛)G 𝑛 . On the other, under the odd forgetful functor, 𝜋𝑜(𝑃𝑛) is a new categorification of the 𝑛th Jones-Wenzl projector in Chom(𝑛)G 𝑜 . They both agree after reduction to Z/2Z-coefficients: 𝔣(𝑃𝑜 𝑛) = 𝔣(𝑃CK 𝑛 ). 4 We will write 𝑃𝑜 𝑛 to denote 𝜋𝑜(𝑃𝑛). We remark that Cooper and Krushkal’s projectors actually live in Bar-Natan’s category Kom(𝑛), but it is known that this category is equivalent to Khovanov’s categorification of 𝑇 𝐿𝑛, Kom(𝐻𝑛PMod). Following Section 6.4 of [NP20], we define a (diskular) tangle invariant Kh𝑞 in §7.2 which specializes to unified Khovanov homology when the tangle is a closed link. The caveat is that Kh𝑞 lives in a category Chom(𝑛)𝑞 𝑅; in general, Kh is not a tangle invariant in the category Chom(𝑛)G 𝑅 , so we must “collapse” the G -grading to an integral 𝑞-grading (see §7.2.1—the term “collapse” is slightly misleading). Regardless, by construction Kh𝑞 specializes to the even Khovanov tangle invariant, denoted Kh𝑒 𝑞, along with an odd Khovanov tangle invariant Kh𝑜 𝑞. In analogy with Section 5 of [CK12], the existence of these tangle invariants, together with Theorem A, is immediately useful. Namely, as the Jones-Wenzl projectors are vital to the construc- tion of the colored Jones polynomials 𝐽(𝐿; m)(𝑞), the existence of categorified projectors quickly implies the existence of a categorification of the colored Jones polynomial. Using the new categori- fication of the Jones-Wenzl projectors (compatible with odd Khovanov homology), we construct a new, “odd” categorification of the colored Jones polynomial. First, if 𝐿 is an 𝑛-component link and m = (𝑚1, . . . , 𝑚𝑛) ∈ N𝑛, denote by 𝑇 m 𝐿 the result of taking 𝑚𝑖 parallel copies of the 𝑖th component of 𝐿 for each 𝑖 = 1, . . . , 𝑛 and then removing a small diskular region from each of the original components (see Figure 1.1). Then, set Πm(𝐿) := (𝑃𝑚1, . . . , 𝑃𝑚𝑛) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑛 ) Kh𝑞(𝑇 m 𝐿 ) where each of the 𝑃𝑚𝑖 is viewed as an object of Chom(𝑚𝑖)𝑞 projectors into the tangle diagram 𝑇 m 𝑅. This has the effect of inserting 𝐿 ; again, consult Figure 1.1 for a schematic. See §1.2 for introductory remarks regarding this tensor product. Let Πm 𝑒 (𝐿) and Πm 𝑜 (𝐿) denote the complexes obtained by specializing 𝑅 by 𝑋 = 𝑌 = 𝑍 = 1, and 𝑋 = 𝑍 = 1, 𝑌 = −1 respectively. We call each of Πm(𝐿), Πm 𝑒 (𝐿), and Πm 𝑜 (𝐿) the unified, even, and odd m-colored Khovanov complexes of 𝐿, respectively. Finally, we define the unified, even, and odd m-colored Khovanov or link homologies of 𝐿 to be the homology of these complexes; we denote them by H (𝐿; m), H𝑒(𝐿; m), and H𝑜(𝐿; m) respectively. We emphasize that we define the 5 Figure 1.1 Schematic for Πm(𝐿), where 𝐿 is the 3-component link L11n314 of the Thistelthwaite link table, and m = (3, 2, 2). even and odd colored link homology by specializing 𝑅 before taking homology. Also, notice that H (𝐿; m) has coefficients in 𝑅, while H𝑒(𝐿; m) and H𝑜(𝐿; m) have coefficients in Z. Let 𝜒𝑞 denote the graded Euler characteristic which records only the 𝑞-grading associated to a particular G -grading or G -grading shift. Then, the following is proven in §8.6. Theorem B. For any colored link (𝐿; m), the chain-homotopy equivalence type of the m-colored Khovanov complex Πm(𝐿) is an invariant of (𝐿; m). Thus, the m-colored Khovanov homologies H (𝐿; m), H𝑒(𝐿; m), and H𝑜(𝐿; m) are invariants of (𝐿; m). Moreover, the even and odd homologies categorify the colored Jones polynomial in the sense that 𝜒𝑞(H𝑒(𝐿; m)) = 𝐽(𝐿; m)(𝑞) = 𝜒𝑞(H𝑜(𝐿; m)). On one hand, H𝑒(𝐿; m) is the colored link homology of Cooper and Krushkal. However, there are colored links (𝐿; m) for which H𝑜(𝐿; m) ≠ H𝑒(𝐿; m), so we obtain a new categorification of the colored Jones polynomial. To see that the two categorifications are distinct, we compute the unified Khovanov homology of the full trace of 𝑃2 (see §8.4.1 and, in particular, Equation (8.4.2)), which coincides with the unified colored link homology of the 2-colored unknot. We obtain the even and odd colored link homologies of the 2-colored unknot by taking homology after specializing the complex of Equation (8.4.1) to the even and odd settings. See Table 1.1 for the even (left) and odd (right) colored link homologies of the 2-colored unknot, where we have expressed the homology in terms of quantum grading 𝑞 and homological grading ℎ. 6 0 Z Z ℎ 𝑞 2 0 −2 −4 −6 −8 −10 −1 −2 −3 −4 −5 Z Z/2 Z Z Z/2 Z . . . 0 Z Z ℎ 𝑞 2 0 −2 −4 −6 −8 −10 −1 −2 −3 −4 −5 Z Z Z Z Z Z Z Z . . . Table 1.1 H𝑒(𝑈; 2) and H𝑜(𝑈; 2), respectively, where ℎ is homological grading and 𝑞 is quantum grading Interestingly, the 2-colored unknot has no torsion. However, using computations provided by Schütz (see Theorem 8.2 and Figure 9 of [Sch22]), the odd colored homology of the 3-colored unknot, H𝑜(𝑈; 3), contains Z/3Z-torsion, whereas H𝑒(𝑈; 3) contains no Z/3Z-torsion. Finally, note that the graded Euler characteristics of both sides agree. 1.2 A gluing theorem for diskular tangles The majority of this thesis is devoted to developing a framework for the construction and calculation of unified projectors. This will entail setting up a tangle theory that is both compatible with unified Khovanov homology and will also allow for a very flexible notion of composition for tangles. Thankfully, the work of Naisse and Putyra [NP20] (to which we will keep returning) accomplishes the former goal—thus, our goal is a generalization of their work which allows for this “more flexible gluing property.” To be clear, recall that Khovanov’s theory for knots and links has been extended to tangles via at least two methods, by both Khovanov [Kho02] and Bar-Natan [BN02, BN05] (see Chapter 2 for a review). In the former, Khovanov extended his work to tangles with an even number of endpoints, showing that the homotopy type of the complex he associates to each tangle is an invariant of the tangle. Furthermore, for each tangle 𝑇, the complex Kh(𝑇) has an interpretation as a graded dg- bimodule over the so-called arc algebras, 𝐻𝑛. Paramount among the properties of these bimodules is the gluing result, which states that, for stackable tangles 𝑇 and 𝑆, Kh(𝑇) ⊗𝐻𝑛 Kh(𝑆) (cid:27) Kh(𝑇 𝑆). 7 While Khovanov and Bar-Natan were able to describe an up-to-homotopy invariant complex associated to a tangle soon after the discovery of Khovanov homology, an analogue for odd Khovanov homology remained elusive for thirteen years after its discovery. Our work will employ the first known solution, provided by Naisse and Putyra in [NP20]. Before detailing their solution, we remark that, in [Vaz20], Vaz constructed a supercategory and derived from it a homological invariant of tangles which supercategorified the Jones polynomial. While he proved that his invariant was distinct from even Khovanov homology, it was not evident that his theory was isomorphic to odd Khovanov homology when restricted to links until the recent work of Schelstraete and Vaz in [SV23]. There, Schelstraete-Vaz provided another lift of odd Khovanov homology to tangles (indeed, their work succeeded in providing the first representation theoretic construction of odd Khovanov homology) which coincided with the “not even Khovanov homology” of [Vaz20]. Naisse and Putyra conjectured that their tangle invariant is isomorphic to Vaz’s, and thus to Schelstraete-Vaz’s, but this remains an open question. Naisse and Putyra’s lift of odd Khovanov homology to tangles [NP20] involves the introduction of objects called grading categories which allow one to define categories of (dg-) bimodules graded by a selected grading category. The grading category for the problem at hand is a category G whose morphisms are given by a pair of a flat tangle (with even inputs and even outputs) and an element of Z × Z. Viewing the unified arc algebra as a G-graded algebra, 𝐻𝑛 becomes graded-associative (associativity fails before this change; see §3.2 and 3.3, and [NV18] for more detail). In the context of grading categories, it is more difficult to define what is meant by a grading shift. In order to accomplish this, Naisse and Putyra implement shifting systems which can be assigned to a grading category; in the case of G, a shifting system is provided by a pair of a chronological cobordism and a shift in the Z × Z-grading. See §3.3 for a more thorough introduction to grading categories and shifting systems. For Naisse and Putyra, all this work meant that one could mimic the constructions of Khovanov in [Kho02] in a graded-associative context, yielding a tangle version of unified Khovanov homology which respects the gluing property. Continuing the analogy, the goal of the majority of this thesis 8 is to provide a generalization of the gluing result of [NP20] in the spirit of Bar-Natan’s canopolies [BN05] or of Jones’s planar algebras [Jon22]. While the extension is minor and well known in the even setting (see a description in Section 4 of [LLS22]), realizing the analogous result in the odd setting, in this thesis, means adapting the flat tangles of Naisse and Putyra to planar arc diagrams. In particular, the grading category G is upgraded to what we call a grading multicategory, denoted G . Then, the work of Naisse and Putyra provide us with a roadmap for proving what we refer to as “multigluing,” Theorem 6.2.4. The following is a statement of multigluing in lesser generality than we prove it. Recall that F is the unified chronological TQFT. Theorem C. Suppose 𝑇 is a diskular tangle of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛) (see Definition 4.0.1) and 𝑇𝑖 is a tangle diagram in a disk with 2𝑚𝑖 points on its boundary for each 𝑖 = 1, . . . , 𝑘. Then there is an isomorphism (cid:0)F (𝑇1), . . . , F (𝑇𝑘 )(cid:1) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) F (𝑇) (cid:27) F (𝑇(𝑇1, . . . , 𝑇𝑘 )). The notation ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ), as well as the map inducing this isomorphism, is described in depth in Chapter 4. The idea of this theorem is that, given a tangle with some holes punched out, and compatible tangles 𝑇1, . . . , 𝑇𝑘 , we can define a tensor product so that some tensor product of the dg-modules associated to 𝑇1, . . . , 𝑇𝑘 (denoted by (𝑇1, . . . , 𝑇𝑘 )) tensored with the multimodule associated to 𝑇 is isomorphic (as G -graded dg-modules) to the dg-module associated to 𝑇 filled by the tangles 𝑇1, . . . , 𝑇𝑘 . See Figure 1.2. 1.3 Other applications While our main motivation for this thesis is a proof of existence for unified projectors and a new categorification of the colored Jones polynomial, there are other notable benefits of a more flexible gluing theorem; we will describe a few in our paper. To start, we can use Theorem 6.2.4 to define operations on G -graded dg-modules (e.g., a vertical stacking operation ⊗, juxtaposition ⊔, and a partial trace Tr) in exactly the same way as [SW24], see §8.1. Defining these operations is essential as, without them, we cannot define categorified projectors. Of particular interest are our lifts of well-known adjunction statements provided by Hogancamp [Hog20, Hog19]. 9 Figure 1.2 Multigluing schematic. Here, we assume 𝑇1, 𝑇2, and 𝑇3 are each tangles in disks with 4 points on their boundary. Theorem 8.1.5. If 𝐴 and 𝐵 are G -graded dg-modules coming from tangle diagrams on 𝑛 − 1 and 𝑛 strands respectively, then Ñ é Hom𝑛 𝐴 ⊔ 1, 𝜑Å B , (0,1) ã𝐵 (cid:27) Hom𝑛−1(𝐴, Tr(𝐵){−1, 0}). See the statement of Theorem 8.1.5 found in Chapter 8 for more details. The notation Hom𝑛 denotes the complex of maps of homogeneous bidegree; see §5.3 and 6.1. Notice the G -grading shift which is invisible to 𝑞-degree. We also obtain a more familiar statement, which we use in the proof of uniqueness for unified projectors: Hom𝑛(𝐴 ⊗ F (𝛿), 𝐵) (cid:27) Hom𝑛(𝐴, 𝐵 ⊗ F (𝛿∨)) where 𝛿 is a flat tangle. It comes as a corollary of another familiar “duality” statement; see Theorem 8.2.3. We remark that, in §8.5, we construct 𝑃𝑛 as arising from an infinite torus braid, as in [Roz14] and Section 5 of [SW24]. This description awards us with another, inductive description of 𝑃𝑛 as a filtered chain complex, inducing a spectral sequence. Explicitly, if 𝑃𝑛 and 𝑃𝑛−1 are projectors, we 10 have that 𝑃𝑛 = · · · 𝑃𝑛−1 · · · ... where the wrap-around repeats indefinitely. It follows that 𝑃𝑛 is the colimit of a filtered chain complex of the following form. · · · 𝑃𝑛−1 · · · · · · 𝑃𝑛−1 · · · · · · · · · 𝑃𝑛−1 · · · · · · 𝑃𝑛−1 · · · · · · The filtration on 𝑃𝑛 induces one on its full trace, and using results of §8.3, we conclude that Hom𝑛(𝑃𝑛, 𝑃𝑛) is a filtered complex. We will investigate the associated graded of this filtration in future work. 1.4 Future goals Further motivation for our work was provided by some questions left unanswered in this thesis. We conclude the introduction by outlining a few of them. Periodicity of projectors and a GOR conjecture We note (see Corollary 8.3.7) that the existence of unified projectors (Theorem 8.5.3), together with an adjunction statement (Theorem 8.1.5), implies that 𝐻∗Hom𝑛(𝑃𝑛, 𝑃𝑛) (cid:27) 𝑞−𝑛H (𝑈; 𝑛). (1.4.1) In [Hog19], Hogancamp uses the specialization of Equation (1.4.1) to the even setting in order to construct particular elements 𝑈𝑛 ∈ Hom𝑛(𝑃CK 𝑛 , 𝑃CK 𝑛 ) to make substantial progress toward a conjecture of Gorsky-Oblomkov-Rasmussen [GOR13, GORS14]. The chain maps 𝑈𝑛 take the form 𝑡2−2𝑛𝑞2𝑛𝑃CK 𝑛 → 𝑃CK 𝑛 and satisfy Cone(𝑈𝑛) ≃ 𝑄𝑛 (for a particular complex 𝑄𝑛), showing that 𝑃CK 𝑛 is a periodic chain complex built from copies of 𝑄𝑛. Interestingly, in [Sch22], Schütz computes the first few odd projectors 𝑃𝑜 2 and 𝑃𝑜 3 algorithmically and shows that, while odd 𝑃𝑜 2 is 11 also periodic of period 2, odd 𝑃𝑜 3 which has period 4 (cf. Section 4.4 of [CK12]). We hope to use results of this thesis to prove that 𝑃𝑛 remains periodic in 3 is periodic of period 8, unlike even 𝑃CK the unified and odd settings, and to determine the period of 𝑃𝑛 for arbitrary 𝑛. Odd Khovanov spectra for tangles The idea for generalizing the work of Naisse-Putyra via dg-multimodules associated to diskular tangles came largely from observations of the utility of spectral multimodules in the work of Lawson-Lipshitz-Sarkar [LLS23, LLS22] and Stoffregen-Willis [SW24]. Now, an odd (indeed, unified) Khovanov homotopy type is known [SSS20], but it has yet to be lifted to the setting of tangles—we hope that our work might be melded with that of [LLS23] and [SSS20] to produce a unified homotopy type for tangles. If this is accomplished, it is also our hope that the work here will allow for the arguments of [LLS22] to lift, proving that homotopy functoriality holds in higher generality. It is also interesting to note that the spectral projector on three strands of [SW24] is periodic of period 8, like the odd projector on three strands of [Sch22], but unlike the three-stranded even projector. Investigating functoriality The last two chapters in this Thesis are addenda regarding the investigation of functoriality for odd Khovanov homology. In the fall of 2024, Migdail and Wehrli [MW24] gave the first proof that odd Khovanov homology is functorial up-to-sign, without passing to a tangle theory. In forthcoming work [Spy25], introduced in Chapter 9, we note that there is a natural definition of a C -graded bar resolution and Hochschild homology for C -graded algebras. Using this result, we may mimic Khovanov’s proof of functoriality [Kho02] (see [LLS22] for an excellent outline) in the unified setting to obtain a second proof of up-to-unit functoriality for unified Khovanov homology, and thus up-to-sign functoriality for odd Khovanov homology. Perhaps more interesting (especially if aiming for a Lasagna-type invariant coming from a functorial invariant of links in 𝑆3 [MWW22, MWW24]) is the development of a functorial “oriented model” [Bla10] for odd Khovanov homology. Such a model is provided in [SV23], but the question of functoriality remains open. In forthcoming joint work with Matthew Stoffregen [SS25], we 12 consider an alternative approach to obtaining a functorial-with-signs model for odd Khovanov homology, using the existence of a spectral sequence from odd Khovanov homology to plane Floer homology [Dae15] (the latter is known to be functorial). Indeed, in [SS25], we prove a conjecture of Migdail and Wehrli from [MW24]: that odd Khovanov homology any 2-knot 𝜁 counts the number of spin𝑐-structures on the branched double cover of 𝜁 branched along 𝑆4. We omit a discussion of our work from this thesis. 1.5 Outline Chapters 2 and 3 of this thesis are preparatory and intended as introductions for the uninitiated; experts should feel free to skip them. In Chapter 2, we start by recalling the definition of the Temperley-Lieb algebra and its special elements, the Jones-Wenzl projectors. Then, we recall basic features of the categorifications of 𝑇 𝐿𝑛 by both of Bar-Natan [BN02, BN05] and Khovanov [Kho02]. Finally we review some facts about the first categorification of Jones-Wenzl projectors, following [CK12]. Chapter 3 is devoted to reviewing some of the work of Putyra regarding categories of chronological cobordisms [Put10, Put14]. We end Chapter 3 by presenting an outline of C-graded structures, for a grading category C, as in [NP20]—the hope is that §3.3 might give the reader a bird’s-eye view of the goals of Chapters 4, 5, and 6. Chapters 4, 5, and 6 are the technical heart of this thesis, wherein we introduce grading multi- categories, shifting 2-systems for those grading multicategories, and apply the general framework constructed to prove multigluing, Theorem 6.2.4. Again, see §3.3 for a more complete outline. In Chapter 7, we use multigluing to obtain an invariant of (diskular) tangles, slightly generalizing a result of [NP20]. As in the cited paper, the grading system is, perhaps, too sensitive for the complex associated to a (diskular) tangle diagram to be invariant under each of the Reidemeister moves (see Lemmas 7.2.3, 7.2.4, and 7.2.6). However, it is invariant up to a grading shift in which the number of saddles in the cobordism component is equal to the sum of the entries of the Z × Z component. Hence, we can “collapse” the G -degree to an integral 𝑞-grading in to obtain a tangle invariant. We remark that, however slight the generalization, the added flexibility is necessary for our final result in §8.6 (additionally, we believe the differences in our proof to be notable). 13 Finally, in Chapter 8, we define and prove the existence and uniqueness of categorifications of the Jones-Wenzl projectors living in a category of G -graded dg-modules, specializing to the projectors of [CK12], but also to “odd” projectors which, prior to this thesis, had only been computed up to three or so strands (cf. [Sch22]). Other highlights of this section are the proofs of the aforementioned duality and adjunction results, which we hope to be useful in future work. In conclusion, we point out that the existence of unified projectors, together with multigluing and the tangle invariant of Chapter 7, imply the existence of a unified colored link homology, specializing to the colored link homology of, say, [CK12], but also to a new, “odd” categorification of the colored Jones polynomial. The final chapter is an addendum initiating further investigation into C-graded structures, especially motivated by questions related to the functoriality of odd Khovanov homology, since [MW24]. In Chapter 9, we provide a careful study of grading categories to develop a general theory of Hochschild homology for algebras graded by grading categories. This chapter is a portion of the forthcoming work [Spy25], in which we apply the general framework introduced here to the odd arc algebras. 14 CHAPTER 2 CLASSICAL CATEGORIFICATIONS OF 𝑇 𝐿𝑛 AND PROJECTORS In this chapter, we survey attributes of the even setting which we hope to lift—in one way or another—to the odd setting. In §2.1, we briefly discuss the decategorified setting. In §2.2, we recall the even categorifications of the Temperley-Lieb algebras due to Bar-Natan [BN05] and Khovanov [Kho02]. We conclude by providing Cooper and Krushkal’s categorification of the Jones-Wenzl projectors in §2.3, as we hope to compare their results with our work in §8. 2.1 Temperley-Lieb algebras and Jones-Wenzl projectors The Temperley-Lieb algebras 𝑇 𝐿𝑛 arise naturally as the 𝑈𝑞(𝔰𝔩2)-equivariant endomorphisms of 𝑛-fold tensor powers of the fundamental representation of 𝑈𝑞(𝔰𝔩2). As a unital Z[𝑞, 𝑞−1]-algebra, 𝑇 𝐿𝑛 is generated by 𝑛 elements 1𝑛, 𝑒1, . . . , 𝑒𝑛−1 subject to the relations 1. 𝑒𝑖𝑒 𝑗 = 𝑒 𝑗 𝑒𝑖 if |𝑖 − 𝑗 | ≥ 2, 2. 𝑒𝑖𝑒𝑖±1𝑒𝑖 = 𝑒𝑖, and 3. 𝑒2 𝑖 = (𝑞 + 𝑞−1)𝑒𝑖. The first relation is referred to as “distant commutativity.” We will make use of the quantum integer notation 𝑞𝑘 − 𝑞−𝑘 𝑞 − 𝑞−1 so that, for example, the third relation can be rewritten 𝑒2 [𝑘] = 𝑖 = [2]𝑒𝑖. 𝑇 𝐿𝑛 can be given a diagramatic description, where the generating elements are presented by 1𝑛 = · · · and 𝑒𝑖 = · · · · · · 𝑖 𝑖 + 1 with multiplication given by top-to-bottom vertical stacking. Therefore, 𝑇 𝐿𝑛 can be viewed as the linear skein of the disk with 2𝑛 distinguished points on its boundary, where we regard this disk as a square with 𝑛 marked points on the top and 𝑛 marked points on the bottom. It is in this way 15 that every (𝑛, 𝑛)-tangle may be assigned an element of 𝑇 𝐿𝑛; indeed, given an oriented tangle, the relations = 𝑞 − 𝑞2 and = 𝑞−2 − 𝑞−1 yield the Jones polynomial up to normalization. In [Lic93], it was shown that the Witten-Reshetikhin-Turaev 3-manifold invariants ([Wit89, RT91]) may be constructed combinatorially via the Kauffman bracket. Key ingredients of this construction are the Jones-Wenzl projectors, which we recall now. Definition 2.1.1. The Jones-Wenzl projectors, denoted by 𝑝𝑛, are particular elements of 𝑇 𝐿𝑛, defined by the recursion 𝑝1 = 11 and 𝑝𝑛+1 = (𝑝𝑛 ⊔ 1) − [𝑛] [𝑛 + 1] (𝑝𝑛 ⊔ 1)𝑒𝑛−1(𝑝𝑛 ⊔ 1). It is common to depict 𝑝𝑛 by a box 𝑝𝑛 = 𝑛 in which case the recursion appears as 1 = and 𝑛 + 1 = 𝑛 − [𝑛] [𝑛 + 1] 𝑛 𝑛 . The Jones-Wenzl projectors are well-studied. They may be defined equivalently as the unique elements of 𝑇 𝐿𝑛 for which (JW1) (𝑝𝑛 − 1𝑛) belongs to the algebra generated by {𝑒1, . . . , 𝑒𝑛−1}, and (JW2) 𝑝𝑛𝑒𝑖 = 𝑒𝑖 𝑝𝑛 = 0 for all 𝑖 = 1, . . . , 𝑛 − 1. These properties immediately imply that the projectors are idempotents. One can also check that upon taking the Markov closure of the projectors, the Kauffman bracket evaluates them as a quantum integer: 𝑝𝑛⟩ = [𝑛 + 1]. ⟨(cid:99) 16 The purpose of listing these well-known properties of the Jones-Wenzl projectors is that their categorifications satisfy analogues in the categorified setting. We will use these properties fre- quently in what follows. 2.2 Categorifications of the Temperley-Lieb algebra We start by reviewing a construction of Bar-Natan [BN02, BN05] which categorifies 𝑇 𝐿𝑛. Consequently, we may determine the Khovanov complex for a tangle, which turns out to be a tangle invariant up to homotopy. Afterwards, we describe another categorification of Khovanov, which has a known analogue in the odd setting. In the broader context of this thesis, we wish to review Bar-Natan’s categorification to motivate our grading multicategory G , defined in Chapter 4. Recall that a pre-additive category C is a category such that 1. for every 𝑋, 𝑌 ∈ ob(C), HomC(𝑋, 𝑌 ) is an abelian group, and 2. morphism composition distributes over the abelian group’s addition rule. Additionally, a monoidal category C is a category endowed with a functor ⊗ : C × C → C, a distinguished object 1 ∈ ob(C), and natural isomorphisms 𝛼 (called the associator) and left- and right-unitors 𝜆 and 𝜌 satisfying the triangle and pentagon identities. Given a pre-additive category C, we may define the (split) Grothendieck group of C to be the free abelian group generated by isomorphism classes in C, with the added relation that [𝐴 ⊕ 𝐵] = [𝐴] + [𝐵]: 𝐾0(C) = Z⟨C⟩ (cid:44)   [𝐴] = [𝐵] if 𝐴 (cid:27) 𝐵   .  [𝐴 ⊕ 𝐵] = [𝐴] + [𝐵]  It is common to take the Grothendieck group of pre-addivive monoidal categories—in this case, the tensor product induces an algebra structure on 𝐾0(C). For us, to categorify 𝑇 𝐿𝑛 means to define a pre-additive monoidal category C for which 𝐾0(C) (cid:27) 𝑇 𝐿𝑛. Here is an outline of the construction provided by Bar-Natan. Step 1: Let pre-Cob(𝑛) denote the cateory whose 17 • objects are isotopy classes of formally 𝑞-graded Temperley-Lieb diagrams with 2𝑛 boundary points, and • Hom(𝑞𝑖 𝐴, 𝑞 𝑗 𝐵) is the free Z-module spanned by isotopy classes of orientable cobordisms from 𝐴 to 𝐵. Note that pre-Cob(𝑛) is pre-additive by definition. All of our cobordisms will be oriented upwards (from bottom to top). It is also naturally monoidal via stacking in 𝑇 𝐿𝑛. It is clear that if 𝐶 : 𝐴 → 𝐵 and 𝐶′ : 𝐴′ → 𝐵′, then there is a cobordism 𝐶 ⊗ 𝐶′ : 𝐴 ⊗ 𝐴′ → 𝐵 ⊗ 𝐵′. Definition 2.2.1. The degree of a cobordism 𝐶 : 𝑞𝑖 𝐴 → 𝑞 𝑗 𝐵 is the value deg(𝐶) = deg𝑡(𝐶) + deg𝑞(𝐶) where (i) deg𝑡(𝐶) = 𝜒(𝐶) − 𝑛 is called the topological degree of 𝐶, and (ii) deg𝑞(𝐶) = 𝑗 − 𝑖 is called the quantum degree of 𝐶. It is common practice to fix 𝑞-gradings on the Temperley-Lieb elements so that deg(𝐶) is always zero. There are a few special cobordisms which we highlight here. Their frequent use necessitates additional (but commonplace) notation. (1) Cobordisms in this category may be decorated by dots, which correspond to hollow handle attachments up to multiplication by 2. =: 2 • = 2 • 18 (2) Saddles will have the following shorthand. = Note that, for example Ñ é deg𝑡 • = 𝜒 Ä𝑆1 ∨ 𝑆1ä − 1 = −2 since the dotted identity has the same homotopy type as the punctured torus, and Ñ é deg𝑡 = 𝜒 ÄD2ä − 2 = −1. Therefore, we will take dots to increase quantum degree 2 and saddles to increase quantum degree 1. Step 2: Pass to the matrix category Mat(pre-Cob(𝑛)), whose objects are vectors of objects in pre-Cob(𝑛) and whose morphisms are matrices of morphisms in pre-Cob(𝑛). Observing the defining relations in 𝑇 𝐿𝑛, to construct a category C for which 𝐾0(C) (cid:27) 𝑇 𝐿𝑛, the object represented by ⃝ in C must be isomorphic to the sum of two empty objects in degree ±1: ⃝ (cid:27) 𝑞−1 ∅ ⊕ 𝑞 ∅. We accomplish this by defining delooping operations. Consider the morphisms and à í • 𝜑 : ⃝ −−−−−−−−−−→ 𝑞−1 ∅ ⊕ 𝑞 ∅ Ö è 𝜓 : 𝑞−1 ∅ ⊕ 𝑞 ∅ • −−−−−−−−−−−−−−−→ ⃝. 19 We impose the isomorphism above by defining the relations implied by 𝜑 ◦ 𝜓 = idZ⊗Z and 𝜓 ◦ 𝜑 = id⃝. On one hand, On the other, à í 𝜑 ◦ 𝜓 = 𝜓 ◦ 𝜑 = • • • • . • + • . In conclusion, we define Cob(𝑛) to be the quotient of pre-Cob(𝑛) by the relations = 0, • = 1, •• = 0, and • + • = . The first three relations are called the sphere relations (referred to as S0, S1, and S2 respectively), and the last relation is called the tube-cutting relation. Interestingly, the sphere with three dots does not have an evaluation. The most general remedy is cosmetic, and it is treated as a free variable. Explicitly, in Cob(𝑛), we declare a fourth sphere relation by setting However, in what follows, we will take 𝛼 to be zero; that is, we will replace the last sphere relation ••• = 𝛼. (S2) with the relation • • = 0. Lemma 2.2.2. There is an isomorphism of Z[𝑞, 𝑞−1]-algebras 𝐾0(Cob(𝑛)) (cid:27) 𝑇 𝐿𝑛. Proof. Multiplication by 𝑞 defines an endofunctor Cob(𝑛) → Cob(𝑛), which in turn determines an endomorphism on 𝐾0(Cob(𝑛)), making it a Z[𝑞, 𝑞−1]-algebra. Then the result is immediate. □ 20 Step 3: Finally, we’d like a way to assign to a tangle in the 3-ball with 2𝑛 marked points some collection of objects in Cob(𝑛). Definition 2.2.3. Let Kom(𝑛) = Kom(Mat(Cob(𝑛))) denote the category of partially bounded chain complexes of finite direct sums of objects in Cob(𝑛). In this thesis, we allow complexes with unbounded negative homological degree in keeping with [Hog19], but opposed to, for example, [CK12]. The tensor product of chain complexes extends ⊗ in Cob(𝑛) to Kom(𝑛): schematically, (cid:32) 𝐶 ⊗ 𝐷 = · · · 𝐶2 𝐶1 𝐶0 ⊗ · · · 𝐷2 𝐷1 𝐷0 (cid:33) (cid:32) (cid:33) = · · · 𝐶2 𝐷0 ⊕ 𝐶1 𝐷1 ⊕ 𝐶0 𝐷2 𝐶1 𝐷0 ⊕ 𝐶0 𝐷1 𝐶0 𝐷0 . Indeed, passing to the homotopy category of a pre-additive category does not change the Grothendieck group up to isomorphism; see Section 2.7.1. of [CK12] for a full discussion. Lemma 2.2.4. There is an isomorphism of Z[𝑞] (cid:74) 𝑞−1 -algebras (cid:75) 𝐾0(Kom(𝑛)) (cid:27) 𝑇 𝐿𝑛. denote the complex corresponding to a tangle 𝑇, obtained by the skein relations Let 𝑇 (cid:74) (cid:75) (cid:114) (cid:122) = 𝑞 𝑞2 and (cid:114) (cid:122) = 𝑞−2 𝑞−1 where the underlined term is in homological degree zero. For example, (cid:117) (cid:119) (cid:118) (cid:125) (cid:127) (cid:126) = 𝑞−1 Ö è⊤ á Ö ë 𝑞0 ⊕ è 𝑞 21 Notice that there is a free loop in homological grading zero, hence we may apply the delooping operations to yield the complex 𝑞−1 𝐴⊤ 𝑞−1 ⊕ 𝑞0 ⊕ 𝑞1 𝐵 𝑞 (2.2.1) where and Ç 𝐴⊤ = å⊤ Ç id ◦ 𝜑 = å⊤ Ç 𝐵 = 𝜓 ◦ å Ç = å id . 2.2.1 Chain homotopy lemmas In [BN07], delooping was introduced alongside the following lemma from homotopy theory to simplify computations in Khovanov homology. Lemma 2.2.5 (Simultaneous Gaussian elimination). Suppose A is a pre-additive category, and let 𝐾∗ be an object of Kom(A) of the form 𝐴0 ⊕ 𝐶0 𝑀0 𝐴1 ⊕ 𝐵1 ⊕ 𝐶1 𝑀1 𝐴2 ⊕ 𝐵2 ⊕ 𝐶2 𝑀2 · · · à 𝑎0 𝑐0 𝑑0 𝑓0 where 𝑀0 = í à and 𝑀𝑖 = í 𝑎𝑖 𝑏𝑖 𝑑𝑖 𝑒𝑖 𝑐𝑖 𝑓𝑖 for all 𝑖 > 0. If 𝑎2𝑖 : 𝐴2𝑖 → 𝐴2𝑖+1 and 𝑗𝑖 : 𝐵2𝑖+1 → 𝐵2𝑖+2 are isomorphisms for all 𝑖 ≥ 0, then the chain complex 𝐾∗ is homotopy 𝑔𝑖 ℎ𝑖 𝑔0 𝑗0 𝑒2𝑖1 equivalent to the complex 𝐶0 𝑄0 𝐶1 𝑄1 𝐶2 𝑄2 · · · where   𝑄2𝑖 = 𝑗2𝑖 − 𝑔2𝑖𝑎−1  𝑄2𝑖+1 = 𝑗2𝑖+1 − ℎ2𝑖+1𝑒−1 2𝑖 𝑐2𝑖 2𝑖+1 𝑓2𝑖+1 . Proof. This is an application of the simpler “Gaussian elimination,” see [CK12]. □ 22 As an application, note that we may apply simultaneous Gaussian elimination to the complex (2.2.1). The result is that the complex (cid:115) (cid:123) is homotopy equivalent (hereinafter written ≃) to the chain complex 0 0 ; i.e., the complex is invariant, up to chain homotopy equivalence, 𝑇 (cid:74) (cid:75) under Reidemeister II moves for tangles. The following is due to Bar-Natan. Theorem 2.2.6 (Theorem 1 of [BN05]). The homotopy class of the complex Kom(𝑛) is an invariant of the tangle 𝑇. regarded in 𝑇 (cid:74) (cid:75) To conclude this subsection, we note that there is a notion of a zero object in Kom(𝑛): we call a chain complex 𝐾∗ contractible if 𝐾∗ ≃ 0. The following is well known. Lemma 2.2.7 (Big collapse). A chain complex 𝐾∗ of contractible chain complexes is, itself, contractible. 2.2.2 Khovanov’s arc algebras Another categorification, provided by Khovanov [Kho02], is given by the category of complexes of 𝐻𝑛-modules, where 𝐻𝑛 is the 𝑛th arc algebra, described below. These can be generalized to the unified setting; see [NV18] for a thorough discussion. We will use arc algebras to describe odd Khovanov complexes for tangles, following [Put14] and [NP20]. A large portion of this thesis is devoted to providing a small generalization Naisse-Putyra’s construction, allowing one to perform Bar-Natanesque computations in a particular category of 𝐻𝑛-modules. Consider the Temperley-Lieb 2-category T L, whose • objects are natural numbers, • 1-morphisms HomT L(𝑚, 𝑛) are isotopy classes of crossingless tangles embedded in the square with 2𝑚 marked points on the [0, 1] × {0} axis and 2𝑛 marked points on the [0, 1] × {1} axis, and • 2-morphisms HomT L(𝑡, 𝑠) are cobordisms with corners from the crossingless tangle 𝑡 to 𝑠. 23 Write 𝐵𝑛 𝑚 = HomT L(𝑚, 𝑛). In the case that 𝑚 = 0, we write 𝐵𝑛 (respectively, 𝑛 = 0 is written 𝐵𝑚); this is the collection of crossingless matchings of 𝑛 points fixed on the top axis (resp., 𝑚 on the bottom axis). We will write |𝑎| = 𝑛 for 𝑎 ∈ 𝐵𝑛. Composition of 1-morphisms is given by stacking: 𝐵 𝑝 𝑛 × 𝐵𝑛 𝑚 is given by (𝑠, 𝑡) ↦→ 𝑡𝑠. There is also a mirroring operation, · : 𝐵𝑛 𝑚 → 𝐵 𝑝 𝑚 → 𝐵𝑚 𝑛 , which flips tangles about the line [0, 1] × {1/2}. Let 𝑎 ∈ 𝐵𝑚, 𝑏 ∈ 𝐵𝑛, and 𝑡 ∈ 𝐵𝑛 𝑚. Then 𝑎𝑡𝑏 is a closed 1-manifold. Let 𝑠 ∈ 𝐵 𝑝 𝑛 and 𝑐 ∈ 𝐵𝑝. Consider the cobordism (𝑎𝑡𝑏)(𝑏𝑠𝑐) → 𝑎(𝑡𝑠)𝑐 given by contracting symmetric arcs of 𝑏𝑏. We denote this cobordism by 𝑊𝑎𝑏𝑐(𝑡, 𝑠). It is minimal in the sense that its Euler characteristic is −|𝑏|. The last ingredient required for defining the arc algebra is Khovanov’s Frobenius TQFT. Let 𝑉 = Z⟨𝑣+, 𝑣−⟩ denote the free abelian group generated by 𝑣+ and 𝑣−, and impose a grading on 𝑉 by |𝑣+| = 1 and |𝑣−| = −1. Consider the functor F𝑒 : Pre-Cob(0) → ZMod defined as follows. On objects, F𝑒(⃝ ⊔ · · · ⊔ ⃝ (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:123)(cid:122) 𝑛 ) = 𝑉 ⊗𝑛. For morphisms, recall that any surface decomposes into a sequence of 2-dimensional 0-, 1- and 2-handles. There are two types of 1-handles, which we refer to as merges and splits; they are evaluated by F𝑒 as listed below. Ñ é : 𝑉 ⊗ 𝑉 → 𝑉 = Ñ é : 𝑉 → 𝑉 ⊗ 𝑉 = F𝑒 F𝑒       𝑣+ ⊗ 𝑣+ ↦→ 𝑣+, 𝑣+ ⊗ 𝑣− ↦→ 𝑣−, 𝑣− ⊗ 𝑣− ↦→ 0, 𝑣− ⊗ 𝑣+ ↦→ 𝑣−, 𝑣+ ↦→ 𝑣− ⊗ 𝑣+ + 𝑣+ ⊗ 𝑣−, 𝑣− ↦→ 𝑣− ⊗ 𝑣−. Additionally, 0- and 2-handles, called births and deaths respectively, have the following evaluation 24 by F𝑒. (cid:16) F𝑒 (cid:17) : Z → 𝑉 = (cid:16) F𝑒 (cid:17) : 𝑉 → Z = ® 1 ↦→ 𝑣+,    𝑣+ ↦→ 0, 𝑣− ↦→ 1. For example, a cylinder with a hole in it can be decomposed into a split followed by a merge. Clearly, this maps 𝑣+ ↦→ 2𝑣− and 𝑣− ↦→ 0. So, altering Cob so that objects can be decorated by dots, we have that Ñ é F𝑒 • : 𝑉 → 𝑉 =    𝑣+ ↦→ 𝑣−, 𝑣− ↦→ 0. F𝑒 extends to Mat(Pre-Cob(𝑛)), and one can easily verify that F𝑒 satisfies the each of the sphere and tube-cutting relations. Let 𝑡 ∈ 𝐵𝑛 𝑚. The arc space of 𝑡 is defined F𝑒(𝑡) = (cid:202) F𝑒(𝑎𝑡𝑏). 𝑎∈𝐵𝑚,𝑏∈𝐵𝑛 Given another tangle 𝑠 ∈ 𝐵 𝑝 𝑛 , define the composition map 𝜇[𝑡, 𝑠] : F𝑒(𝑎𝑡𝑏) ⊗ F𝑒(𝑏′𝑠𝑐) → F𝑒(𝑎(𝑡𝑠)𝑐) by 𝜇[𝑡, 𝑠] =    0 if 𝑏 ≠ 𝑏′ F𝑒(𝑊𝑎𝑏𝑐(𝑡, 𝑠)) if 𝑏 = 𝑏′ for 𝑏′ ∈ 𝐵𝑛 and 𝑐 ∈ 𝐵𝑝. Definition 2.2.8. The arc algebra 𝐻𝑛 is the arc space with multiplication 𝜇[1𝑛, 1𝑛]. 𝐻𝑛 = F (1𝑛) = (cid:202) F𝑒(𝑎1𝑛𝑏) 𝑎∈𝐵𝑚,𝑏∈𝐵𝑛 25 It is more work, but the category of left 𝐻𝑛-modules provides another categorification of the Temperley-Lieb algebra; see Section 5.2 of [Kho02] for details. Lemma 2.2.9. There is an isomorphism of Z[𝑞, 𝑞−1]-algebras 𝐾0(𝐻𝑛PMod) (cid:27) 𝑇 𝐿𝑛 and 𝐾0(Kom(𝐻𝑛PMod)) (cid:27) 𝑇 𝐿𝑛. for 𝐻𝑛PMod the category of projective 𝐻𝑛-modules. 2.3 Cooper-Krushkal projectors The first categorification of Jones-Wenzl projectors was described by Cooper and Krushkal in [CK12]. Their definition mirrors that of the Jones-Wenzl projectors, and they are uniquely defined in Kom(𝑛) (that is, up to homotopy equivalence). Everything presented here still holds if we replace Kom(𝑛) with Kom(𝐻𝑛PMod). Definition 2.3.1. A negativiely graded chain complex (𝐶∗, 𝑑∗) ∈ Kom(𝑛) with degree zero differ- ential and is called a Cooper-Krushkal projector if it satisfies the following axioms: (CK1) 𝐶0 = 1𝑛 and the identity does not appear in any other homological degree. (CK2) 𝐶∗ is contractible under turnbacks: for any 𝑒𝑖 ∈ 𝑇 𝐿𝑛, 𝐶∗ ⊗ 𝑒𝑖 ≃ 𝑒𝑖 ⊗ 𝐶∗ ≃ 0. The second axiom is referred to as “turnback killing.” Notice that, by construction, if 𝐶 ∈ Kom(𝑛) is a Cooper-Krushkal projector, then [𝐶] ∈ 𝐾0(Kom(𝑛)) (cid:27) 𝑇 𝐿𝑛 satisfies (JW1) and (JW2), so [𝐶] = 𝑝𝑛 ∈ 𝑇 𝐿𝑛. Like the Jones-Wenzl projectors, homotopy uniqueness of the Cooper-Krushkal projectors follows from little work. The main tool is the following generalization of idempotence (whose analogue also holds for Jones-Wenzl projectors). Proposition 2.3.2. Suppose 𝐶 ∈ Kom(𝑚) and 𝐷 ∈ Kom(𝑛) are Cooper-Krushkal projectors with 0 ≤ 𝑚 ≤ 𝑛. Then 𝐶 ⊗ (𝐷 ⊔ 1𝑛−𝑚) ≃ 𝐶 ≃ (𝐷 ⊔ 1𝑛−𝑚) ⊗ 𝐶. 26 Homotopy idempotence and uniqueness are then corollaries. Proof. See Proposition 3.3 of [CK12]. □ The main theorem of [CK12] is the following. Theorem 2.3.3 (Theorem 3.2 of [CK12]). For each 𝑛 > 0, there exists a chain complex 𝐶 ∈ Kom(𝑛) that is a Cooper-Krushkal projector. We will write 𝑃CK 𝑛 to denote the 𝑛th Cooper-Krushkal projector (or a representative of it), so that [𝑃CK 𝑛 ] = 𝑝𝑛. We represent Cooper-Krushkal projectors via numbered boxes, as we did the Jones-Wenzl projectors. For example, here is a Jones-Wenzl projector when 𝑛 = 2: 2 = · · · 𝐶−4 𝑞−5 𝐶−3 𝑞−3 𝐶−2 𝑞−1 𝐶−1 where 𝐶𝑖 =    𝑖 = −1 • − • 𝑖 = −2𝑘 • + • 𝑖 = −2𝑘 − 1 for all positive integers 𝑘. It is straightforward to check that this is an element of Kom(𝑛), and that it satisfies axioms (CK1) And (CK2). This categorification succeeds in possessing many properties analogous to the original object. In particular, if Tr𝑛 denotes the (complete) Markov trace applied to each entry and differential in the chain complex, we have that the graded Euler characteristic of the homology of the trace of each projector is a quantum integer; i.e., 𝜒(𝐻∗(Tr𝑛(𝑃CK 𝑛 ))) = [𝑛 + 1]. 27 For example, it is also straightforward to verify that, for 𝑘 a positive integer and 𝛼 ≡ 0, 𝐻𝑛(Tr2(𝑃CK 2 )) =    𝑞2Z ⊕ Z 0 𝑛 = 0 𝑛 = −1 𝑞−4𝑘+2Z ⊕ 𝑞−4𝑘 Z/2Z 𝑛 = −2𝑘 𝑞−4𝑘−2Z 𝑛 = −2𝑘 − 1 It is interesting that the homology of Tr𝑛(𝑃CK 𝑛 ) is not spanned only by classes which correspond to coefficients of the graded Euler characteristic. This turns out to be the case for the projectors of odd Khovanov homology as well. Moreover, the two homologies disagree (for example, there is no torsion for the odd, 2-stranded projector) but their graded Euler characteristics coincide. 28 CHAPTER 3 THE ODD SETTING: CHRONOLOGIES AND G-GRADED STRUCTURES In this chapter, we provide a modern introduction to odd Khovanov homology. That is, rather than detailing the projective TQFT of Ozsváth-Rasmussen-Szabó, we discuss Putyra’s 2-category of chronological cobordisms and its linearlization over the ground ring 𝑅 := Z[𝑋, 𝑌 , 𝑍 ±1]/(𝑋 2 = 𝑌 2 = 1) in §3.1. In §3.2, we attempt to mimic the constructions of [Kho02], as outlined briefly in §2.2.2. Here, we discover the challenges motivating the next few chapters of our work: unified arc algebras are not associative in this context, and the composition maps 𝜇 are not degree-preserving. Finally, in §3.3, we give a description of the solution posed by Naisse and Putrya in [NP20]. We hope that §3.3 serves as a roadmap and extended outline of Chapters 4 and 5. 3.1 Chronological cobordisms and changes of chronology First introduced by Putyra [Put10, Put14], we will proceed using the definition of chronological cobordisms provided by Schütz in [Sch22]. Definition 3.1.1. A chronological cobordism between closed 1-manifolds 𝑆0 and 𝑆1 is a cobordism 𝑊 between 𝑆0 and 𝑆1 embedded into R2 × [0, 1] such that (i) there is an 𝜖 > 0 such that 𝑊 ∩ (R2 × [0, 𝜖]) = 𝑆0 × [0, 𝜖] and 𝑊 ∩ (R2 × [1 − 𝜖, 1]) = 𝑆1 × [1 − 𝜖, 1] and (ii) the height function 𝜏 : 𝑊 → [0, 1] given by projection onto the third coordinate is a Morse function for which #(𝜏−1({𝑐}) ∩ 𝐶) = 1 whenever 𝑐 is a critical value of 𝜏 and 𝐶 is the collection of critical points for 𝜏. We call such a Morse function separative. Next, a framing on a chronological cobordism is a choice of orientation of a basis for each unstable manifold 𝑊𝑝 ⊂ 𝑊, for 𝑝 a critical point of 𝜏 of index 1 or 2. We will assume all chronological cobordisms to be framed. Since a framing is determined by a choice of tangent vector on each unstable manifold determined by a critical point, it is standard to visualize the framing by 29 an arrow through critical points. We’ll adapt the 2-dimensional notation to 1-dimensional diagrams appropriately; for example, = and = . Naturally, two chronological cobordisms are considered equivalent if they can be related by a diffeotopy 𝐻𝑡, 𝑡 ∈ [0, 1], so that projection of 𝐻𝑡(𝑊) onto the third coordinate is a separative Morse function at each time 𝑡. This is a much more strict equivalence relation than that of the even case. To account for this, Putyra introduces the following action/relation. A change of chronology is a diffeotopy 𝐻𝑡 such that projection of 𝐻𝑡(𝑊) onto the third coordinate is a generic homotopy of Morse functions, together with a smooth choice of framings on 𝐻𝑡(𝑊). Two changes of chronology between equivalent cobordisms are equivalent if they are homotopic in the space of oriented Igusa functions after composing with the equivalences of cobordisms; for a thorough description, consult [Put14]. We write 𝐻 : 𝑊1 ⇒ 𝑊2 for a change of chronology 𝐻 between chronological cobordisms 𝑊1 and 𝑊2. Definition 3.1.2. A change of chronology 𝐻 on a chronological cobordism 𝑊 is called locally vertical if there is a finite collection of cylinders {𝐶𝑖}𝑖 in R2 × 𝐼 such that 𝐻 is the identity on 𝑊 − (cid:208)𝑖 𝐶𝑖. We will use locally vertical changes of chronology frequently. Their main utility stems from the fact that they are unique up to homotopy. Proposition 3.1.3 (Proposition 4.4 of [Put14]). If 𝐻 and 𝐻′ are locally vertical changes of chronol- ogy (with respect to the same cylinders) with the same source and target, then they are homotopic in the space of framed diffeotopies. There are two different ways of composing changes of chronology. First, given a sequence of cobordisms 𝐴 𝑊 −→ 𝐵 𝑊 ′ −−→ 𝐶, and changes of chronology 𝐻 on 𝑊 and 𝐻′ on 𝑊 ′, there is a change of chronology 𝐻′ ◦ 𝐻 on 𝑊 ′ ◦ 𝑊. Second, given a sequence of changes of chronology 𝑊 𝐻 ==⇒ 𝑊 ′ 𝐻′ ==⇒ 𝑊 ′′, we will denote their composition by 𝐻′ ★ 𝐻. 30 On the other hand, we may completely describe the elementary chronological cobordisms between closed 1-manifolds: with an additional twisting (transposing) identity cobordism. Together, these observations im- ply that we may decompose all changes of chronology into sequences of elementary changes of chronologies. These are exactly those pairs of cobordisms described in the commutation chart (Figure 2) of [ORS13]. At this point, we have defined a 2-category whose objects are closed 1-manifolds, with chrono- logical cobordisms as 1-morphisms and changes of chronology as 2-morphisms. This 2-category is simplified by the following procedure: for 𝑅 = Z[𝑋, 𝑌 , 𝑍 ±1]/(𝑋 2 = 𝑌 2 = 1), define the map 𝜄 which assigns to each elementary change of chronology a monomial, as pictured in Figure 3.1.1 Indeed, 𝜄(𝐻′ ◦ 𝐻) = 𝜄(𝐻′)𝜄(𝐻) and 𝜄(𝐻′ ★ 𝐻) = 𝜄(𝐻′)𝜄(𝐻) so 𝜄 assigns to every change of chronology a monomial in 𝑅; for more on the map 𝜄 (e.g., well- definedness and multiplicativity), see [Put14]. Finally, as in the even case, we will eventually allow chronological cobordisms to be decorated by finitely many dots as long as each dot never shares the same level set as another dot or critical point. Precisely, let 𝐶 denote the critical points of 𝜏 and 𝐷 denote the dots on 𝑊. Both are taken to be finite. Then, a dotted chronological cobordism is a chronological cobordism for which 𝜏(𝑥) ≠ 𝜏(𝑦) whenever 𝑥, 𝑦 ∈ 𝐶 ∪ 𝐷 are distinct. In [Put14], Putyra shows that if 𝐻 is a change of chronology which does nothing but move one dot past another with respect to the Morse function, then 𝜄(𝐻) = 𝑋𝑌 . A subtle but important distinction of the setup is the degree; define the Z × Z-degree of a cobordism 𝑊 by |𝑊 | = (#births − #merges − #dots, #deaths − #splits − #dots). 1For those elementary cobordisms 𝐻 with 𝜄(𝐻) = 𝑍, it is assumed that 𝐻 takes a merge followed by a split to a split followed by a merge. If the opposite is true, 𝜄(𝐻) = 𝑍 −1. 31 𝑋 𝑍 𝑌 1 𝑋𝑌 Figure 3.1 This is the collection of elementary changes of chronologies, together with their eval- uation by 𝜄. Notice that taking 𝑋 = 𝑍 = 1 and 𝑌 = −1 yields the commutation chart of [ORS13]. Framings are omitted if evaluation by 𝜄 does not depend on them. Note that the sum of the entries of |𝑊 | is the topological degree det𝑡(𝑊) from §2.2. Moreover, define 𝜆 : (Z × Z)2 → 𝑅 to be the bilinear map given by 𝜆((𝑥1, 𝑦1), (𝑥2, 𝑦2)) = 𝑋 𝑥1𝑥2𝑌 𝑦1𝑦2 𝑍 𝑥1𝑦2−𝑦1𝑥2. Suppose 𝐻 is a change of chronology moving two cobordisms 𝑊 and 𝑊 ′ past one another; e.g., 𝐻 looks like Then, 𝑊 𝐻 ===⇒ 𝑊 ′ 𝑊 ′ or 𝑊 𝑊 ′ 𝑊 𝑊 ′ 𝐻 ===⇒ . 𝑊 𝜄(𝐻) = 𝜆 Ä |𝑊 | ,(cid:12) (cid:12)𝑊 ′(cid:12) (cid:12) ä . Note that this agrees with and generalizes the statement about changes of chronologies which move dots past one another. Putyra also provides the following, extremely helpful change of framing 32 relations. = 𝑋 = 𝑌 = 𝑌 In summary, we let ChCob•(0) (or just ChCob•) denote the graded monoidal category whose • objects are formally Z × Z-graded closed 1-manifolds (i.e., a pair of a closed 1-manifold and an element of Z × Z) and • Hom((𝑥1, 𝑦1)𝐴, (𝑥2, 𝑦2)𝐵) is the free Z-module spanned by isotopy classes of (dotted) chono- logical cobordisms 𝑊 from 𝐴 to 𝐵 with degree |𝑊 | = (𝑥1 − 𝑥2, 𝑦1 − 𝑦2), modulo the change of framing and change of chronology relations: 𝑊 ′ = 𝜄(𝐻)𝑊 for each change of chronology 𝐻 : 𝑊 ⇒ 𝑊 ′. 3.2 Unified arc algebras In this section, we consider the unified arc algebras 𝐻𝑛 over 𝑅, as provided by [NV18] and [NP20] (there, referred to as “covering” arc algebras). This is done in spirit of [Kho02], as in §2.2.2, using the “chronological TQFT” provided in [Put14]. There are a number of challenges presented by this construction: for example, the unified arc algebras are non-associative, and the composition map 𝜇[𝑡, 𝑠] do not preserve Z × Z-degree. The solution we study, provided in [NP20], is to use the structure of a grading category, described in §3.3. We must be a bit more careful when setting up the unified arc algebras. Still, for 𝑎 ∈ 𝐵𝑚, 𝑏 ∈ 𝐵𝑛, and 𝑡 ∈ 𝐵𝑛 𝑚, 𝑎𝑡𝑏 is a closed 1-manifold; for 𝑠 ∈ 𝐵 𝑝 𝑛 and 𝑐 ∈ 𝐵𝑝, we can still define a cobordism (𝑎𝑡𝑏)(𝑏𝑠𝑐) → 𝑎(𝑡𝑠)𝑐 but we specify a chronology when we do so. The cobordism is still obtained by contracting symmetric arcs of 𝑏𝑏, and we fix the chronology by taking saddles from right-to-left and choosing the “upwards” framing. This is the chronological cobordism denoted by 𝑊𝑎𝑏𝑐(𝑡, 𝑠). 33 Next, define the “chronological” TQFT F : ChCob• → 𝑅Mod. Set F (⃝ ⊔ · · · ⊔ ⃝ (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:123)(cid:122) 𝑛 ) = 𝑉 ⊗𝑛. where 𝑉 = 𝑅⟨𝑣+, 𝑣−⟩ and, now in this Z × Z-graded landscape, we set deg𝑅(𝑣+) = (1, 0) and deg𝑅(𝑣−) = (0, −1). so that deg𝑅(𝑢) = (#𝑣+(𝑢), −#𝑣−(𝑢)) where 𝑣±(𝑢) denotes the collection of copies of 𝑣± appearing in 𝑢. Finally, on elementary cobordisms, set Ñ é : 𝑉 ⊗ 𝑉 → 𝑉 = Ñ é : 𝑉 → 𝑉 ⊗ 𝑉 = F F (cid:16) F (cid:17) : 𝑅 → 𝑉 = Å ã F : 𝑉 → 𝑅 = 𝑣+ ⊗ 𝑣+ ↦→ 𝑣+, 𝑣+ ⊗ 𝑣− ↦→ 𝑣−, 𝑣− ⊗ 𝑣− ↦→ 0, 𝑣− ⊗ 𝑣+ ↦→ 𝑋 𝑍𝑣−, and 𝑣+ ↦→ 𝑣− ⊗ 𝑣+ + 𝑌 𝑍𝑣+ ⊗ 𝑣−, 𝑣− ↦→ 𝑣− ⊗ 𝑣−       ® 1 ↦→ 𝑣+,    𝑣+ ↦→ 0, 𝑣− ↦→ 1. To obtain a complete description on elementary chronological cobordisms, we apply the change of framing local relations and map the twisting cobordism to a symmetry 𝜏, defined by 𝜏(𝑎 ⊗ 𝑏) = 𝜆(deg𝑅(𝑎), deg𝑅(𝑏))𝑏 ⊗ 𝑎. For more on 𝜏, see Section 3.3 of [NP20]; in addition, see Section 10 of [Put14] for a definition of chronological Frobenius systems. Now, notice that a cylinder with a hole evaluates to either    𝑣+ ↦→ 𝑍(𝑋 + 𝑌 )𝑣− 𝑣− ↦→ 0    𝑣+ ↦→ 𝑍(𝑋𝑌 + 1)𝑣− 𝑣− ↦→ 0 or 34 depending on the framing. Therefore, unfortunately, we are not able to think of dots as 1/2 of a hole anymore; we define F on dots by setting Ñ é F • : 𝑉 → 𝑉 =    𝑣+ ↦→ 𝑣−, 𝑣− ↦→ 0. as before. Again, it is easy to check that F observes the sphere and tube cutting relations. Finally, for 𝑡 ∈ 𝐵𝑛 𝑚, the unified arc space is defined F (𝑡) = (cid:202) F (𝑎𝑡𝑏). 𝑎∈𝐵𝑚,𝑏∈𝐵𝑛 Given another tangle 𝑠 ∈ 𝐵 𝑝 𝑛 , define the composition map 𝜇[𝑡, 𝑠] : F (𝑎𝑡𝑏) ⊗ F (𝑏′𝑠𝑐) → F (𝑎(𝑡𝑠)𝑑) by 𝜇[𝑡, 𝑠] =    0 if 𝑏 ≠ 𝑏′ F (𝑊𝑎𝑏𝑐(𝑡, 𝑠)) if 𝑏 = 𝑏′ where 𝑏′ ∈ 𝐵𝑛 and 𝑐 ∈ 𝐵𝑝. Note that, as promised, 𝜇[𝑡, 𝑠] does not preserve Z × Z-degree. Definition 3.2.1. The unified arc algebra, which we still denote 𝐻𝑛, is the unified arc space 𝐻𝑛 = F (1𝑛) = (cid:202) F (𝑎1𝑛𝑏) 𝑎∈𝐵𝑚,𝑏∈𝐵𝑚 with multiplication 𝜇[1𝑛, 1𝑛]. 3.3 A brief outline of C-graded structures In this section, we review the motivation for and construction of G-graded 𝑅-modules given in [NP20]. In the following chapters, we provide a thorough description of a slight generalization of the procedure introduced here. It has been shown (cf. [NV18] Proposition 3.2) that the multiplication as defined above is not associative in the unified arc algebra. This presents the main difficulty—in [Kho02], Khovanov provides that F (𝑡) ⊗𝐻𝑛 F (𝑠) (cid:27) F (𝑡𝑠) 35 declaring that (𝑢′ · ℎ) ⊗ 𝑢 = 𝑢′ ⊗ (ℎ · 𝑢). The assumption that multiplication in 𝐻𝑛 is associative is implicit here. On the other hand, the failure of associativity is controlled by the cobordisms involved. Explic- itly, observe the square 𝜇𝑎𝑏𝑐[𝑡, 𝑡′](𝑥, 𝑦) ∈ F𝑐(𝑎𝑡𝑡′𝑐) 𝑧 ∈ F𝑐(𝑐𝑡′′𝑑) 𝑥 ∈ F𝑐(𝑎𝑡𝑏) 𝑦 ∈ F𝑐(𝑏𝑡′𝑐) 𝑧 ∈ F𝑐(𝑐𝑡′′𝑑) In general, 𝑥 ∈ F𝑐(𝑎𝑡𝑏) 𝜇𝑏𝑐𝑑[𝑡′, 𝑡′′](𝑦, 𝑧) ∈ F𝑐(𝑏𝑡′𝑡′′𝑑) 𝜇𝑎𝑐𝑑[𝑡𝑡′, 𝑡′′](𝜇𝑎𝑏𝑐[𝑡, 𝑡′](𝑥, 𝑦), 𝑧) 𝜇𝑎𝑏𝑑[𝑡, 𝑡′𝑡′′](𝑥, 𝜇𝑏𝑐𝑑[𝑡′, 𝑡′′](𝑦, 𝑧)) ∈ F𝑐(𝑎𝑡𝑡′𝑡′′𝑑) 𝜇𝑎𝑐𝑑[𝑡𝑡′, 𝑡′′] ◦ (𝜇𝑎𝑏𝑐[𝑡, 𝑡′] ⊗ 1𝑧) ≠ 𝜇𝑎𝑏𝑑[𝑡, 𝑡′𝑡′′] ◦ (1𝑥 ⊗ 𝜇𝑏𝑐𝑑[𝑡′, 𝑡′′]), but the failure is witnessed by the degree of the cobordisms involved: 𝑊𝑎𝑐𝑑(𝑡𝑡′, 𝑡′′) and 𝑊𝑎𝑏𝑐(𝑡, 𝑡′), and 𝑊𝑎𝑏𝑑(𝑡, 𝑡′𝑡′′) and 𝑊𝑏𝑐𝑑(𝑡′, 𝑡′′). The degree of elements also have effect. In the literature, Majid and Albuquerque [AM99] show that the octonions O, while non- assoicative, admit a grading by the group (Z/2Z)3, and the gradings witness the failure of associa- tivity. That is, they show that O is quasi-associative; in general, a 𝐺-graded K-algebra 𝐴 is called quasi-associative (or graded associative) if there is a 3-cocycle 𝛼 : 𝐺[3] → K× for which 𝑎 · (𝑏 · 𝑐) = 𝛼 (cid:0)|𝑎| ,|𝑏| ,|𝑐|(cid:1) (𝑎 · 𝑏) · 𝑐 for all homogeneous elements 𝑎, 𝑏, 𝑐 ∈ 𝐴 (here, | · | : 𝐴 → 𝐺 is the grading). Naisse and Putyra [NP20] generalize the notion of quasi-associativity. Remarking that the 3-cocycle condition is exactly the pentagon relation for a monoidal category, their first goal is to provide similar definitions for modules and algebras graded by categories. 36 Definition 3.3.1. By a grading category, we will mean a category C endowed with a 3-cocycle 𝛼 : C[3] → K×, referred to as the associator. Then, a C-graded K-module is a K-module 𝑀 which admits a decomposition 𝑀 = (cid:202) 𝑀𝑔. 𝑔∈Mor(C) By “𝑔 ∈ Mor(C)”, we just mean that 𝑔 is any morphism of C. This generalizes gradings by a group by delooping: we can view any group 𝐺 as a category with a single object • with End(•) = 𝐺. A C-graded map 𝑓 : 𝑀 → 𝑁 between C-graded modules is just one which preserves grading: 𝑓 (𝑀𝑔) ⊂ 𝑁𝑔. Define the category ModC of C-graded K-modules with morphisms being graded maps. It is a monoidal category where the decomposision of 𝑀′ ⊗ 𝑀 = (cid:201) 𝑔∈Mor(C)(𝑀′ ⊗ 𝑀)𝑔 is given by (𝑀′ ⊗ 𝑀)𝑔 = (cid:202) 𝑔=𝑔2◦𝑔1 𝑀′ 𝑔2 ⊗K 𝑀𝑔1 for composable 𝑔1 and 𝑔2 (revealing a slightly different feature of the C-graded setting). The coherence isomorphism is then given by the associator: (𝑀3 ⊗ 𝑀2) ⊗ 𝑀1 𝛼 −→ 𝑀3 ⊗ (𝑀2 ⊗ 𝑀1) (cid:12)𝑦(cid:12) (𝑧 ⊗ 𝑦) ⊗ 𝑥 ↦→ 𝛼(|𝑧| ,(cid:12) (cid:12) ,|𝑥|) 𝑧 ⊗ (𝑦 ⊗ 𝑥) for homogeneous elements 𝑥, 𝑦, and 𝑧. The C-graded K-module (cid:201) 𝑋∈Ob(C) KId𝑋 is the unit object, and the unitors for this tensor product may also be defined via the associator. We will describe this process explicitly in slightly more generality later on. With this language, Naisse and Putyra are able to define C-graded algebras and bimodules as well. First, a C-graded K-algebra 𝐴 is a C-graded K-module with a graded associative multiplication map 𝐴 ⊗ 𝐴 → 𝐴 such that 𝐴𝑔 · 𝐴𝑔′ ⊂ 𝐴𝑔′◦𝑔, where 𝐴𝑔′◦𝑔 = {0} whenever 𝑔′ ◦ 𝑔 is undefined. Similarly, for two C-graded algebras 𝐴1 and 𝐴2, a C-graded 𝐴2–𝐴1-bimodule 𝑀 is a C-graded module 𝑀 with graded, K-linear left and right actions 𝐴2 ⊗ 𝑀 → 𝑀 and 𝑀 ⊗ 𝐴1 → 𝑀 satisfying the usual bimodule conditions, twisted by the associator: for example, these actions respect (𝑦 · 𝑚) · 𝑥 = 𝛼 Ä(cid:12) (cid:12)𝑦(cid:12) (cid:12) ,|𝑚| ,|𝑥| ä 𝑦 · (𝑚 · 𝑥) 37 for all 𝑦 ∈ 𝐴2, 𝑚 ∈ 𝑀 and 𝑥 ∈ 𝐴1. In this section, we will denote the category of C-graded 𝐴2– 𝐴1-bimodules by BimodC(𝐴2, 𝐴1). The morphisms of this category will be graded maps between 𝐴2–𝐴1-bimodules which preserve the left and right actions. We employ the associator to see that, given 𝑀′ ∈ BimodC(𝐴3, 𝐴2) and 𝑀 ∈ BimodC(𝐴2, 𝐴1), 𝑀′ ⊗K 𝑀 ∈ BimodC(𝐴3, 𝐴1): the left and right actions are the horizontal maps making the following diagrams commute. 𝐴3 ⊗ (𝑀′ ⊗ 𝑀) 𝑀′ ⊗ 𝑀 (𝑀′ ⊗ 𝑀) ⊗ 𝐴1 𝑀′ ⊗ 𝑀 𝛼−1 (𝐴3 ⊗ 𝑀′) ⊗ 𝑀 𝛼 𝑀′ ⊗ (𝑀 ⊗ 𝐴1) Then, we can define the tensor product over the intermediary algebra 𝐴2 via the coequalizer: explicitly, 𝑀′ ⊗𝐴2 𝑀 = 𝑀′ ⊗K 𝑀 (cid:46) (cid:16) (𝑚′ · 𝑥) ⊗ 𝑚 − 𝛼 Ä(cid:12) (cid:12)𝑚′(cid:12) (cid:12) ,|𝑥| ,|𝑚| (cid:17) ä 𝑚′ ⊗ (𝑥 · 𝑚) with left 𝐴3- and right 𝐴1-actions induced by the ones on 𝑀′ ⊗K 𝑀. Now, with the goal of showing that the unified arc algebra 𝐻𝑛 is graded associative, we must build a suitable grading category (G, 𝛼). Let 𝐵• = (cid:195)𝑛≥0 𝐵𝑛 denote the collection of all crossingless matchings. Given a flat tangle 𝑡, we write (cid:98)𝑡 or 𝑡∧ to mean the tangle 𝑡 with all free loops removed; (cid:98)𝐵𝑛 𝑚 denotes the collection of planar tangles with no free loops. Let G denote the category where • Ob(G) = 𝐵•, and whose • morphisms are formally Z × Z-graded planar tangles; that is, HomG(𝑎, 𝑏) = (cid:98)𝐵𝑛 𝑚 × Z2 for any 𝑎 ∈ 𝐵𝑚 and 𝑏 ∈ 𝐵𝑛. The composition, for (𝑡, 𝑝) ∈ HomG(𝑎, 𝑏) and (𝑡′, 𝑝′) ∈ HomG(𝑏, 𝑐), is defined (𝑡′, 𝑝′) ◦ (𝑡, 𝑝) = (“𝑡𝑡′, 𝑝 + 𝑝′ +(cid:12) (cid:12)𝑊𝑎𝑏𝑐(𝑡, 𝑡′)(cid:12) (cid:12)) ∈ HomG(𝑎, 𝑐). 38 Note that, since 𝑊𝑎𝑏𝑐(𝑡, 𝑡′) consists of only saddle moves, (cid:12)𝑊𝑎𝑏𝑐(𝑡, 𝑡′)(cid:12) (cid:12) (cid:12) = (−#merges in 𝑊𝑎𝑏𝑐(𝑡, 𝑡′), −#splits in 𝑊𝑎𝑏𝑐(𝑡, 𝑡′)). So, it follows that the identity morphism for any crossingless matching 𝑎 ∈ 𝐵𝑚 is Id𝑎 = (1𝑚, (𝑚, 0)). Henceforth, to make life easier, given objects 𝑎 ∈ 𝐵𝑚 and 𝑏 ∈ 𝐵𝑛, we’ll write 𝑎𝑡𝑏 when, really, we mean 𝑎𝑡𝑏. We will omit a description of the associator until defining our own in the generalized setting—it will be apparent how to specialize ours to the current situation. Instead, we describe the way in which way elements of 𝐻𝑛, or F (𝑡) in general, are G-graded. For 𝑢 ∈ F (𝑎𝑡𝑏), we set degG(𝑢) = ((cid:98)𝑡, deg𝑅(𝑢)) ∈ HomG(𝑎, 𝑏). Hopefully this explains the choice to remove free loops from tangles: they are not involved in composition maps between arc algebras, and are extraneous information in light of the second entry of the grading. Secondly, this presents a solution to the first problem for unified arc algebras: 𝜇𝑎𝑏𝑐[𝑡, 𝑠] preserves the G-grading. Suppose 𝑢 ∈ F (𝑎𝑡𝑏) and 𝑣 ∈ F (𝑏𝑠𝑐), so degG(𝑢) = (𝑡, deg𝑅(𝑢)) ∈ Hom(𝑎, 𝑏) and degG(𝑣) = (𝑠, deg𝑅(𝑣)) ∈ Hom(𝑏, 𝑐). Their composition in unified arc spaces is given by the map 𝜇𝑎𝑏𝑐[𝑡, 𝑠]. Recall that in the definition of the chronological TQFT F , each merge decreases the number of copies of 𝑣+ by 1, and each split increases the number of copies of 𝑣− by 1; consequently degG(𝜇𝑎𝑏𝑐[𝑡, 𝑠](𝑢, 𝑣)) = (cid:0) (cid:98)𝑡𝑠, deg𝑅(𝑢) + deg𝑅(𝑣) +|𝑊𝑎𝑏𝑐(𝑡, 𝑠)|(cid:1) = degG(𝑣) ◦ degG(𝑢) as desired. Finally, we can prove that 𝜇𝑎𝑐𝑑[𝑡𝑡′, 𝑡′′] (cid:0)𝜇𝑎𝑏𝑐[𝑡, 𝑡′](𝑥, 𝑦), 𝑧(cid:1) = 𝛼 Ä (cid:12)𝑦(cid:12) |𝑥| ,(cid:12) (cid:12) ,|𝑧| ä 𝜇𝑎𝑏𝑑[𝑡, 𝑡′𝑡′′] (cid:0)𝑥, 𝜇𝑎𝑏𝑑[𝑡′, 𝑡′′](𝑦, 𝑧)(cid:1) for any 𝑥 ∈ F (𝑎𝑡𝑏), 𝑦 ∈ F (𝑏𝑡′𝑐) and 𝑧 ∈ F (𝑐𝑡′′𝑑). In particular, Naisse and Putyra provide the following (for a discussion on unitality, see [NP20] Proposition 6.2). Proposition 3.3.2. 𝐻𝑛 is a unital, associative, G-graded 𝑅-algebra. 39 It is routine to check that, for 𝑡 ∈ 𝐵𝑛 𝑚, F (𝑡) is an (𝐻𝑚, 𝐻𝑛)-bimodule: the left 𝐻𝑚-action is given by 𝜇[1𝑚, 𝑡] and the right 𝐻𝑛-action is given by 𝜇[𝑡, 1𝑛]. Naisse and Putyra then provide the desired properties of these bimodules, in the sense that it mirrors results of [Kho02]. Proposition 3.3.3. Let 𝑡 ∈ 𝐵𝑛 𝑚. Then F (𝑡) is an (𝐻𝑚, 𝐻𝑛)-bimodule. It is also sweet as an (𝐻𝑚, 𝐻𝑛)- bimodule; that is, it is projective as a left 𝐻𝑚-module and as a right 𝐻𝑛-module. Moreover, given 𝑠 ∈ 𝐵 𝑝 𝑛 , there is an isomorphism F (𝑡) ⊗𝐻𝑛 F (𝑠) (cid:27) F (𝑡𝑠) induced by 𝜇[𝑡, 𝑠] : F (𝑡) ⊗𝑅 F (𝑠) → F (𝑡𝑠). 3.3.1 G-shifting system So far, we have successfully defined the relevant algebraic objects in the G-graded setting. 𝑡, 𝑠 ∈ 𝐵𝑛 However, we have glossed over the important discussion of graded maps. In particular, given 𝑚, so that F (𝑡), F (𝑠) ∈ Ob ÄBimodG(𝐻𝑚, 𝐻𝑛)ä, can we describe those relevant morphisms between F (𝑡) and F (𝑠) in this category? Of course, any cobordism 𝑊 : 𝑡 → 𝑠 induces a map F (𝑊) : F (𝑡) → F (𝑠), but this map is clearly not graded! There must be a fix if we are to interpret cubes of resolutions with this approach; in particular, the only graded map between F (cid:16) (cid:17) and (cid:16) F (cid:17) is the zero map. The solution of Naisse and Putyra is the introduction of grading shifting functors via a G-shifting system. Here is the idea of a C-shifting system; a more precise, expanded definition is given in Section 5. Definition 3.3.4. A C-shifting system is a pair (𝐼, Φ) consisting of a monoid (𝐼, •, 𝑒) and a collection Φ = {𝜑𝑖}𝑖∈𝐼 of families of maps 𝜑𝑖 = {𝜑𝑋,𝑌 𝑖 : D 𝑋,𝑌 𝑖 → HomC(𝑋, 𝑌 )}𝑋,𝑌 ∈Ob(C) 𝑋,𝑌 for D 𝑖 ⊂ HomC(𝑋, 𝑌 ). These families of maps 𝜑𝑖 are called C-grading shifts, and they are required to satisfy the property that, for each 𝑖, 𝑗 ∈ 𝐼 and 𝑋, 𝑌 ∈ Ob(C), the following diagram 40 commutes. HomC(𝑌 , 𝑍) × HomC(𝑋, 𝑌 ) 𝜑 𝑗 ×𝜑𝑖 HomC(𝑌 , 𝑍) × HomC(𝑋, 𝑌 ) ◦ ◦ HomC(𝑋, 𝑍) 𝜑 𝑗•𝑖 HomC(𝑋, 𝑍) It is not immediate that a C-shifting system (𝐼, Φ) is compatible with the associator 𝛼; a major portion of [NP20], and now our work, has to do with this observation. If 𝑆 = (𝐼, {𝜑𝑖}𝑖∈𝐼) is a C-shifting system compatible with 𝛼, then for each 𝑖 ∈ 𝐼, 𝜑𝑖 : ModC → ModC is a functor, called the grading shift functor, and is defined as follows. For 𝑀 = (cid:201) 𝑔∈Mor(C) 𝑀𝑔 ∈ Ob(ModC), put 𝜑𝑖(𝑀) = (cid:202) 𝜑𝑖(𝑀)𝜑𝑖(𝑔) 𝑔∈D𝑖 where 𝜑𝑖(𝑀)𝜑𝑖(𝑔) = 𝑀𝑔. In other words, this grading shift functor turns elements of degree 𝑔 ∈ D𝑖 into elements of degree 𝜑𝑖(𝑔); elements whose degree is not in D𝑖 are sent to zero. We will see that the witnesses to compatibility between a given C-shifting system and associator imply the existence of canonical isomorphisms 𝜑 𝑗 (𝑀′) ⊗ 𝜑𝑖(𝑀) → 𝜑 𝑗•𝑖(𝑀′ ⊗ 𝑀). Indeed, there is a natural transformation 𝜑 𝑗 (−) ⊗ 𝜑𝑖(−) ⇒ 𝜑 𝑗•𝑖(− ⊗ −). From here, under a certain assumption, it is easy to define shifted bimodules. In summary, this is to say that the shifting functor 𝜑𝑖 : ModC → ModC further induces a shifting functor 𝜑𝑖 : BimodC(𝐴2, 𝐴1) → BimodC(𝐴2, 𝐴1). The shifting functor also respects tensor products: for 𝑀′ ∈ BimodC(𝐴3, 𝐴2) and 𝑀 ∈ BimodC(𝐴2, 𝐴1), 𝜑 𝑗 (𝑀′) ⊗𝐴2 𝜑𝑖(𝑀) (cid:27) 𝜑 𝑗•𝑖(𝑀′ ⊗𝐴2 𝑀). Returning to the situation at hand, our goal is to define a G-shifting system (compatible with 𝛼). The G-shifting system we will use is given simply by weighted cobordisms (𝑊, 𝑣) where 𝑣 ∈ Z × Z. Explicitly, to construct the monoid in this shifting system, recall that given two cobordisms 𝑊1 : 𝑡 → 𝑡′ for 𝑡, 𝑡′ ∈ 𝐵𝑛 𝑚 and 𝑊2 : 𝑠 → 𝑠′ for 𝑠, 𝑠′ ∈ 𝐵𝑛 ℓ , we obtain a cobordism 𝑊1 • 𝑊2 : 𝑡𝑠 → 𝑡′𝑠′ by horizontal stacking. 41 Now, given weighted cobordisms (𝑊1, 𝑣1) and (𝑊2, 𝑣2), define (𝑊1, 𝑣1) • (𝑊2, 𝑣2) to be (𝑊1 • 𝑊2, 𝑣1 + 𝑣2) whenever 𝑊1 •𝑊2 is defined, and zero otherwise. The monoid of the G-shifting system will be the collection of weighted cobordisms together with formal identity absorbing elements {(𝑊, 𝑣)} ⊔ {𝑒, 0} under the operation •. Finally, given 𝑡, 𝑡′ ∈ 𝐵𝑛 𝑚 and (𝑊 : 𝑡 → 𝑡′, 𝑣), given any 𝑎 ∈ 𝐵𝑚 and 𝑏 ∈ 𝐵𝑛 we define 𝜑𝑎,𝑏 (𝑊,𝑣)((cid:98)𝑡, 𝑝) = ((cid:98)𝑡′, 𝑝 + 𝑣 +|1𝑎𝑊1𝑏 |) where 1𝑎𝑊1𝑏 is the cobordism 𝑊 capped off by 𝑎 ×[0, 1] on one side and 𝑏 ×[0, 1] (really, 𝑏 ×[0, 1]) on the other. Since Ob(G) = 𝐵•, we can write 𝜑(𝑊,𝑣) = notation and write 𝜑(𝑊,𝑣) when it does not present confusion. Clearly, the domain of 𝜑𝑎,𝑏 (𝑊,𝑣) is (𝑊,𝑣) = {((cid:98)𝑡, 𝑝) ∈ HomG(𝑎, 𝑏) : 𝑝 ∈ Z × Z}. We’ll write 𝜑𝑊 sometimes when 𝑣 can be left ambiguous; however, in computations, this notation means 𝑣 = (0, 0). Finally, for a flat tangle 𝑡, let ; we will often abuse simply D 𝑎∈𝐵𝑚,𝑏∈𝐵𝑛 (𝑊,𝑣) 𝑎,𝑏 ¶𝜑𝑎,𝑏 © 1𝑡 denote the identity cobordism on 𝑡. Consider the collection of identity cobordisms 1 = {1𝑡 }𝑡. Then there is an identity shift functor given by 𝜑id = (cid:201) 1 𝜑1𝑡 . In practice, it is beneficial to view weighted cobordisms (𝑊, 𝑣) as two separate shifts; the first on a given planar tangle and the second on the Z × Z degree associated to that tangle. Unfortunately, to determine compatibility maps one must choose an order: we will always shift first by the chronological cobordism 𝑊 and second by the Z × Z-degree. The opposite choice can also be made, and leads to small differences in the theory—for example, see Proposition 7.1.5. In this way, Naisse and Putyra show that this G-shifting system is compatible with the associator defined above; for more details, see [NP20]. Of course, there is also the possibility of vertically composing cobordisms. This is to say that the G-shifting system may be extended to a shifting 2-system (again, defined by Naisse-Putyra). Explicitly, in the monoid defined above, we define vertical composition in the same spirit as horizontal composition: for 𝑊1 : 𝑡 → 𝑡′ and 𝑊2 : 𝑠 → 𝑠′, (𝑊2, 𝑣2) ◦ (𝑊1, 𝑣1) =    (𝑊2 ◦ 𝑊1, 𝑣2 + 𝑣1) if 𝑡′ = 𝑠 otherwise. 0 42 Compatibility maps are constructed via the change of chronology 𝐻 : (𝑊 ′ 2 ◦ 𝑊2) • (𝑊 ′ 1 ◦ 𝑊1) ⇒ (𝑊 ′ 2 • 𝑊 ′ 1) ◦ (𝑊2 • 𝑊1). With this structure in place, we will see that any cobordism with corners 𝑊 : 𝑡 → 𝑠 induces a graded map F (𝑊) : 𝜑𝑊 (F (𝑡)) → F (𝑠), as desired. 43 CHAPTER 4 GRADING MULTICATEGORIES AND PLANAR ARC DIAGRAMS In this chapter, we generalize the work of Naisse and Putyra to provide a category compatible with “multigluing”; i.e., a framework for replacing flat tangles 𝑡 with planar arc diagrams 𝐷. We note that the content of this chapter and the next will come as little surprise to readers familiar with [NP20], outside of complications and additional structure associated with multicategories. We start by extending the definition of F to planar arc diagrams, defined momentarily. In §4.1, we review multicategories, define grading multicategories, and construct the grading multicategory G utilized throughout this thesis. In §4.2, we verify that G is indeed a grading multicategory. Then, §4.3 is dedicated to establishing some properties of modules graded by multicategories which we use extensively. We conclude with §4.4, wherein we list consequences of observations made in §4.3 for G -graded multimodules associated to planar arc diagrams by F . Definition 4.0.1. An (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-planar arc diagram 𝐷 is a disk 𝐷 with 𝑘 interior disks removed, together with a proper embedding of disjoint circles and closed intervals, so that there are 2𝑚𝑖 endpoints on the boundary component corresponding to the 𝑖th removed disk, and 2𝑛 endpoints on the outer boundary of 𝐷. Note that planar arc diagram 𝐷 comes with an ordering on the removed inner disks. Each boundary component carries a basepoint, disjoint from the endpoints of intervals, denoted by ×. We say that 𝐷 is oriented if the embedded circles and intervals are oriented. Both oriented and unoriented planar arc diagrams are considered up to planar isotopy. The collection of planar arc diagrams of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛) is denoted by D(𝑚1,...,𝑚𝑘;𝑛). Similarly “D(𝑚1,...,𝑚𝑘;𝑛) is the collection of (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-planar arc diagrams with free loops removed. For example, pictured below is an oriented (1, 1, 1, 2; 3)-planar arc diagram. We can compose planar arc diagrams by filling the 𝑖th empty region of one planar arc diagram with a (· · · ; 𝑚𝑖) planar arc diagram. That is, given planar arc diagrams 𝐷𝑖 of type (ℓ𝑖1, . . . , ℓ𝑖𝛼𝑖 ; 𝑚𝑖) for 𝑖 = 1, . . . , 𝑘 and 𝐷 of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛), we set 𝐷 ◦ (𝐷1, . . . , 𝐷 𝑘 ) = 𝐷(𝐷1, . . . , 𝐷 𝑘 ; ∅). 44 There is also a pairwise composition 𝐷 ◦𝑖 𝐷𝑖 = 𝐷(∅, . . . , 𝐷𝑖, . . . , ∅; ∅). To the author’s knowledge, this notation was first introduced in [LLS22] (we will adapt this definition to diskular tangles in Section 6.2). Note that the two notions of composition are related by 𝐷 ◦ (𝐷1, . . . , 𝐷 𝑘 ) = (· · · ((𝐷 ◦𝑘 𝐷 𝑘 ) ◦𝑘−1 𝐷 𝑘−1) ◦𝑘−2 · · · ) ◦1 𝐷1 If 𝐸 is a planar arc diagram with an interior boundary component with 2𝑛 endpoints, we’ll write 𝐷(𝐷1, . . . , 𝐷 𝑘 ; 𝐸) to denote the resulting planar arc diagram. Otherwise, we frequently drop the last ∅ from the notation. 3× 1 × 4× 2 × × On one hand, it is clear that any crossingless matching 𝑎 ∈ 𝐵𝑛 uniquely defines a planar arc diagram of type (; 𝑛). We choose the association • • • • • • • • × ⇝ so that, if we are being careful, the inner disks of a planar arc diagram can be filled with crossingless matchings belonging to 𝐵• and can be closed on the outside by a crossingless matching belonging to 𝐵•. Thus, if 𝐷 is a (𝑚1, . . . , 𝑚𝑘 ; 𝑛) planar arc diagram, we define F (𝐷) = (cid:202) F (𝐷(𝑥1, . . . , 𝑥𝑘 ; 𝑦)) 𝑥𝑖 ∈𝐵𝑚𝑖 :𝑖=1,...,𝑘 𝑦∈𝐵𝑛 45 where F is the unified chronological TQFT. It is a (𝐻𝑚1 ⊗ · · · ⊗ 𝐻𝑚𝑘 , 𝐻𝑛)-bimodule by the compositions 𝜇[(1𝑚1, . . . , 1𝑚𝑘 ); 𝐷] and 𝜇[𝐷; 1𝑛]. These composition maps are defined just as before: for compatible 𝐷𝑖, we define 𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] : 𝑘 (cid:204) 𝑖=1 F (𝐷𝑖) ⊗ F (𝐷) → F (𝐷(𝐷1 . . . , 𝐷 𝑘 )) component-wise, as follows. For the time being, all tensor products are taken over 𝑅. Working with planar arc diagrams necessitates some burdensome notation. Notice that potentially far more closures are necessary: each 𝐷𝑖 requires, say, 𝛼𝑖-many inner closures which we denote by 𝑥(𝑖,1), . . . , 𝑥(𝑖,𝛼𝑖), and one outer closure 𝑦𝑖. On the other hand, 𝐷 requires 𝑘 inner closures 1, . . . , 𝑦′ 𝑦′ {𝑥(1,1), . . . , 𝑥(𝑘,𝛼𝑘)}. 𝑘 and one outer closure 𝑧. Let (cid:174)𝑥 denote the entire collection of crossingless matchings In the future, (cid:174)· will always denote the entire collection of crossingless parings of that label. If (cid:174)· has a subscript 𝑖, we mean all corssingless parings of that label whose first entry of their subscript is 𝑖; e.g., (cid:174)𝑥𝑖 = {𝑥(𝑖,1), . . . , 𝑥(𝑖,𝛼𝑖)}. With this notation in place, we define 𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] component-wise by 𝜇(cid:174)𝑥 (cid:174)𝑦𝑧[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] : 𝑘 (cid:204) 𝑖=1 F (𝐷𝑖((cid:174)𝑥𝑖; 𝑦𝑖)) ⊗ F (𝐷( (cid:174)𝑦′; 𝑧)) → F (𝐷(𝐷1((cid:174)𝑥1), . . . , 𝐷 𝑘 ((cid:174)𝑥𝑘 ); 𝑧) where we can interpret 𝐷𝑖((cid:174)𝑥𝑖) := 𝐷𝑖((cid:174)𝑥𝑖; ∅) as a crossingless matching, and 𝜇(cid:174)𝑥 (cid:174)𝑦𝑧[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] =    0 if 𝑦𝑖 ≠ 𝑦′ 𝑖 for some 𝑖; F (𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷 𝑘 ); 𝐷)) if 𝑦𝑖 = 𝑦′ 𝑖 for all 𝑖. Elements of Ä(cid:203)𝑘 𝑖=1 F (𝐷𝑖)ä ⊗ F (𝐷) are written (𝑢1, . . . , 𝑢𝑘 ) ⊗ 𝑢 or, frequently, (cid:174)𝑢 ⊗ 𝑢. The last thing we must do is describe the chronological cobordism 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷 𝑘 ), 𝐷). This cobordism is (as one would expect, comparing to Sections 2.2.2 and 3.2) defined by contracting the symmetric arcs of 𝑦𝑖. The chronology is chosen by moving counter-clockwise from the basepoint of the 𝑖th removed disk of 𝐷 and contracting symmetric arcs outwardly, starting at 𝑖 = 1 and progressing to 𝑖 = 𝑘. Use Figure 4.1 for reference. In this example, 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, 𝐷2, 𝐷3), 𝐷) is the chronological 46 𝑧 • • • • • • • • 𝐷 • • 𝑦1 1 𝐷1 𝑥(1,1) • • • × • • 𝑥(1,2) • • × • × × • • 2 𝑦2 • • 𝐷2 3 • • • × 𝑥(2,1) • • × × × • 𝑦3 • • 4 𝐷3 • × • • 𝑥(3,2) • • • 𝑥(3,1) • • 5 • × × × • • • Figure 4.1 An example of a chronological coboridm 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, 𝐷2, 𝐷3), 𝐷). cobordism obtained by contracting the symmetric arcs of (cid:174)𝑦 as specified by the gray arrows in the numbered order. So, it is a merge, followed by a split, and then three more merges. Notice that 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷𝑛), 𝐷) has Euler characteristic − (cid:205)𝑖 (cid:12) (cid:12)𝑦𝑖 (cid:12)𝑦(cid:12) (cid:12) (recall that (cid:12) (cid:12) (cid:12) = 𝑐 whenever 𝑦 ∈ 𝐵𝑐). 47 As we proceed, we will use the notation (cid:174)𝑦𝐷𝑧 to mean 𝐷((cid:174)𝑦; 𝑧). This seems redundant, but it is especially helpful to write (cid:174)𝑥(𝐷1, . . . , 𝐷 𝑘 )(cid:174)𝑦, or even (cid:174)𝑥 (cid:174)𝐷′(cid:174)𝑦 for (cid:174)𝐷′ = (𝐷1, . . . , 𝐷 𝑘 ), rather than (𝐷1((cid:174)𝑥1; 𝑦1), 𝐷2((cid:174)𝑥2; 𝑦2), . . . , 𝐷 𝑘 ((cid:174)𝑥𝑘 ; 𝑦𝑘 )). Let (cid:174)𝐷′ = (𝐷1, . . . , 𝐷 𝑘 ). We will frequently refer to the chronolonological cobordism above via the (upwardly oriented) schematic (cid:174)𝐷′ 𝐷 where the trivalent vertex represents the cobordism 𝑊(cid:174)𝑥,(cid:174)𝑦,𝑧((𝐷1, . . . , 𝐷 𝑘 ), 𝐷). In following sections, we’ll have to consider the compositions of such cobordisms, but it is not immediately clear how the chronology is defined. Let (cid:174)𝐷′′ = (𝐷(1,1), . . . , 𝐷(1,𝛼1), . . . , 𝐷(𝑘,𝛼𝑘)), so that (cid:174)𝐷′′ 𝑖 are the planar arc diagrams filling 𝐷𝑖. While we can interpret (cid:174)𝐷′′ (cid:174)𝐷′ 𝐷 as a chronological cobordism using our rules above, we’d like to consider compositions of the form (cid:174)𝐷′′ (cid:174)𝐷′ 𝐷 as well. In the latter, notice that the leftmost trivalent vertex is a collection of chronological cobordisms, 𝑊 (cid:174)𝑤𝑖 (cid:174)𝑥𝑖 𝑦𝑖 ((𝐷(𝑖,1), . . . , 𝐷(𝑖,𝛼𝑖))). So, we will define the order of these chronological cobordisms to follow the index 𝑖 = 1, . . . , 𝑘—the same idea applies to larger compositions. Denote 48 the composition of these chronological cobordisms by 𝑊 (cid:174)𝑤,(cid:174)𝑥,(cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′), and the corresponding map as 𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′]. To be explicit, 𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′] : Ñ 𝑘 (cid:204) 𝛼𝑖(cid:204) 𝑖=1 𝑗=1 é F (𝐷𝑖 𝑗 ) ⊗ Ñ 𝑘 (cid:204) 𝑖=1 é F (𝐷𝑖) → 𝑘 (cid:204) 𝑖=1 F (𝐷𝑖(𝐷𝑖1, . . . , 𝐷𝑖𝛼𝑖 )) interpreting 𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′] = (cid:203)𝑘 𝑖=1 𝜇[(𝐷𝑖1, . . . , 𝐷𝑖𝛼𝑖 ), 𝐷𝑖]. We shorten the expression above to 𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′] : F ( (cid:174)𝐷′′) ⊗ F ( (cid:174)𝐷′) → F ( (cid:174)𝐷′( (cid:174)𝐷′′)). Finally, while we will almost always use the composition maps 𝜇(cid:174)𝑥,(cid:174)𝑦,𝑧( (cid:174)𝐷, 𝐷) moving forward, we note that the flexibility of planar arc diagrams allows for a few more composition maps. First, note that one may fill the 𝑖th hole of 𝐷 by 𝐷𝑖, leaving the other holes unchanged, by considering the composition 𝜇[(1𝑛1, . . . , 𝐷𝑖, . . . , 1𝑛𝑘 ); 𝐷]. On the other hand, we could also define a composition map which only fills one hole of 𝐷 without reference to the others. Consider the map 𝜇[𝐷𝑖; 𝐷] : F (𝐷𝑖) ⊗ F (𝐷) → F (𝐷(∅, . . . , 𝐷𝑖, . . . , ∅)) defined componentwise as 𝜇(𝑦′ 1,...,(cid:174)𝑥𝑖,...,𝑦′ 𝑘 ),𝑦𝑖,𝑧[𝐷𝑖; 𝐷] :F (𝐷𝑖((cid:174)𝑥𝑖; 𝑦𝑖)) ⊗ F (𝐷((cid:174)𝑦′; 𝑧)) → F (𝐷(𝑦′ 1, . . . , 𝐷𝑖((cid:174)𝑥𝑖), . . . , 𝑦′ 𝑘 )) 𝜇(𝑦′ 1,...,(cid:174)𝑥𝑖,...,𝑦′ 𝑘 ),𝑦𝑖,𝑧[𝐷𝑖; 𝐷] =    0 if 𝑦′ 𝑖 ≠ 𝑦𝑖 F (𝑊(𝑦′ 1,...,(cid:174)𝑥𝑖,...,𝑦′ 𝑘 ),𝑦𝑖,𝑧(𝐷𝑖; 𝐷)) if 𝑦′ 𝑖 = 𝑦𝑖 where 𝑊(𝑦′ 1,...,(cid:174)𝑥𝑖,...,𝑦′ 𝑘 ),𝑦𝑖,𝑧(𝐷𝑖; 𝐷) is the chronological cobordism which simply contracts symmetric arcs of 𝑦𝑖 𝑦𝑖 counter-clockwise with respect to the basepoint, with closures specified by the other indices. Then, notice that 𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] = 𝜇[𝐷 𝑘 ; 𝐷(𝐷1, . . . , 𝐷 𝑘−1, ∅)] ◦ · · · ◦ (cid:0)𝜇[𝐷2; 𝐷(𝐷1, ∅, . . . , ∅)] ⊗ Id𝐷3 ⊗ · · · ⊗ Id𝐷 𝑘 (cid:1) ◦ (cid:0)𝜇[𝐷1; 𝐷] ⊗ Id𝐷2 ⊗ · · · ⊗ Id𝐷 𝑘 (cid:1) where Id𝐷𝑖 means the identity on elements living in components corresponding to closures of 𝐷𝑖. 49 4.1 (Grading) multicategories Recall that a (small) multicategory C consists of 1. a set of objects Ob(C ), 2. for each 𝑘 ≥ 0 and objects 𝑥1, . . . , 𝑥𝑘 , 𝑦 ∈ Ob(C ), a set Hom(𝑥1, . . . , 𝑥𝑘 ; 𝑦) of multimorphisms from (𝑥1, . . . , 𝑥𝑘 ) to 𝑦, 3. a composition map Hom(𝑦1, . . . , 𝑦𝑘 ; 𝑧) × 𝑘 (cid:214) 𝑖=1 and Hom(𝑥𝑖1, . . . , 𝑥𝑖𝛼𝑖 ; 𝑦𝑖) → Hom(𝑥11, . . . , 𝑥𝑘𝛼𝑘 ; 𝑧), 4. a distinguished element Id𝑥 ∈ Hom(𝑥; 𝑥) for each 𝑥 ∈ Ob(𝑥) called the identity of 𝑥 defined so that composition is associative, in the sense that the following diagram commutes: Hom(𝑦1, . . . , 𝑦𝑘 ; 𝑧) × (cid:206)𝑘 𝑖=1 Hom(𝑥𝑖1, . . . , 𝑥𝑖𝛼𝑖 ; 𝑦𝑖) × (cid:206)𝑘 𝑖=1 (cid:206)𝛼𝑖 𝑗=1 Hom(𝑤𝑖 𝑗1, . . . , 𝑤𝑖 𝑗 𝛽𝑖 𝑗 ; 𝑥𝑖 𝑗 ) Hom(𝑥11, . . . , 𝑥𝑘𝛼𝑘 ; 𝑧) (cid:206)𝛼𝑖 × (cid:206)𝑘 𝑖=1 𝑗=1 Hom(𝑤𝑖 𝑗1, . . . , 𝑤𝑖 𝑗 𝛽𝑖 𝑗 ; 𝑥𝑖 𝑗 ) Hom(𝑦1, . . . , 𝑦𝑘 ; 𝑧) × (cid:206)𝑘 𝑖=1 Hom(𝑤𝑖11, . . . , 𝑤𝑖𝛼𝑖 𝛽𝑖 𝛼𝑖 ; 𝑦𝑖) (cid:47) Hom(𝑤111, . . . , 𝑤 𝑘𝛼𝑘 𝛽𝑘 𝛼𝑘 ; 𝑧). In addition, we require that the identity elements are both right and left identities for composition. Proceeding, for a multimorphism 𝑓 : (𝑥1, . . . , 𝑥𝑘 ) → 𝑦, we set dom( 𝑓 ) := (𝑥1, . . . , 𝑥𝑘 ) and codom( 𝑓 ) := 𝑦. Example. Planar arc diagrams comprise a multicategory important to the work that follows. Let 𝑝T denote the multicategory whose • objects are the natural numbers, including zero, 50 (cid:47) (cid:47) (cid:15) (cid:15) (cid:15) (cid:15) (cid:47) • Hom𝑝T(𝑚1, . . . , 𝑚𝑘 ; 𝑛) is the collection of (𝑚1, . . . , 𝑚𝑘 ; 𝑛) planar arc diagrams, which we will denote by D(𝑚1,...,𝑚𝑘;𝑛). Composition in 𝑝T is composition of planar arc diagrams, as defined at the beginning of this section. It follows immediately that 𝑝T is a multicategory with identity elements 1𝑛, which is just a circle with 𝑛 marked points times the interval. Note that we can view 𝑝T as a multicategory enriched in categories since D(𝑚1,...,𝑚𝑘;𝑛) can be viewed as a category whose morphisms are (potentially chronological) cobordisms between planar arc diagrams of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛). A very similar multicategory, G , will be the main object of study for the rest of this section. The objects of G will be crossingless matchings rather than natural numbers, but the more striking difference between G and 𝑝T is the composition rule. Definition 4.1.1. Define the multicategory G whose • objects are crossingless matchings, Ob(G ) = 𝐵•; • for crossingless matchings 𝑥𝑖 ∈ 𝐵𝑚𝑖 , 𝑖 = 1, . . . , 𝑘, and 𝑦 ∈ 𝐵𝑛, set HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦) = “D(𝑚1,...,𝑚𝑘;𝑛) × Z2. Then, composition Hom(𝑦1, . . . , 𝑦𝑘 ; 𝑧) × Ñ 𝑘 (cid:214) 𝑖=1 é Hom(𝑥𝑖1, . . . , 𝑥𝑖𝛼𝑖 ; 𝑦𝑖) → Hom(𝑥11, . . . , 𝑥𝑘𝛼𝑘 ; 𝑧) is defined by ( “𝐷, 𝑝)◦ Ä(”𝐷1, 𝑝1), · · · , (”𝐷 𝑘 , 𝑝𝑘 )ä = 𝐷(𝐷1, . . . , 𝐷 𝑘 ; ∅)∧, 𝑝 + Ñ é (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷 𝑘 ); 𝐷) (cid:12) (cid:12) (cid:12) (cid:12) 𝑝𝑖 + 𝑘 ∑︁ 𝑖=1 where 𝐷(𝐷1, . . . , 𝐷 𝑘 ; ∅)∧ means 𝐷(𝐷1, . . . , 𝐷 𝑘 ; ∅) with all closed loops removed. Finally, the distinguished identity element Id𝑥 associated to each crossingless matching 𝑥 is given by (1|𝑥|, (|𝑥| , 0)) ∈ Hom(𝑥; 𝑥). Proposition 4.1.2. G is a multicategory; in particular, composition in G is associative. 51 Proof. Consider the following compositions of multimorphisms. (𝑤111, · · · , 𝑤11𝛽11) × · · · × (𝑤1𝛼11, · · · , 𝑤1𝛼1 𝛽1𝛼1 𝐷1𝛼1 ) 𝐷11 (𝑥11, · · · , 𝑥1𝛼1) 𝐷1 × ... × · · · , 𝑥𝑘𝛼𝑘 ) (𝑥𝑘1, 𝐷 𝑘1 (𝑤 𝑘11, · · · , 𝑤 𝑘1𝛽𝑘1) × · · · × (𝑤 𝑘𝛼𝑘1, 𝐷 𝑘𝛼𝑘 · · · , 𝑤 𝑘𝛼𝑘 𝛽𝑘 𝛼𝑘 ) (𝑦1, · · · , 𝑦𝑘 ) 𝐷 𝑧 (4.1.1) 𝐷 𝑘 Our goal is to verify the associativity of these compositions in G ; i.e., 𝑘 (cid:214) 𝛼𝑖(cid:214) (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ) ◦ 𝑖=1 𝑗=1 Ñ 𝑘 (cid:214) 𝑖=1 é (𝐷𝑖, 𝑝𝑖) ◦ (𝐷, 𝑝) = Ñ 𝑘 (cid:214) 𝛼𝑖(cid:214) (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ) ◦ 𝑘 (cid:214) (𝐷𝑖, 𝑝𝑖) é 𝑖=1 𝑗=1 𝑖=1 In either case, the composition yields 𝐷 (cid:0)𝐷1(𝐷11, . . . , 𝐷1𝛼1), 𝐷2(𝐷21, . . . , 𝐷2𝛼2), . . . , 𝐷 𝑘 (𝐷 𝑘1, . . . , 𝐷 𝑘𝛼𝑘 )(cid:1)∧ in the first coordinate. In the former case, the composition yields 𝛼𝑖∑︁ 𝑘 ∑︁ 𝑘 ∑︁ 𝑝𝑖 𝑗 + (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷 𝑘 ); 𝐷) (cid:12) (cid:12) (cid:12) (cid:12) 𝑝 + 𝑝𝑖 + 𝑖=1 𝑖=1 𝑗=1 +(cid:12) (cid:12)𝑊 (cid:174)𝑤 (cid:174)𝑥𝑧((𝐷11, . . . , 𝐷 𝑘𝛼𝑘 ); 𝐷(𝐷1, . . . , 𝐷 𝑘 ))(cid:12) (cid:12) in the second coordinate. In the latter case, the composition yields 𝑝 + 𝑘 ∑︁ 𝑖=1 𝑝𝑖 + 𝑘 ∑︁ 𝛼𝑖∑︁ 𝑖=1 𝑗=1 𝑝𝑖 𝑗 + 𝑘 (cid:12) ∑︁ 𝑊 (cid:174)𝑤𝑖 (cid:174)𝑥𝑖 𝑦𝑖 (cid:12) (cid:12) 𝑖=1 (cid:12) ((𝐷𝑖1, . . . , 𝐷𝑖𝛼𝑖 ); 𝐷𝑖) (cid:12) (cid:12) ◦ (𝐷, 𝑝). (4.1.2) (4.1.3) + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦𝑧((𝐷1(𝐷11, . . . , 𝐷1𝛼1), . . . , 𝐷 𝑘 (𝐷 𝑘1, . . . , 𝐷 𝑘𝛼𝑘 )); 𝐷) (cid:12) (cid:12) (cid:12) (cid:12) in the second coordinate since, for each 𝑖 = 1, . . . , 𝑘, 𝛼𝑖(cid:214) (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ) ◦ (𝐷𝑖, 𝑝𝑖) = Ñ 𝑗=1 𝐷𝑖(𝐷𝑖1, . . . , 𝐷𝑖𝛼𝑖 ), 𝑝𝑖 + 𝛼𝑖∑︁ 𝑗=1 𝑝𝑖 𝑗 + (cid:12) 𝑊 (cid:174)𝑤𝑖 (cid:174)𝑥𝑖 𝑦𝑖 (cid:12) (cid:12) ((𝐷𝑖1, . . . , 𝐷𝑖𝛼𝑖 ), 𝐷𝑖 é . (cid:12) (cid:12) (cid:12) The values (4.1.2) and (4.1.3) are equivalent since the total number of merges and splits of the se- quence of cobordisms is unchanged; otherwise, the minimality condition on the Euler characteristic is contradicted. □ 52 By a multipath, we mean a sequence of collections of composable multimorphisms. Explicitly, a multipath of length 𝑛 is a sequence of sequences of multimorphisms Ä( 𝑓 1 𝑖1 )𝑖1, ( 𝑓 2 𝑖1𝑖2 )𝑖1𝑖2, . . . , ( 𝑓 𝑛 𝑖1𝑖2...𝑖𝑛 )𝑖1𝑖2...𝑖𝑛 ä with ranges 𝑖1 = 1, . . . , 𝑘, 𝑖2 = 1, . . . , 𝑘𝑖1, up to 𝑖𝑛 = 1, . . . , 𝑘𝑖1𝑖2...𝑖𝑛−1 such that dom( 𝑓 𝑡 𝑖1...𝑖𝑡 (cid:16) ) = codom( 𝑓𝑖1...𝑖𝑡 1)𝑡+1, . . . , codom( 𝑓 𝑡+1 𝑖1...𝑖𝑡 𝑘𝑖1...𝑖𝑡 (cid:17) ) for each 𝑡 = 1, . . . , 𝑛. Denote by C [𝑛] the collection of multipaths of length 𝑛. As we proceed, we frequently confound terminology and refer to the sequence of multimorphisms obtained by taking the composites of a multipath as a multipath. For example, suppose that Ä( 𝑓 1 𝑖1 ), ( 𝑓 2 𝑖1𝑖2 ), ( 𝑓 3 𝑖1𝑖2𝑖3 )ä ∈ C [3] with 𝑖1 = 1, . . . , 𝑘, 𝑖2 = 1, . . . 𝑘𝑖1, and 𝑖3 = 1, . . . , 𝑘𝑖1𝑖2. We’ll denote by ( 𝑓 1 𝑖1 ) ◦ ( 𝑓 2 𝑖1𝑖2 ) ◦ ( 𝑓 3 𝑖1𝑖2𝑖3 ) (4.1.4) the sequence of composites ÅÄ 𝑓 2 𝑓 1 1 ◦ 11 ◦ ( 𝑓 3 ÅÄ 𝑓 2 ÅÄ 𝑓 2 𝑓 1 2 ◦ 𝑓 1 𝑘 ◦ 111, . . . , 𝑓 3 11𝑘11 )ä , . . . , (cid:16) 1𝑘11, . . . , 𝑓 3 1𝑘1𝑘1𝑘1 (cid:17)ã ) , 21 ◦ ( 𝑓 3 211, . . . , 𝑓 3 21𝑘21 )ä , . . . , 2𝑘2 ◦ ( 𝑓 3 𝑓 2 2𝑘21, . . . , 𝑓 3 2𝑘2𝑘2𝑘2 (cid:17)ã ) , . . . , 𝑘1 ◦ ( 𝑓 3 𝑘11, . . . , 𝑓 3 𝑘1𝑘 𝑘1 )ä , . . . , (cid:16) 𝑓𝑘 𝑘 𝑘 ◦ ( 𝑓 3 𝑘 𝑘 𝑘1, . . . , 𝑓 3 𝑘 𝑘 𝑘 𝑘 𝑘𝑘𝑘 (cid:17)ã . 1𝑘1 ◦ ( 𝑓 3 𝑓 2 (cid:16) Then, this sequence is frequently referred to as a multipath of length 3, when it is really a composite of such a multipath. Finally, distilling notation further, we’ll write (cid:174)𝑓 1 := ( 𝑓 1 1 , . . . , 𝑓 1 𝑘 ), (cid:174)𝑓 2 𝑖 := ( 𝑓 2 𝑖1, . . . , 𝑓 2 𝑖𝑘𝑖 ) and similarly for (cid:174)𝑓 3 𝑖 𝑗 , and write the sequence of composites of multimorphisms (4.1.4) as Ñ 𝑘 (cid:214) é (cid:174)𝑓 2 𝑖 ◦ (cid:174)𝑓 1 ◦ Ñ 𝑘 (cid:214) 𝑘𝑖(cid:214) é (cid:174)𝑓 3 𝑖 𝑗 . 𝑖=1 𝑖=1 𝑗=1 (4.1.5) We will replace 𝑘𝑖 with the notation 𝛼𝑖, and similarly the notation 𝑘𝑖 𝑗 with the notation 𝛽𝑖 𝑗 . This runs the risk of presenting confusion in light of the associator and compatibility maps introduced momentarily—we hope that the meaning of notation is clear presented in context. 53 We remark that if (cid:174)𝑓 1 is a single multimorphism, then the multipath (4.1.5) can be pictured as (4.1.1) from the previous proof. In general, (cid:174)𝑓 may consist of many multimorphisms, and we can think of a multipath as a collection of such diagrams—in other words, multipaths can be viewed as trees and forests. Definition 4.1.3. A grading multicategory is pair (C , 𝛼) where C is a multicategory and 𝛼 : C [3] → K× is a 3-cocycle, meaning that for all Ö Ñ 𝑘 (cid:214) é (cid:174)𝑔𝑖 , (cid:174)𝑓 , Ñ 𝑘 (cid:214) 𝛼𝑖(cid:214) é (cid:174)ℎ𝑖 𝑗 , Ñ 𝑘 (cid:214) 𝛼𝑖(cid:214) 𝛽𝑖 𝑗 (cid:214) 𝑖=1 𝑖=1 𝑗=1 𝑖=1 𝑗=1 𝑘=1 éè (cid:174)ℓ𝑖 𝑗 𝑘 ∈ C [4] (shortened to (cid:174)𝑓 , (cid:174)𝑔, (cid:174)ℎ, (cid:174)ℓ ∈ C [4]), 𝛼 satisfies the expression 𝑑𝛼( (cid:174)ℓ, (cid:174)ℎ, (cid:174)𝑔, (cid:174)𝑓 ) := 𝛼( (cid:174)ℓ, (cid:174)ℎ, (cid:174)𝑔)𝛼( (cid:174)ℓ, (cid:174)ℎ, (cid:174)𝑓 (cid:174)𝑔)−1𝛼( (cid:174)ℓ, (cid:174)𝑔(cid:174)ℎ, (cid:174)𝑓 )𝛼((cid:174)ℎ (cid:174)ℓ, (cid:174)𝑔, (cid:174)𝑓 )−1𝛼((cid:174)ℎ, (cid:174)𝑔, (cid:174)𝑓 ) = 1. We call such an 𝛼 an associator. 4.2 G as a grading multicategory Our goal is to show that there exists a suitable associator 𝛼 endowing G with the structure of a grading multicategory. We will define 𝛼 to be the product of two values associated to changes of chronologies, one explicit and the other implicit. We’ll use the notation (cid:174)𝐷, (cid:174)𝐷′, and so on to denote collections of planar arc diagrams which form a multipath in 𝑝T. If (cid:174)𝐷 is a single planar arc diagram 𝐷, and (cid:174)𝐷′ = (𝐷1, . . . , 𝐷𝑛), then their composition, which will in general be denoted (cid:174)𝐷( (cid:174)𝐷′), is denoted 𝐷(𝐷1, . . . , 𝐷𝑛). In the general setting, the constituents of a multipath 𝑔, 𝑔′, 𝑔′′, 𝑔′′′ ∈ G [4] will be written 𝑔 = ( (cid:174)𝐷, (cid:174)𝑝) = (cid:214) (𝐷𝑖, 𝑝𝑖) 𝑔′ = ( (cid:174)𝐷′, (cid:174)𝑝′) = 𝑖 (cid:214) (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ) 𝑔′′ = ( (cid:174)𝐷′′, (cid:174)𝑝′′) = 𝑖, 𝑗 (cid:214) 𝑖, 𝑗,𝑘 (𝐷𝑖 𝑗 𝑘 , 𝑝𝑖 𝑗 𝑘 ) 𝑔′′′ = ( (cid:174)𝐷′′′, (cid:174)𝑝′′′) = (cid:214) 𝑖, 𝑗,𝑘,ℓ (𝐷𝑖 𝑗 𝑘ℓ, 𝑝𝑖 𝑗 𝑘ℓ). 54 On one hand, our indexing notation allows us to write (cid:174)𝐷′ collection (𝐷1( (cid:174)𝐷′ as (cid:206)𝑖, 𝑗,𝑘 ( (cid:174)𝐷′′′ 𝑖, 𝑗,𝑘 , (cid:174)𝑝′′′ 𝑖 = (cid:206) 𝑗 (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ). Then, (cid:174)𝐷( (cid:174)𝐷′) denotes the 1), . . . , 𝐷𝑛( (cid:174)𝐷′ 𝑛). We could also use our indexing notation to write, for example, 𝑔′′′ 𝑖, 𝑗,𝑘 ). Finally, we will denote by 𝑃 the sum of the entries of (cid:174)𝑝 (that is, 𝑃 = (cid:205)𝑖 𝑝𝑖) and similarly for the other cases; e.g., 𝑃′′′ = (cid:174)𝑝′′′ · ⟨1, . . . , 1⟩ = (cid:205)𝑖, 𝑗,𝑘,ℓ 𝑝𝑖 𝑗 𝑘ℓ. As we proceed, we will make use of the following lemma. It is implicit in the proof of Proposition 4.1.2, but we restate it here. Lemma 4.2.1. For any multipath of planar arc diagrams (cid:174)𝐷, (cid:174)𝐷′, and (cid:174)𝐷′′ as above, (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) (cid:12) (cid:12) (cid:12) = (cid:12) (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) . Proof. The compositions 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) ◦ 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) and 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷) ◦ 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (we assume the first cobordisms in both composites are the identity elsewhere) have the same source and target. Thus they are isotopic cobordisms—if this were not the case, the minimality condition on the Euler characteristic would be contradicted. □ To construct our associator, consider the change of chronology (cid:174)𝑤 (cid:174)𝐷( (cid:174)𝐷′( (cid:174)𝐷′′))(cid:174)𝑧 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) 𝑊 (cid:174)𝑤 (cid:174)𝑦 (cid:174)𝑧( (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷) (cid:174)𝑤 (cid:174)𝐷′′(cid:174)𝑥 ⊗ (cid:174)𝑥𝐷( (cid:174)𝐷′)(cid:174)𝑧 𝐻𝛼 ===⇒ (cid:174)𝑤 (cid:174)𝐷′( (cid:174)𝐷′′)(cid:174)𝑦 ⊗ (cid:174)𝑦 (cid:174)𝐷 (cid:174)𝑧 1 (cid:174)𝑤 (cid:174)𝐷′′ (cid:174)𝑥 ⊗𝑊 (cid:174)𝑥 (cid:174)𝑦 (cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′)⊗1 (cid:174)𝑦 (cid:174)𝐷 (cid:174)𝑧 (cid:174)𝑤 (cid:174)𝐷′′(cid:174)𝑥 ⊗ (cid:174)𝑥 (cid:174)𝐷′(cid:174)𝑦 ⊗ (cid:174)𝑦 (cid:174)𝐷 (cid:174)𝑧 Define 𝛼1(𝑔′′, 𝑔′, 𝑔) to be the evaluation of this change of chronology 𝜄(𝐻𝛼)—notice that this component of 𝛼 does not see the second coordinates of its inputs. Secondly, take 𝛼2(𝑔′′, 𝑔′, 𝑔) = 𝜆 Ñ (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) é , ∑︁ 𝑖, 𝑗,𝑘 𝑝𝑖 𝑗 𝑘 = 𝜆 (cid:16)(cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) . 55 Then, set 𝛼 = 𝛼2𝛼1. Remark 4.2.2. This definition is clearly motivated by and generalizes the associator presented in [NP20]. A property we will use frequently is that the degree of cobordisms decomposes into a sum of constituents; notice, for example, that 𝛼2(𝑔′′′, 𝑔′′, 𝑔′) = 𝜆 (cid:12) (cid:16)(cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) Ñ , 𝑃′′′(cid:17) = 𝜆 ∑︁ 𝑖 (cid:12) 𝑊 (cid:174)𝑤𝑖 (cid:174)𝑥𝑖 (cid:174)𝑦𝑖 (cid:12) (cid:12) ( (cid:174)𝐷′′ (cid:12) 𝑖 , (cid:174)𝐷𝑖) (cid:12) (cid:12) é , 𝑃′′′ (We could rewrite the last line as (cid:206)𝑖 𝜆 (cid:16)(cid:12) 𝑊 (cid:174)𝑤𝑖 (cid:174)𝑥𝑖 (cid:174)𝑦𝑖 (cid:12) (cid:12) ( (cid:174)𝐷′′ 𝑖 , (cid:174)𝐷𝑖) , 𝑃′′′(cid:17) (cid:12) (cid:12) (cid:12) , invoking the bilinearity of 𝜆, although there might be slight confusion with this rewriting since 𝑃′′′ is a sum involving the index 𝑖—indeed, the second coordinates of each term in this product are equivalent.) Finally, we remark that we can view 𝛼 as coming from the following sequence of schematics, just as in [NP20] (pictured for the case we have just described). 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼1(𝑔′′′, 𝑔′′, 𝑔′) 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼1(𝑔′′′, 𝑔′′, 𝑔′)𝛼2(𝑔′′′, 𝑔′′, 𝑔′) 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 Proposition 4.2.3. The map 𝛼 : G [3] → 𝑅 is a 3-cocycle. Proof. This proof is completely analogous to the proof of Proposition 5.4 in [NP20]—we represent their proof in the context of grading multicategories. As in the original case, 𝑑𝛼(𝑔′′′, 𝑔′′, 𝑔′, 𝑔) 56 computes the difference between the paths of the diagram below. 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼(𝑔′′′,𝑔′′,𝑔′) 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼(𝑔′′′𝑔′′,𝑔′,𝑔) 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼(𝑔′′′,𝑔′′,𝑔′𝑔) 𝛼(𝑔′′′,𝑔′′𝑔′,𝑔) ⇒ 𝑑𝛼 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 𝛼(𝑔′′,𝑔′,𝑔) (4.2.1) 𝑔′′′ 𝑔′′ 𝑔′ 𝑔 On one hand, we are comparing two locally vertical changes of chronology with the same source and target, so the following diagram commutes by Proposition 3.1.3. Ä (cid:174)𝐷′′′, (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷ä 𝛼1 𝛼1( (cid:174)𝐷′′′, (cid:174)𝐷′′, (cid:174)𝐷′) (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 𝛼1( (cid:174)𝐷′′, (cid:174)𝐷′, (cid:174)𝐷) (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 Ä (cid:174)𝐷′′( (cid:174)𝐷′′′), (cid:174)𝐷′, (cid:174)𝐷ä 𝛼1 (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 𝛼1( (cid:174)𝐷′′′, (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 (cid:174)𝐷′′′(cid:174)𝐷′′ (cid:174)𝐷′ (cid:174)𝐷 𝜅 Since the corresponding change of chronology consists only of the sliding of two chronological cobordisms past one another, we know by work in Section 3.1 that 𝜅 is 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) (cid:12) 𝑊(cid:174)𝑣 (cid:174)𝑤 (cid:174)𝑥( (cid:174)𝐷′′′, (cid:174)𝐷′′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) . Thus, the contribution of 𝛼1 in equation (4.2.1) is top = 𝜅 bot. 57 On the other hand, we can compute and compare the contributions of 𝛼2 on the top and bottom path of (4.2.1). The top path evaluates to 𝜆 (cid:12) (cid:16)(cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′(cid:17) · 𝜆 (cid:12) (cid:16)(cid:12) 𝑊 (cid:174)𝑤,(cid:174)𝑦,(cid:174)𝑧( (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′(cid:17) · 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) or, applying bilinearity of 𝜆, 𝜆 (cid:12) (cid:16)(cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′( (cid:174)𝐷′′), (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′(cid:17) · 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) . (4.2.2) The bottom path is slightly trickier to evaluate, since the second coordinate of 𝛼2(𝑔′′′𝑔′′, 𝑔′, 𝑔) requires a computation. As in the proof of Proposition 4.1.2, this comes from summing the second coordinates of 𝑔′′′ and 𝑔′′ and the cobordisms among their coordinates; explicitly, 𝛼2(𝑔′′′𝑔′′, 𝑔′, 𝑔) = 𝜆 Ñ (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′ + 𝑃′′ + (cid:12) 𝑊(cid:174)𝑣(𝑖, 𝑗) (cid:174)𝑤𝑖 𝑗 (cid:174)𝑥𝑖 𝑗 (cid:12) (cid:12) ∑︁ 𝑖, 𝑗 é (cid:12) 𝑖 𝑗 , (cid:174)𝐷′′ 𝑖 𝑗 ) (cid:12) (cid:12) ( (cid:174)𝐷′′′ . The last summation in the second coordinate can be rewritten as in Remark 4.2.2: we find that the bottom path evaluates to 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′ + 𝑃′′ + (cid:12) (cid:12) 𝑊(cid:174)𝑣 (cid:174)𝑤 (cid:174)𝑥( (cid:174)𝐷′′′, (cid:174)𝐷′′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) · 𝜆 (cid:12) (cid:16)(cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′(cid:17) . Decomposing via bilinearity yields 𝜆 , 𝑃′′′(cid:17) (cid:12) (cid:16)(cid:12) (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) , 𝑃′′′(cid:17) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 · 𝜆 . , 𝑃′′(cid:17) · 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) (cid:12) 𝑊(cid:174)𝑣 (cid:174)𝑤 (cid:174)𝑥( (cid:174)𝐷′′′, (cid:174)𝐷′′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) Combining the first and last term, and reordering suggestively, gives the product 𝜆 , (cid:12) (cid:16)(cid:12) (cid:12) (cid:12) 𝑊(cid:174)𝑣 (cid:174)𝑤 (cid:174)𝑥( (cid:174)𝐷′′′, (cid:174)𝐷′′) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) , 𝑃′′(cid:17) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 . (cid:17) · 𝜆 (cid:16)(cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧( (cid:174)𝐷′′, (cid:174)𝐷( (cid:174)𝐷′)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′′(cid:17) In this rewriting, the first term is 𝜅. Moreover, by Lemma 4.2.1, the first coordinate of the second term is equivalent to the fist coordinate of the first term of (4.2.2). Thus, the overall contribution of 𝛼2 in equation (4.2.1) is 𝜅 top = bot. Together, this provides that 𝑑𝛼 = 1, as desired. □ 58 4.3 Generalities on modules graded by grading multicategories Before proceeding with the grading multicategory at hand, we note generalities of C -graded modules. That is, we consider the ways in which results of Section 4 of [NP20] lift to the setting of grading multicategories. Throughout, C is a grading multicategory with associator 𝛼 over a unital, commutative ring K. By a C -graded K-module, we mean a K-module 𝑀 with decomposition 𝑀 = (cid:201) 𝑔∈Mor(C ) 𝑀𝑔 where 𝑔 is a multimorphism of C . As before, we write |𝑥| = 𝑔 whenever 𝑥 ∈ 𝑀𝑔. This generalizes the notion of grading by a category, introduced in [NP20], which in turn generalized the notion of grading by a group (take the category consisting of a single element ★ and End(★) = 𝐺). Of course, we are interested in the case C = G and K = 𝑅. Tensor products in this setting are rather odd in the sense that their graded structure has a few different interpretations. This choice should be clear given the context. In one case, if 𝑀 and 𝑀′ are two C -graded K-modules, then we can define where 𝑀′ ⊗ 𝑀 = (cid:202) (𝑀′ ⊗ 𝑀)ℎ ℎ∈Mor(C ) (𝑀′ ⊗ 𝑀)ℎ = (cid:202) 𝑀′ 𝑔′ ⊗K 𝑀𝑔. ℎ=𝑔◦𝑔′ Notice that this definition does not make full use of the flexibility offered by a grading multicategory. On the other hand, for C -graded modules 𝑀1, . . . , 𝑀𝑘 , 𝑀, we can view the tensor product over K as C -graded by defining (𝑀1 ⊗ · · · ⊗ 𝑀𝑘 ) ⊗ 𝑀 = (cid:202) ℎ∈Mor(C ) [(𝑀1 ⊗ · · · ⊗ 𝑀𝑘 ) ⊗ 𝑀]ℎ where [(𝑀1 ⊗ · · · ⊗ 𝑀𝑘 ) ⊗ 𝑀]ℎ = (cid:202) (𝑀1,𝑔1 ⊗ · · · ⊗ 𝑀𝑘,𝑔𝑘 ) ⊗ 𝑀𝑔. ℎ=𝑔◦(𝑔1,...,𝑔𝑘) Notice that 𝑀1 ⊗ · · · ⊗ 𝑀𝑘 is interpreted as a collection of C -graded modules, but not as a C -graded module itself. Rather, 𝑀1 ⊗ · · · ⊗ 𝑀𝑘 in the above scenario is viewed as C 𝑘 = C × · · · × C -graded 59 in the sense that 𝑀1 ⊗ · · · ⊗ 𝑀𝑘 = (cid:202) 𝑀1,𝑔1 ⊗ · · · ⊗ 𝑀𝑘,𝑔𝑘 . (𝑔1,...,𝑔𝑘)∈Mor(C 𝑘) We will always abbreviate 𝑀1 ⊗ · · · ⊗ 𝑀𝑘 (without interpretation as a C -graded module itself) by (𝑀1, . . . , 𝑀𝑘 ) or, more succinctly, (cid:174)𝑀 to avoid confusion. For example, the above scenario will be written (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 or, succinctly, (cid:174)𝑀 ⊗ 𝑀. Likewise, by (cid:174)𝑀′ ⊗ (cid:174)𝑀 we mean ( (cid:174)𝑀′ 1 ⊗ 𝑀1, . . . , (cid:174)𝑀′ Denote by ModC 𝑘 ⊗ 𝑀𝑘 ). K , or just ModC , the category of C -graded K-modules, whose morphisms are K-linear maps which preserve grading. That is, for 𝑓 : 𝑀 → 𝑁, we have 𝑓 (𝑀𝑔) ⊂ 𝑁𝑔 for each 𝑔. We call such maps C -graded, or just graded. The associator of the grading multicategory C provides a coherence isomorphism ( (cid:174)𝑀′′ ⊗ (cid:174)𝑀′) ⊗ 𝑀 (cid:174)𝑀′′ ⊗ ( (cid:174)𝑀′ ⊗ 𝑀) ( (cid:174)𝑚′′ ⊗ (cid:174)𝑚′) ⊗ 𝑚 𝛼 Ä(cid:12) (cid:12) (cid:174)𝑚′(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑚′′(cid:12) (cid:12) ,|𝑚| ä (cid:174)𝑚′′ ⊗ ( (cid:174)𝑚′ ⊗ 𝑚) where (cid:174)𝑚′ (and, similarly, (cid:174)𝑚′′) is comprised of tensored homogeneous elements 𝑚𝑖 ∈ (𝑀𝑖)|𝑚𝑖 |, and (cid:12) (cid:174)𝑚′(cid:12) (cid:12) (cid:12) = (|𝑚1| , . . . ,|𝑚𝑘 |) is the corresponding collection of multimorphisms (that is, C -gradings). Since the number of modules involved in a tensor product can vary, we have a collection of unit objects, one for each 𝑘, all defined as the tensor product of a single module: let 1 denote the C -graded K-module (cid:201) 𝑋∈Ob(C )(K)1𝑋 . Then, 1⊗𝑘 is a unit object in the sense that there are (graded) isomorphisms (i.e., left- and right-unitors) L : 1⊗𝑘 ⊗ 𝑀 (cid:27) 𝑀 and R : 𝑀 ⊗ 1 (cid:27) 𝑀 of C -graded modules which satisfy the triangle identity Ä(𝑀1, . . . , 𝑀𝑘 ) ⊗ 1⊗𝑘 ä ⊗ 𝑀 𝛼 (𝑀1, . . . , 𝑀𝑘 ) ⊗ Ä1⊗𝑘 ⊗ 𝑀ä (cid:206)𝑖 R𝑖 ⊗id𝑀 (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 (cid:206)𝑖 id𝑀𝑖 ⊗L where R𝑖 means the right unitor applied to 𝑀𝑖. The left- and right-unitors we pick are determined by the associator: if each 𝑚𝑖 in 𝑚1 ⊗ · · · ⊗ 𝑚𝑘 ∈ 𝑀1 ⊗ · · · ⊗ 𝑀𝑘 is homogeneous (with, say, 60 |𝑚𝑖 | : (𝑥𝑖1, . . . , 𝑥𝑖𝛼𝑖 ) ↦→ 𝑦′ 𝑖), and similarly for 𝑐1 ⊗ · · · ⊗ 𝑐𝑘 ∈ 1⊗𝑘 and 𝑚 ∈ 𝑀 (with, say, |𝑚| : (𝑦1, . . . , 𝑦𝑘 ) ↦→ 𝑧), we can choose left-unitor given by (𝑐1 ⊗ · · · ⊗ 𝑐𝑘 ) ⊗ 𝑚 ↦→ 𝛼((1𝑦1, . . . , 1𝑦𝑘 ), (1𝑦1, . . . , 1𝑦𝑘 ),|𝑚|)−1𝑐1 · · · 𝑐𝑘 𝑚 and right-unitor by (𝑚1⊗· · ·⊗𝑚𝑘 )⊗(𝑐1⊗· · ·⊗𝑐𝑘 ) ↦→ 𝛼((|𝑚1| , . . . ,|𝑚𝑘 |), (1𝑦1, . . . , 1𝑦𝑘 ), (1𝑦1, . . . , 1𝑦𝑘 ))𝑚1𝑐1⊗· · ·⊗𝑚𝑘 𝑐𝑘 . To see why this satisfies the triangle identity, take 𝑦𝑖 = 𝑦′ 𝑖 so that (cid:12) (cid:12) (cid:174)𝑚′(cid:12) (cid:12) and |𝑚| are composable multimorphisms, and consider the path of length 4 given by (|𝑚1|,...,|𝑚𝑘 |) −−−−−−−−−→ (cid:174)𝑦 (cid:174)𝑥 (1𝑦1 ,...,1𝑦𝑘 −−−−−−−−→ (cid:174)𝑦 ) Then, the cocycle condition of 𝛼 establishes that (1𝑦1 ,...,1𝑦𝑘 −−−−−−−−→ (cid:174)𝑦 |𝑚| −−→ 𝑧. ) 1 = 𝑑𝛼((cid:12) (cid:12) (cid:174)𝑚′(cid:12) (cid:12) , 1(cid:174)𝑦, 1(cid:174)𝑦,|𝑚|) (cid:12) (cid:174)𝑚′(cid:12) (cid:12) , 1(cid:174)𝑦, 1(cid:174)𝑦)𝛼((cid:12) (cid:12) (cid:174)𝑚′(cid:12) = 𝛼((cid:12) (cid:12) , 1(cid:174)𝑦,|𝑚|)−1𝛼(1(cid:174)𝑦, 1(cid:174)𝑦,|𝑚|). This gives the triangle identity after re-arranging. Since the cocyle requirement of the associator of a grading multicategory is exactly the pen- tagonal relation of monoidal categories, it follows from the work above that ModC K has a structure resembling a monoidal category. Finally, we briefly describe two important types of C -graded modules: algebras and multimod- ules. A C -graded algebra is a C -graded K-module 𝐴 = (cid:201) 𝑔∈Mor(C ) 𝐴𝑔, supported only in gradings 𝑔 which are single-input multimorphisms (i.e., morphisms) of C , with a K-linear multiplication map 𝜇 : 𝐴 ⊗ 𝐴 → 𝐴 and a unit 1𝑋 ∈ 𝐴Id𝑋 for each 𝑋 ∈ Ob(C ) such that (i) 𝜇 is graded: 𝜇(𝐴𝑔′, 𝐴𝑔) ⊂ 𝐴𝑔◦𝑔′ for all 𝑔′, 𝑔 ∈ C , (cid:12)𝑦(cid:12) (ii) 𝜇 is graded-associative: 𝜇(𝜇(𝑧, 𝑦), 𝑥) = 𝛼(|𝑧| ,(cid:12) (cid:12) ,|𝑥|)𝜇(𝑧, 𝜇(𝑦, 𝑥)), and (iii) 𝜇(1𝑌 , 𝑥) = L(Id𝑌 ,|𝑥|) 𝑥 and 𝜇(𝑥, 1𝑋) = R(|𝑥| , Id𝑋) 𝑥 for all 𝑥 ∈ 𝐴|𝑥|:𝑋→𝑌 . 61 Before proceeding, we emphasize that C -graded algebras are supported by single-input multimor- phisms exclusively—really, C -graded algebras are hardly different than the C-graded algebras (C a category) of [NP20]. We’ll write 𝜇(𝑥, 𝑦) as 𝑥 · 𝑦 when it is clear which multiplication is in use. Going on, we will only consider the tensor product (𝐴1, . . . , 𝐴𝑘 )—that is, 𝐴1 ⊗ · · · ⊗ 𝐴𝑘 viewed as C 𝑘 -graded—with mul- tiplication (𝑎′ 1, . . . , 𝑎′ 𝑘 , 𝑎𝑘 )). Suppose 𝐴1, . . . , 𝐴𝑘 , 𝐵 are C -graded algebras. Then, a C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule 𝑘 ) · (𝑎1, . . . , 𝑎𝑘 ), or, concisely, (cid:174)𝑎′ · (cid:174)𝑎, defined as (𝜇 𝐴1(𝑎′ 1, 𝑎1), . . . , 𝜇 𝐴𝑘 (𝑎′ is a C -graded K-module 𝑀 = (cid:201) 𝑔∈Mor(C ) 𝑀𝑔 with graded, K-linear left and right actions 𝜌𝐿 : (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝑀 → 𝑀 and 𝜌𝑅 : 𝑀 ⊗ 𝐵 → 𝑀 such that (i) 𝜌𝐿(( (cid:174)𝑎′ · (cid:174)𝑎), 𝑚) = 𝛼((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎′(cid:12) (cid:12) ,|𝑚|)𝜌𝐿( (cid:174)𝑎′, 𝜌𝐿( (cid:174)𝑎, 𝑚)), (ii) 𝜌𝑅(𝜌𝑅(𝑚, 𝑏′), 𝑏) = 𝛼(|𝑚| ,|𝑏|′ ,|𝑏|)𝜌𝑅(𝑚, 𝑏′ · 𝑏), (cid:12) (cid:174)𝑎(cid:12) (iii) 𝜌𝑅(𝜌𝐿( (cid:174)𝑎, 𝑚), 𝑏) = 𝛼((cid:12) (cid:12) ,|𝑚| ,|𝑏|)𝜌𝐿( (cid:174)𝑎, 𝜌𝑅(𝑚, 𝑏)), and (iv) 𝜌𝐿((1𝑌 , . . . , 1𝑌 ), 𝑚) = L((1𝑌 , . . . , 1𝑌 ),|𝑚|)𝑚 and 𝜌𝑅(𝑚, 1𝑋) = R(|𝑚| , Id𝑋) for all 𝑚 ∈ 𝑀|𝑚|:𝑋→𝑌 for all (cid:174)𝑎′, (cid:174)𝑎 ∈ (𝐴1, . . . , 𝐴𝑘 ), 𝑏′, 𝑏 ∈ 𝐵, and 𝑚 ∈ 𝑀. One should take caution: again, we are viewing (𝐴1, . . . , 𝐴𝑘 ) as a collection of C -graded algebras, not as a single C -graded object. In particular, a C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule is, perhaps surprisingly, not equivalent to the notion of a C -graded (𝐴1 ⊗K · · · ⊗K 𝐴𝑘 , 𝐵)-bimodule. In particular, the left action 𝜌𝐿 is graded in the sense that 𝜌𝐿((𝐴1,𝑔1 ⊗ · · · ⊗ 𝐴𝑘,𝑔𝑘 ) ⊗ 𝑀𝑔) ⊂ 𝑀𝑔◦(𝑔1,...,𝑔𝑘) and not in the sense that 𝜌𝐿((𝐴1,𝑔1 ⊗ · · · ⊗ 𝐴𝑘,𝑔𝑘 ) ⊗ 𝑀𝑔) ⊂ 𝑀𝑔◦𝑔𝑘◦···◦𝑔1. 62 We define a C -graded (𝐴, 𝐵)-bimodule as a C -graded (𝐴; 𝐵)-multimodule for C -graded algebras 𝐴 and 𝐵. A graded map of (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodules is a graded, K-linear map satisfying 𝑓 (𝜌𝐿( (cid:174)𝑎, 𝑚)) = 𝜌𝐿( (cid:174)𝑎, 𝑓 (𝑚)) and 𝑓 (𝜌𝑅(𝑚, 𝑏)) = 𝜌𝑅( 𝑓 (𝑚), 𝑏) for all (cid:174)𝑎, 𝑚, and 𝑏. Denote the category of C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodules, cumbersomely, by MultiModC 𝑅 (𝐴1, . . . , 𝐴𝑘 ; 𝐵). As always, if it is clear what algebras we’re working over, we denote this category by MultiModC . Take 𝑀 ∈ MultiModC (𝐵1, . . . , 𝐵𝑘 ; 𝐶) and 𝑀𝑖 ∈ MultiModC (𝐴𝑖1, . . . , 𝐴𝑖ℓ𝑖 ; 𝐵𝑖) for each 𝑖 = 1, . . . , 𝑘. Then (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 has the structure of a C -graded (𝐴11, . . . , 𝐴𝑘ℓ𝑘 ; 𝐶)-multimodule by defining left- and right-actions so that the diagrams (𝐴11, . . . , 𝐴𝑘ℓ𝑘 ) ⊗ ((𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀) (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 𝛼−1 (cid:206) 𝜌𝐿 ⊗1 ((𝐴11, . . . , 𝐴𝑘ℓ𝑘 ) ⊗ (𝑀1, . . . , 𝑀𝑘 )) ⊗ 𝑀 and ((𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀) ⊗ 𝐶 (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 𝛼 1⊗𝜌𝑅 (𝑀1, . . . , 𝑀𝑘 ) ⊗ (𝑀 ⊗ 𝐶) commute, interpreting ((𝐴11, . . . , 𝐴𝑘ℓ𝑘 ) ⊗ (𝑀1, . . . , 𝑀𝑘 )) as ((𝐴11, . . . , 𝐴1ℓ1) ⊗ 𝑀1, . . . , (𝐴𝑘1, . . . , 𝐴𝑘ℓ𝑘 ) ⊗ 𝑀𝑘 ). Explicitly, the left action is given by (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (𝑎11, . . . , 𝑎𝑘ℓ𝑘 )·( (cid:174)𝑚⊗𝑚) := 𝛼−1((cid:12) (cid:12) ,|𝑚|)(𝜌1 𝐿((𝑎11, . . . , 𝑎1ℓ1), 𝑚1), . . . , 𝜌𝑘 𝐿((𝑎𝑘1, . . . , 𝑎𝑘ℓ𝑘 ), 𝑚𝑘 ))⊗𝑚 where 𝜌𝑖 𝐿 is meant to denote the left action for the multimodule 𝑀𝑖. The right is just ( (cid:174)𝑚 ⊗ 𝑚) · 𝑐 := 𝛼((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚| ,|𝑐|) (cid:174)𝑚 ⊗ (𝜌𝑅(𝑚, 𝑐)). 63 Finally, we note that the tensor product of (𝑀1, . . . , 𝑀𝑘 ) with 𝑀 over C -graded algebras (𝐵1, . . . , 𝐵𝑘 ), denoted (𝑀1, . . . , 𝑀𝑘 ) ⊗(𝐵1,...,𝐵𝑘) 𝑀, is defined as (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:14)(cid:0)(𝜌1 𝑅(𝑚𝑘 , 𝑏𝑘 )) ⊗ 𝑚 𝑅(𝑚1, 𝑏1), . . . , 𝜌𝑘 (cid:12) (cid:12) (cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) , (cid:174)𝑏 (cid:12) (cid:12) (cid:12) − 𝛼((cid:12) ,|𝑚|)(𝑚1, . . . , 𝑚𝑘 ) ⊗ 𝜌𝐿((𝑏1, . . . , 𝑏𝑘 ), 𝑚)(cid:1) where 𝜌𝑖 𝑅 is meant to denote the right action for the multimodule 𝑀𝑖. This is to say that the tensor product of (𝑀1, . . . , 𝑀𝑘 ) with 𝑀 over (𝐵1, . . . , 𝐵𝑘 ) is defined as the coequializer of the diagram (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ (𝐵1, . . . , 𝐵𝑘 )(cid:1) ⊗ 𝑀 (cid:206) 𝜌𝑖 𝑅 ⊗1𝑀 𝛼 (cid:206) 1𝑀𝑖 ⊗𝜌𝐿 (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 (𝑀1, . . . , 𝑀𝑘 ) ⊗ (cid:0)(𝐵1, . . . , 𝐵𝑘 ) ⊗ 𝑀(cid:1) in the category of C -graded modules. Given 𝑓 : 𝑀 → 𝑁 and 𝑓𝑖 : 𝑀𝑖 → 𝑁𝑖 for all 𝑖 = 1, . . . , 𝑘, we define the tensor product of maps ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 : (𝑀1, . . . , 𝑀𝑘 ) ⊗(𝐵1,...,𝐵𝑘) 𝑀 → (𝑁1, . . . , 𝑁𝑖) ⊗(𝐵1,...,𝐵𝑘) 𝑁 by (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) (cid:0)(𝑚1, . . . , 𝑚𝑘 ) ⊗ 𝑚(cid:1) = ( 𝑓1(𝑚1), . . . , 𝑓𝑘 (𝑚𝑘 )) ⊗ 𝑓 (𝑚). 4.4 G -graded arc modules If 𝐷 is a planar arc diagram of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛), F (𝐷) is a G -graded 𝑅-multimodule where, for 𝑢 ∈ F (𝐷(𝑥1, . . . , 𝑥𝑘 ; 𝑦)) ⊂ F (𝐷), degG (𝑢) = ( “𝐷, deg𝑅(𝑢)) ∈ HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦). Then, the following lemmas are apparent. Lemma 4.4.1. The composition maps 𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] preserve G -grading. Proof. This is by definitions: recall the composition maps 𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷] : (cid:0)F (𝐷1), . . . , F (𝐷 𝑘 )(cid:1) ⊗ F (𝐷) → F (𝐷(𝐷1, . . . , 𝐷 𝑘 )) 64 from the beginning of this section. Now, an element (𝑢1, . . . , 𝑢𝑘 ) ⊗ 𝑢 living in the source has degree (𝐷∧, deg𝑅(𝑢))◦ (cid:0)(𝐷∧ 1 , deg𝑅(𝑢1)), . . . , (𝐷∧ Ñ 𝑘 , deg𝑅(𝑢𝑘 ))(cid:1) = 𝐷(𝐷1, . . . , 𝐷 𝑘 )∧, deg𝑅(𝑢) + deg𝑅(𝑢𝑖) + (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((𝐷1, . . . , 𝐷 𝑘 ); 𝐷) (cid:12) (cid:12) (cid:12) (cid:12) é 𝑘 ∑︁ 𝑖=1 where |𝑢𝑖 | : (cid:174)𝑥𝑖 → 𝑦𝑖 and |𝑢| : (cid:174)𝑦 → 𝑧. On the other hand, degG (cid:0)𝜇[(𝐷1, . . . , 𝐷 𝑘 ); 𝐷]((𝑢1, . . . , 𝑢𝑘 ) ⊗ 𝑢)(cid:1) is, by the definition of the degree of cobordisms, the second coordinate of the pair above. □ Lemma 4.4.2. For 𝑢𝑖 𝑗 ∈ F (𝐷𝑖 𝑗 ), 𝑢𝑖 ∈ F (𝐷𝑖), and 𝑢 ∈ F (𝐷), 𝜇[ (cid:174)𝐷′( (cid:174)𝐷′′), 𝐷] Ä𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′]((cid:174)𝑢′′, (cid:174)𝑢′), 𝑢ä = 𝛼 Ä(cid:12) (cid:12)(cid:174)𝑢′(cid:12) (cid:12) ,(cid:12) (cid:12)(cid:174)𝑢′′(cid:12) (cid:12) ,|𝑢| ä 𝜇[ (cid:174)𝐷′′, 𝐷( (cid:174)𝐷′)] Ä (cid:174)𝑢′′, 𝜇[ (cid:174)𝐷′, 𝐷]((cid:174)𝑢′, 𝑢)ä . Proof. This is immediate by the construction of the 𝜇 composition maps and the associator 𝛼, recalling that ChCob• has the relation that 𝑊 ′ = 𝜄(𝐻)𝑊 for each change of chronology 𝐻 : 𝑊 ⇒ 𝑊 ′. □ Proposition 4.4.3. The arc algebra F (1𝑛) = 𝐻𝑛 is unital and associative as a G -graded 𝑅-algebra. Proof. Recall that the multiplication in 𝐻𝑛 is 𝜇[1𝑛, 1𝑛], so Lemma 4.4.1 implies that the multiplica- tion in 𝐻𝑛 is G -graded, while Lemma 4.4.2 implies that it is graded associative. Since we defined the left- and right-unitors via the associator, the third requirement of G -graded algebras is also satisfied by Lemma 4.4.2, and we conclude that 𝐻𝑛 is a G -graded algebra. Associativity follows from Lemma 4.4.2 as well; for a proof of unitality, see the proof of Proposition 6.2 in [NP20]. □ Proposition 4.4.4. Suppose 𝐷 is a planar arc diagram of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛). Then F (𝐷) is a G -graded (𝐻𝑚1, . . . , 𝐻𝑚𝑘 ; 𝐻𝑛)-multimodule with left action 𝐿 = 𝜇[(1𝑚1, . . . , 1𝑚𝑘 ), 𝐷] : (𝐻𝑚1, . . . , 𝐻𝑚𝑘 ) ⊗ F (𝐷) → F (𝐷) 𝜌𝐷 and right action 𝑅 = 𝜇[𝐷, 1𝑛] : F (𝐷) ⊗ 𝐻𝑛 → F (𝐷). 𝜌𝐷 65 Proof. Just as the previous proposition, this follows by applying Lemmas 4.4.2 and 4.4.1, now knowing that 𝐻𝑛 is a G -graded algebra for each 𝑛. □ Recall that if (cid:174)𝐷 = (𝐷1, . . . , 𝐷 𝑘 ) is a collection of planar arc diagrams of type (ℓ𝑖1, . . . , ℓ𝑖𝛼𝑖 ; 𝑚𝑖) for each 𝑖 = 1, . . . , 𝑘, then each of F (𝐷𝑖) in F ( (cid:174)𝐷) = (cid:0)F (𝐷1), . . . , F (𝐷 𝑘 )(cid:1) is a G -graded Ä𝐻ℓ𝑖1, . . . , 𝐻ℓ𝑖 𝛼𝑖 ; 𝐻𝑚𝑖 ä-multimodule with left-aciton and right action 𝜇[(1ℓ𝑖1, . . . , 1ℓ𝑖𝛼𝑖 ); 𝐷𝑖] 𝜇[𝐷𝑖; 1𝑚𝑖 ]. Then, using results of §4.3, we can view (cid:0)F (𝐷1), . . . , F (𝐷 𝑘 )(cid:1)⊗F (𝐷) as an Ä𝐻ℓ11, . . . , 𝐻ℓ𝑘 𝛼𝑘 ; 𝐻𝑚ä- multimodule. Similarly, comparing with the general case, we can define the tensor product F ( (cid:174)𝐷) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) F (𝐷) as F ( (cid:174)𝐷) ⊗ F (𝐷) quotiented by (𝜇[𝐷1, 1𝑚1](𝑢1, 𝑥1), . . . , 𝜇[𝐷 𝑘 , 1𝑚𝑘 ](𝑢𝑘 , 𝑥𝑘 )) ⊗ 𝑢 (cid:12)(cid:174)𝑥(cid:12) (cid:12) ,(cid:12) (cid:12)(cid:174)𝑢(cid:12) − 𝛼((cid:12) (cid:12) ,|𝑢|)(𝑢1, . . . , 𝑢𝑘 ) ⊗ 𝜇[(1𝑚1, . . . , 1𝑚𝑘 ); 𝐷]((cid:174)𝑥, 𝑢) (4.4.1) for (cid:174)𝑢 ∈ F ( (cid:174)𝐷′), (cid:174)𝑥 ∈ (𝐻𝑛1, . . . , 𝐻𝑛𝑘 ), and 𝑢 ∈ F (𝐷). Mimicking [Kho02], we note each of the following. See also Section 6.1 of [NP20]. The proofs of these statements are essentially identical to those found in Sections 2.6 and 2.7 of Khovanov’s paper, and would take us too far afield to prove here—we leave them to the reader. Proposition 4.4.5. F (𝐷) is sweet: it is projective as a left (𝐻𝑚1, . . . , 𝐻𝑚𝑘 )-module and as a right 𝐻𝑛-module. Proposition 4.4.6. If 𝐷𝑖 is a planar arc diagram of type (ℓ𝑖1, . . . , ℓ𝑖𝛼𝑖 ; 𝑚𝑖) for each 𝑖 = 1, . . . , 𝑘 and 𝐷 is a planar arc diagram of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛), then there is an isomorphism of G -graded (𝐻ℓ11, . . . , 𝐻ℓ𝑘 𝛼𝑘 , 𝐻𝑛)-multimodules é F (𝐷𝑖) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) F (𝐷) (cid:27) F (𝐷(𝐷1, . . . , 𝐷 𝑘 ; ∅)) Ñ 𝑘 (cid:204) 𝑖=1 66 induced by 𝜇[(𝐷1, . . . , 𝐷 𝑘 ), 𝐷]. (The first collection of tensor products in the formula above are taken over 𝑅.) We note that the sweetness proposition is important for the proof of the latter; again, see Sections 2.6 and 2.7 of [Kho02]. Note also that 𝜇[ (cid:174)𝐷′, 𝐷] : F ( (cid:174)𝐷′) ⊗ F (𝐷) → F (𝐷( (cid:174)𝐷′)) induces a maps F ( (cid:174)𝐷′) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) F (𝐷) → F (𝐷( (cid:174)𝐷′)) by the universal property of the coequalizer. To see this, use Lemma 4.4.2: for (cid:174)𝑢′ ∈ F ( (cid:174)𝐷′), (cid:174)𝑥 ∈ (𝐻𝑚1, . . . , 𝐻𝑚𝑘 ), and 𝑢 ∈ F (𝐷), we have that 𝜇[ (cid:174)𝐷′, 𝐷](cid:0)𝜇[ (cid:174)𝐷′, (1𝑚1, . . . , 1𝑚𝑘 )]((cid:174)𝑢′, (cid:174)𝑥), 𝑢(cid:1) = 𝛼((cid:12) (cid:12)(cid:174)𝑥(cid:12) (cid:12) ,(cid:12) (cid:12)(cid:174)𝑢′(cid:12) (cid:12) ,|𝑢|)𝜇[ (cid:174)𝐷′, 𝐷](cid:0)(cid:174)𝑢′, 𝜇[(1𝑚1, . . . , 1𝑚𝑘 ), 𝐷](𝑥, 𝑢)(cid:1). Then, compare with equation (4.4.1). Sometimes, we will write “⊗𝐻” as shorthand when its meaning is clear given context. For example, in the lemma below, the “⊗𝐻” on the left means “⊗(𝐻ℓ11 ,...,𝐻ℓ𝑘 𝛼𝑘 ) right means “⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ).” We will also sometimes write “𝜇[ (cid:174)𝐷′, 𝐷]” to mean “the isomorphism ” and the “⊗𝐻” on the of G -graded bimodules induced by 𝜇[ (cid:174)𝐷′, 𝐷].” Lemma 4.4.7. The following diagram commutes for all (cid:174)𝐷′′, (cid:174)𝐷′, and 𝐷. Ä F ( (cid:174)𝐷′′) ⊗𝐻 F ( (cid:174)𝐷′)ä ⊗𝐻 F (𝐷) 𝜇[ (cid:174)𝐷′′, (cid:174)𝐷′]⊗1 F ( (cid:174)𝐷′( (cid:174)𝐷′′)) ⊗𝐻 F (𝐷) 𝛼 F ( (cid:174)𝐷′′) ⊗𝐻 Ä F ( (cid:174)𝐷′) ⊗𝐻 F (𝐷)ä 1⊗𝜇[ (cid:174)𝐷′,𝐷] F ( (cid:174)𝐷′′) ⊗𝐻 F (𝐷( (cid:174)𝐷′)) 𝜇[ (cid:174)𝐷′( (cid:174)𝐷′′),𝐷] F (𝐷( (cid:174)𝐷′( (cid:174)𝐷′′))) 𝜇[ (cid:174)𝐷′′,𝐷( (cid:174)𝐷′)] Proof. This is immediate from the definition of 𝛼 and 𝜇, following Lemma 4.4.2 in the language of Proposition 4.4.6. □ 67 CHAPTER 5 C -SHIFTING SYSTEMS AND COBORDISMS Usually, grading shifts for graded algebraic objects are defined by way of the additive structure of Z. This raises the question of how one should define grading shifts in a C -graded setting. In the G -graded case, we will see that the naive guess (i.e., a chronological cobordism in the first entry plus a Z × Z-shift in the second) is adequate. The general definition of a C -shifting system is rather dense, so we introduce the more concrete G -shifting system alongside the general definition, hoping it gives a helpful model for the reader. These definitions are provided in §5.1, wherein we also describe the compatibility conditions required of a shifting system associated to a particular grading category. In §5.2, we address generalities of shifting systems before investigating the theory of homogeneous maps for C -graded multimodules in §5.3 (indeed, what does it mean for a map 𝑓 : 𝑀 → 𝑁 of C -graded multimodules to be homogeneous?). This includes the extension of our shifting system to a so-called “shifting 2-system” so that, in our context, we can interpret a composition of grading shifts as related to the grading shift associated to a composition of chronological cobordisms. Finally, G -shifting systems are peculiar in the sense that changes of chronology induce natural transformations of grading shifts, which we detail in §5.4. 5.1 A system of grading shifting functors for G Suppose Δ : 𝐷 → 𝐷′ is a chronological cobordism of planar arc diagrams 𝐷, 𝐷′ ∈ D(𝑚1,...,𝑚𝑘;𝑛). If 𝑥𝑖 ∈ 𝐵𝑚𝑖 for all 𝑖 = 1, . . . , 𝑘 and 𝑦 ∈ 𝐵𝑛, then Δ induces a map from some subset of HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦) to HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦). Explicitly, given 𝑣 ∈ Z × Z, the pair (Δ, 𝑣) induces a map defined by 𝜑(Δ,𝑣) : {(𝐷, 𝑝) ∈ HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦)} → HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦) 𝜑(Δ,𝑣)(𝐷, 𝑝) = (𝐷′, 𝑝 + 𝑣 +(cid:12) (cid:12)Δ(1𝑥1, . . . , 1𝑥𝑘 ; 1𝑦)(cid:12) (cid:12)) where Δ(1𝑥1, . . . , 1𝑥𝑘 ; 1𝑦) is the cobordism Δ corked by thickenings of the relevant crossingless matchings. We will see that any cobordism of planar arc diagrams (potentially paired with a Z × Z- 68 degree, in which case we call the cobordism weighted) constitutes what we will call a G -grading shift. In general, a collection ((Δ1, 𝑣1), . . . , (Δ𝑘 , 𝑣 𝑘 )) of chronological cobordisms of planar arc dia- grams induces a grading shift on ((𝐷1, 𝑝1), . . . , (𝐷 𝑘 , 𝑝𝑘 )). Viewing the former as a disjoint union of chronological cobordisms, there is ambiguity as to what chronology to pick. Hereafter, we fix a chronology which applies Δ1 on its component, then Δ2, . . . , then Δ𝑘 , followed by the iden- tity cobordism weighed by 𝑣1 on its component, then 𝑣2, . . . , and finally 𝑣 𝑘 . A picture is more descriptive of the situation: 𝑣 𝑘 Δ𝑘 𝑣2 Δ2 · · · 𝑣1 Δ1 This is the chronology we mean when we write ((cid:174)Δ, (cid:174)𝑣). We choose this particular chronology so that our arguments appear similar to those found in [NP20]. Later on, we’ll denote Δ(1𝑥1, . . . , 1𝑥𝑘 ; 1𝑦) by 1(cid:174)𝑥Δ1𝑦. Again, this is especially helpful when dealing with a collection of cobordisms (Δ1, . . . , Δ𝑛). The degree (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥(Δ1, . . . , Δ𝑛)1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) Now, for each 𝑖 = 1, . . . , 𝑘, suppose Δ𝑖 : 𝐷𝑖 → 𝐷′ is defined as the sum (cid:205)𝑛 (cid:12) (cid:12)1(cid:174)𝑥𝑖 Δ𝑖1𝑦𝑖 (cid:12) (cid:12). 𝑖=1 𝑖 is a chronological chobordism for 𝐷𝑖, 𝐷′ 𝑖 ∈ D(ℓ𝑖1,...ℓ𝑖 𝛼𝑖 ;𝑚𝑖). We denote by (Δ1, . . . , Δ𝑘 ) • Δ the chronological cobordism (Δ1, . . . , Δ𝑘 ) • Δ : 𝐷(𝐷1, . . . , 𝐷 𝑘 ) → 𝐷′(𝐷′ 1, . . . , 𝐷′ 𝑘 ) with chronology, as usual, dependant on indexing (first Δ, then Δ1, and so on). If each of these cobordisms has a Z × Z weight, we’ll set ((Δ1, 𝑣1), . . . , (Δ𝑘 , 𝑣 𝑘 )) • (Δ, 𝑣) = (Δ1, . . . , Δ𝑘 ) • Δ, 𝑣 + Ñ é ∑︁ 𝑣𝑖 𝑖 for cases like the one above. Otherwise, ((Δ1, 𝑣1), . . . , (Δ𝑘 , 𝑣 𝑘 )) • (Δ, 𝑣) = 0. This multiplication defines what we will call a multimonoid, whose elements are cobordisms of planar arc diagrams together with a neutral element 𝑒 and absorbing element 0, with composition defined as above. 69 The collection of maps induced by cobordisms of planar arc diagrams {𝜑(Δ,𝑣)} constitutes a generalization of a shifting system, in the sense of [NP20]. Explicitly, suppose C is a grading multicategory; a C -grading shift 𝜑 is a collection of maps 𝜑 = {𝜑 (cid:174)𝑋→𝑌 : D (cid:174)𝑋→𝑌 ⊂ HomC ( (cid:174)𝑋; 𝑌 ) → HomC ( (cid:174)𝑋; 𝑌 )} (cid:174)𝑋,𝑌 ∈Ob(C ) where (cid:174)𝑋 = (𝑋1, . . . , 𝑋𝑘 ) for 𝑋1, . . . , 𝑋𝑘 ∈ Ob(C ). We write 𝜑(𝑔) to mean 𝜑 (cid:174)𝑋→𝑌 (𝑔) whenever 𝑔 ∈ D (cid:174)𝑋→𝑌 . We write D to stand for “domain”, and use the sans serif font to differentiate these from our notation for planar arc diagrams. In addition, let Σmin denote the category obtained from C by purging all multimorphisms besides the commuting endomorphisms: that is, • Ob(Σmin) = Ob(C ) and • HomΣmin(𝑋1, . . . , 𝑋𝑘 ; 𝑌 ) =    ∅ 𝑘 > 1 or 𝑋1 ≠ 𝑌 Z(EndC (𝑌 ; 𝑌 )) otherwise Where Z stands for the center. Finally, by a multimonoid I , we mean a set equipped with an associative multiplication law • : I 𝑘 × I → I for each 𝑘 ≥ 1, and a neutral element 𝑒 so that 𝑒𝑘 • 𝑖 = 𝑖 for each 𝑘 and 𝑖 • 𝑒 = 𝑖 for all 𝑖 ∈ I . A multimonoid may also have an absorbing element 0, so that ( 𝑗1, . . . , 𝑗𝑘 ) •𝑖 = 0 if any of 𝑗1, . . . , 𝑗𝑘 , 𝑖 are 0. Definition 5.1.1. Suppose Σ is a wide subcategory of C with at least all the morphisms of Σmin. A C -shifting system 𝑆 = (I , Φ) relative Σ for a grading multicategory C is a multimonoid I and a collection of C -grading shifts Φ = {𝜑𝑖}𝑖∈I such that • 𝜑𝑒, called the neutral shift, has (cid:174)𝑋→𝑌 𝑒 D = HomΣ( (cid:174)𝑋; 𝑌 ) and 𝜑 (cid:174)𝑋→𝑌 𝑒 = Id D (cid:174)𝑋→𝑌 𝑒 ; 70 • given 𝜑 (𝑥11,...,𝑥1𝛼1 𝑗1 ;𝑦1) , . . . , 𝜑(𝑥𝑛1,...,𝑥𝑛𝛼𝑛 ;𝑦𝑛) 𝑗𝑛 and 𝜑(𝑦1,...,𝑦𝑛;𝑧) 𝑖 , we have that (𝑦1,...,𝑦𝑛;𝑧) D 𝑖 ◦ 𝑛 (cid:214) 𝑘=1 (𝑥𝑘1,...,𝑥𝑘 𝛼𝑘 𝑗𝑘 D ;𝑦𝑘) ⊂ D (𝑥11,...,𝑥𝑛𝛼𝑛 ;𝑧) ( 𝑗1,..., 𝑗𝑛)•𝑖 and the diagram (𝑦1,...,𝑦𝑛;𝑧) D 𝑖 × (cid:206)𝑛 𝑘=1 D (𝑥𝑘1,...,𝑥𝑘 𝛼𝑘 𝑗𝑘 ;𝑦𝑘) ◦ (𝑥11,...,𝑥𝑛𝛼𝑛 ;𝑧) ( 𝑗1,..., 𝑗𝑛)•𝑖 D (𝜑𝑖, (cid:206)𝑘 𝜑 𝑗𝑘 ) 𝜑( 𝑗1,..., 𝑗𝑛)•𝑖 Hom(𝑦1, . . . , 𝑦𝑛; 𝑧) × (cid:206)𝑛 𝑘=1 Hom(𝑥𝑘1, . . . , 𝑥𝑘𝛼𝑘 ; 𝑦𝑘 ) ◦ Hom(𝑥11, . . . , 𝑥𝑛𝛼𝑛; 𝑧) commutes; • there is a subset Iid ⊂ I such that for all 𝑘 and all 𝑋1, . . . , 𝑋𝑘 , 𝑌 ∈ Ob(C ) there is a partition HomC (𝑋1, . . . , 𝑋𝑘 ; 𝑌 ) = (𝑋1,...,𝑋𝑘)→𝑌 D 𝑖 (cid:196) 𝑖∈Iid for which 𝜑𝑖 = Id D (cid:174)𝑋→𝑌 𝑖 for all 𝑖 ∈ Iid; • if I contains an absorbing element 0, then 𝜑0, called the null shift, always has D (cid:174)𝑋→𝑌 0 = ∅. (cid:174)𝑋→ (cid:174)𝑌 Remark 5.1.2. We will frequently write D (cid:174)𝑖 (cid:174)𝑋ℓ →𝑌ℓ , or just D(cid:174)𝑖, to denote (cid:206)ℓ D 𝑖ℓ . Then, writing 𝑔 ∈ D(cid:174)𝑖 means 𝑔 is an ordered tuple of morphisms as one expects. Similarly, 𝜑(cid:174)𝑖(𝑔) is understood component-wise. Also, we note that 𝜑𝑒 is assumed only to preserve Σ. We refer the reader to Remark 4.10 of [NP20] for a more detailed discussion. For example, take I to be the multimonoid {(Δ, 𝑣)}Δ,𝑣 ⊔ {𝑒, 0} with multiplication • defined above. Taking C = G , notice that Σmin is the subcategory whose objects are crossingless matchings and whose morphisms are identity (𝑛; 𝑛)-planar arc diagrams (1𝑛, 𝑝) : 𝑎 → 𝑎 for 𝑎 ∈ 𝐵𝑛, viewed only as endomorphisms. We will take Σ to be the slightly larger category which allows for morphisms (1𝑛, 𝑝) : 𝑎 → 𝑏 for potentially distinct 𝑎, 𝑏 ∈ 𝐵𝑛. Using the notation of the above definition, to a chronological cobordism of planar arc diagrams Δ : 𝐷 → 𝐷′, 𝐷, 𝐷′ ∈ D(𝑚1,...,𝑚𝑘;𝑛) and 𝑣 ∈ Z × Z, we have a G -grading shift 𝜑(Δ,𝑣) so that for any crossingless matchings 𝑥1, . . . , 𝑥𝑘 , 𝑦 (cid:12)𝑦(cid:12) with |𝑥𝑖 | = 𝑚𝑖 and(cid:12) (cid:12) = 𝑛, D (𝑥1,...,𝑥𝑘)→𝑦 (Δ,𝑣) = {(𝐷∧, 𝑝) ∈ HomG (𝑥1, . . . , 𝑥𝑘 ; 𝑦) : 𝑝 ∈ Z × Z}. 71 Proposition 5.1.3. The multimonoid I = {(Δ, 𝑣)}Δ,𝑣 ⊔ {𝑒, 0} together with the induced G -grading shifts {𝜑𝑖}𝑖∈I form a G -shifting system. Proof. We define 𝜑𝑒 and 𝜑0 so that the first and last points are satisfied. The second point is straight- forward. Finally, for the third point, we take Iid = {(1𝐷∧, (0, 0)) : 𝐷 is a planar arc diagram}, where 1𝐷 is the identity cobordism on 𝐷. □ The definition of a C -shifting system made no reference to the associator of the grading category C . We say that a C -shifting system 𝑆 is compatible with the associator 𝛼 of C if there is a family of maps 𝛽(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧 ( (cid:174)𝑗1,..., (cid:174)𝑗𝑛),(cid:174)𝑖 : 𝑛 (cid:214) 𝑘=1 D ((cid:174)𝑥𝑘1,...,(cid:174)𝑥𝑘 𝛼𝑘 (cid:174)𝑗𝑘 ;(cid:174)𝑦𝑘) ((cid:174)𝑦1,...,(cid:174)𝑦𝑛;(cid:174)𝑧) × D (cid:174)𝑖 → K×, for each (cid:174)𝑤, (cid:174)𝑥, (cid:174)𝑦, (cid:174)𝑧 consisting of objects of C and (cid:174)𝑖, (cid:174)𝑗 consisting of objects in I , called compatibility maps granted they satisfy the relations (𝑔′′𝑔′, 𝑔)𝛽(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧 (cid:174)𝑘, (cid:174)𝑗 (𝑔′′, 𝑔′) 𝛼(𝑔′′, 𝑔′, 𝑔)𝛽 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧 (cid:174)𝑘• (cid:174)𝑗,(cid:174)𝑖 = 𝛽 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧 (cid:174)𝑘, (cid:174)𝑗•(cid:174)𝑖 (𝑔′′, 𝑔′𝑔)𝛽 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦 (cid:174)𝑗,(cid:174)𝑖 Å (𝑔′, 𝑔)𝛼 𝜑 (cid:174)𝑦(cid:174)𝑧 (cid:174)𝑘 (𝑔′′), 𝜑(cid:174)𝑥 (cid:174)𝑦 (cid:174)𝑗 (𝑔′), 𝜑 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑖 ã (𝑔) , (5.1.1) for all valid 𝑔′′, 𝑔, 𝑔 and (cid:174)𝑖, (cid:174)𝑗, (cid:174)𝑘, and 𝛽𝑒,𝑒 = 𝛽(𝑒,...,𝑒),(𝑒,...,𝑒) = 1. Diagrammatically, this is to say that the following picture commutes (here, the boxed number 𝑛 refers to the C -grading shift, and we suppress burdensome indices). 3 𝑔′′ 2 𝑔′ 𝛼(𝜑3(𝑔′′), 𝜑2(𝑔′), 𝜑1(𝑔)) 3 𝑔′′ 2 𝑔′ 1 𝑔 1 𝑔 𝛽(𝑔′′, 𝑔′) 3 • 2 𝑔′′ 𝑔′ 1 𝑔 3 • 2 • 1 𝛽(𝑔′′𝑔′, 𝑔) 𝑔′′ 𝑔′ 𝑔 𝛼(𝑔′′, 𝑔′, 𝑔) (5.1.2) 3 • 2 • 1 𝑔′′ 𝑔′ 𝑔 3 𝑔′′ 𝛽(𝑔′, 𝑔) 2 • 1 𝛽(𝑔′′, 𝑔′𝑔) 𝑔′ 𝑔 72 For (G , 𝛼), we will define the compatibility maps 𝛽 in a way analogous to the presentation in [NP20]. Suppose and 𝑔′ = ( (cid:174)𝐷′, (cid:174)𝑝′) = (cid:214) ( (cid:174)𝐷′ 𝑖, (cid:174)𝑝′ 𝑖) = (cid:214) (𝐷𝑖 𝑗 , 𝑝𝑖 𝑗 ) 𝑖 𝑖, 𝑗 𝑔 = ( (cid:174)𝐷, (cid:174)𝑝) = (cid:214) (𝐷𝑖, 𝑝𝑖) 𝑖 constitute a multipath of length two; 𝑔 ◦ 𝑔′ ∈ G [2]. Again, denote by 𝑃 ∈ Z × Z the sum of the entries of (cid:174)𝑝 and 𝑃′ the sum of the entries of (cid:174)𝑝′. Write (cid:174)𝐷 = (𝐷1, . . . , 𝐷 𝑘 ), and suppose that ((cid:174)Δ, (cid:174)𝑣) = ((Δ1, 𝑣1), . . . , (Δ𝑘 , 𝑣 𝑘 )) is a collection of cobordisms for 𝑔. We’ll write ((cid:174)Δ, (cid:174)𝑣)( (cid:174)𝐷, (cid:174)𝑝) = ((cid:174)Δ( (cid:174)𝐷), (cid:174)𝑣 + (cid:174)𝑝) where (cid:174)Δ( (cid:174)𝐷) = (Δ1(𝐷1), . . . , Δ𝑘 (𝐷 𝑘 )) and Δ𝑖(𝐷𝑖) denotes the boundary of Δ𝑖 other than 𝐷𝑖. Finally, 𝑉 and 𝑉 ′ ∈ Z × Z will denote the sums of the entries of (cid:174)𝑣 and (cid:174)𝑣′ respectively. The value 𝛽 will be defined as the product of four values. First, consider the change of chronology (cid:174)𝑥 (cid:174)Δ( (cid:174)𝐷) Ä(cid:174)Δ′( (cid:174)𝐷′)ä (cid:174)𝑧 Ä(cid:174)Δ′•(cid:174)Δ ä1(cid:174)𝑧 1 (cid:174)𝑥 𝑊 (cid:174)𝑥 (cid:174)𝑦 (cid:174)𝑧( (cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:174)𝑥 (cid:174)𝐷( (cid:174)𝐷′)(cid:174)𝑧 𝐻𝛽 ===⇒ (cid:174)𝑥 (cid:174)Δ′( (cid:174)𝐷′)(cid:174)𝑦 ⊗ (cid:174)𝑦 (cid:174)Δ( (cid:174)𝐷)(cid:174)𝑧 𝑊 (cid:174)𝑥 (cid:174)𝑦 (cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) 1 (cid:174)𝑥 (cid:174)Δ′1 (cid:174)𝑦⊔1 (cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:174)𝑥 (cid:174)𝐷′(cid:174)𝑦 ⊗ (cid:174)𝑦 (cid:174)𝐷 (cid:174)𝑧 Set 𝛽1 = 𝜄(𝐻𝛽). Then, set , 𝑉 ′ + 𝑉 (cid:17) , 𝛽2 = 𝜆 𝛽3 = 𝜆 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) (cid:17) (cid:12) (cid:12) , and , 𝑉 (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:17) . + 𝑉 (cid:12) (cid:12) (cid:12) (cid:16) 𝛽4 = 𝜆 𝑃′, 73 We define Naisse and Putyra describe this shift diagrammatically as follows. 𝛽 = 𝛽4𝛽3𝛽2𝛽1. 𝑉 ′ Δ′ 𝑔′ 𝑉 Δ 𝑔 𝑉 ′ Δ′ = 𝛽4 𝑉 = 𝛽4𝛽3 Δ 𝑔′ 𝑔 𝑉 ′ Δ′ 𝑔′ 𝑉 Δ 𝑔 = 𝛽4𝛽3𝛽2 𝑉 ′ + 𝑉 Δ′ 𝑔′ Δ 𝑔 = 𝛽4𝛽3𝛽2𝛽1 𝑉 ′ + 𝑉 Δ′ • Δ 𝑔′ 𝑔 Lemma 5.1.4. If 𝐷 ∈ D(𝑚1,...,𝑚𝑘;𝑛) and (cid:174)𝐷 = (𝐷1, . . . , 𝐷 𝑘 ) for 𝐷𝑖 ∈ D(ℓ𝑖1,...,ℓ𝑖 𝛼𝑖 ;𝑚𝑖). Moreover, let Δ be a cobordism on 𝐷 and (cid:174)Δ′ be a collection of cobordisms for (cid:174)𝐷′. Then, for any (cid:174)𝑥 ∈ (cid:206)𝑖, 𝑗 𝐵ℓ𝑖 𝑗 , (cid:174)𝑦 ∈ (cid:206)𝑖 𝐵𝑚𝑖 and 𝑧 ∈ 𝐵𝑛, we have that (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦Δ1𝑧 (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((cid:174)Δ′( (cid:174)𝐷′), Δ(𝐷)) (cid:12) (cid:12) (cid:12) = (cid:12) (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧( (cid:174)𝐷′, 𝐷) (cid:12) (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥((cid:174)Δ′ • Δ)1𝑧 (cid:12) (cid:12) (cid:12) Notice that this lemma immediately applies to the cases when 𝐷 is actually a collection (cid:174)𝐷 = (cid:206)𝑖 𝐷𝑖 and (cid:174)𝐷′ = (cid:206)𝑖, 𝑗 𝐷𝑖 𝑗 . Proof. Exactly as in [NP20], there is a diffeomorphism between the cobordisms below, so they must have the same degree. (cid:174)Δ′ (cid:174)𝐷′ (cid:174)Δ (cid:174)𝐷 (cid:174)Δ′ • (cid:174)Δ (cid:27) (cid:174)𝐷′ (cid:174)𝐷 Proposition 5.1.5. The G -shifting system of Proposition 5.1.3 is compatible with the associator of Proposition 4.2.3 through the compatibility map 𝛽 defined above. Proof. The following follows closely the proof of [NP20]. We will show that 𝛽 satisfies equation (5.1.1). The first step is to document the contributions of 𝛼1 and 𝛽1. To do this, consider the □ 74 following diagram of cobordisms. 3 • 2 𝛽1 1 𝜂 3 3 𝛼1 2 2 1 1 𝜁 2 3 3 • 2 1 𝛽1 3 • 2 • 1 𝛼1 3 • 2 • 1 3 2 • 1 𝛽1 𝛽1 1 where 𝜁 = 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:17) (cid:12) (cid:12) and 𝜂 = 𝜆 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) . Comparing with diagram (5.1.2), we see that 𝜁 × (Left of (5.1.1)) = 𝜂 × (Right of (5.1.1)). Next, we have to compare the contributions of 𝛼2, 𝛽2, 𝛽3, and 𝛽4. First, for the left side of (5.1.1) (or, the top-and-then-down path in (5.1.2)), we have the following. 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦((cid:174)Δ′′( (cid:174)𝐷′′), (cid:174)Δ′( (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧(((cid:174)Δ′′ • (cid:174)Δ′)( (cid:174)𝐷′( (cid:174)𝐷′′)), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′(cid:17) · 𝜆 · 𝜆 (cid:16) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑦 (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:12) (cid:12) (cid:12) (cid:17) , 𝑉 ′(cid:17) (cid:16) · 𝜆 𝑃′′, (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 1 (cid:174)𝑤((cid:174)Δ′′ • (cid:174)Δ′)1(cid:174)𝑦 , 𝑉 (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (∗) · 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) + 𝑉 ′(cid:17) (cid:17) (cid:16) 𝑃′ + 𝑃′′ + · 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:123)(cid:122) (∗∗) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 + 𝑉 (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:17) (cid:12) (cid:12) (cid:12) , · 𝜆 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) 𝛽2,3,4(𝑔′′, 𝑔′) 𝛽2,3,4(𝑔′′𝑔′, 𝑔) 𝛼2(𝑔′′, 𝑔′, 𝑔) Turn your attention to the term marked (∗). By application of Lemma 5.1.4, we can write 𝜆 (cid:16) (cid:12) (cid:12) (cid:12) 1 (cid:174)𝑤((cid:174)Δ′′ • (cid:174)Δ′)1(cid:174)𝑦 (cid:17) , 𝑉 (cid:12) (cid:12) (cid:12) = 𝜆 (cid:12) (cid:12) (cid:12) + 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:16) (cid:12) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 (cid:12) (cid:12) (cid:12) (cid:17)−1 , 𝑉 . · 𝜆 (cid:17) · 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦((cid:174)Δ′′( (cid:174)𝐷′′), (cid:174)Δ′( (cid:174)𝐷′)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) , 𝑉 75 On the other hand, using linearity, we can rewrite the term marked (∗∗) as (cid:16) 𝜆 𝑃′ + 𝑃′′ + (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) + 𝑉 = 𝜆 (cid:16) 𝑃′ + 𝑃′′ + , (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) (cid:17) , 𝑉 · 𝜆 (cid:0)𝑃′ + 𝑃′′, 𝑉(cid:1) · 𝜆 The last terms in the past two expansions cancel, and we can rewrite the contributions of 𝛼2, 𝛽2, 𝛽3, and 𝛽4 on the left side of (5.1.1) as 𝜆 (cid:17) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦((cid:174)Δ′′( (cid:174)𝐷′′), (cid:174)Δ′( (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧(((cid:174)Δ′′ • (cid:174)Δ′)( (cid:174)𝐷′( (cid:174)𝐷′′)), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 (cid:0)𝑃′ + 𝑃′′, 𝑉(cid:1) · 𝜆 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 𝑃′ + 𝑃′′, (cid:16) · 𝜆 (cid:12) (cid:12) (cid:12) , 𝑉 ′(cid:17) · 𝜆 (cid:16) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑦 (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 · 𝜆 (cid:16)(cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦( (cid:174)𝐷′′, (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:123)(cid:122) 𝜁 (cid:12) (cid:17) (cid:12) (cid:12) + (cid:16) · 𝜆 𝑃′′, 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) , (cid:12) (cid:12) (cid:12) + 𝑉 ′(cid:17) (cid:17) , 𝑉 · 𝜆 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) . The process above could be described as “simplifying” 𝛽2,3,4(𝑔′′𝑔′, 𝑔). Likewise, on the right side of (5.1.1) (or, the down-and-then-bottom path in (5.1.2)), we have the following. (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) (cid:17) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) , 𝑃′′ + 𝑉 ′′ + 𝜆 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (★) 𝛼2(𝜑(𝑔′′), 𝜑(𝑔′), 𝜑(𝑔)) (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) , 𝑉 ′ + 𝑉 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:17) (cid:123)(cid:122) (★★) ·𝜆 (cid:16) (cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) . , 𝑉 · 𝜆 . (cid:17) (cid:16) · 𝜆 𝑃′, + 𝑉 (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧((cid:174)Δ′′( (cid:174)𝐷′′), ((cid:174)Δ′ • (cid:174)Δ)( (cid:174)𝐷( (cid:174)𝐷′))) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥((cid:174)Δ′ • (cid:174)Δ)1(cid:174)𝑧 · 𝜆 (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) , 𝑉 + 𝑉 ′(cid:17) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:16) (cid:12) (cid:12) (cid:12) 𝑃′′, · 𝜆 (cid:16) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:123)(cid:122) (★★★) (cid:17) + 𝑉 + 𝑉 ′(cid:17) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) 𝛽2,3,4(𝑔′, 𝑔) 𝛽2,3,4(𝑔′′, 𝑔′𝑔) Notice that the term (★) has three “parts.” The 𝑉 ′′ part can be absorbed into the term (★★); the rest can be written 𝜆 (cid:16) (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) · 𝜆 (cid:16) (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , (cid:12) (cid:12) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:17) (cid:12) (cid:12) (cid:12) . 76 The (★ ★ ★) term decomposes into parts (cid:16) 𝜆 𝑃′′, (cid:12) 1(cid:174)𝑥((cid:174)Δ′ • (cid:174)Δ)1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) · 𝜆 (cid:0)𝑃′′, 𝑉 + 𝑉 ′(cid:1) . Again, applying Lemma 5.1.4, we can write (cid:16) 𝜆 𝑃′′, (cid:12) 1(cid:174)𝑥((cid:174)Δ′ • (cid:174)Δ)1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) (cid:16) 𝑃′′, = 𝜆 (cid:12) (cid:12) (cid:12) + (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) . (cid:16) · 𝜆 𝑃′′, − (cid:12) (cid:17) (cid:12) (cid:12) (cid:16) 𝑃′′, · 𝜆 (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) The middle term after the equality cancels with first term in the rewriting of (★). The last term can be rewritten as 𝜆 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑃′′(cid:17) . All together, this means that we can rewrite the contributions of 𝛼2, 𝛽2, 𝛽3, and 𝛽4 on the right side of (5.1.1) as (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:12) (cid:12) (cid:17) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) , (cid:123)(cid:122) 𝜂 · 𝜆 · 𝜆 , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧((cid:174)Δ′′( (cid:174)𝐷′′), ((cid:174)Δ′ • (cid:174)Δ)( (cid:174)𝐷( (cid:174)𝐷′))) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 (cid:0)𝑃′′, 𝑉 + 𝑉 ′(cid:1) · 𝜆 (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) 𝑃′′, (cid:16) + (cid:17) · 𝜆 (cid:16) (cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:12) (cid:17) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 (cid:12) (cid:17) (cid:12) (cid:12) · 𝜆 (cid:17) , 𝑉 (cid:16) 𝑃′, · 𝜆 (cid:12) (cid:12) (cid:12) · 𝜆 (cid:16) (cid:12) 1 (cid:174)𝑤 (cid:174)Δ′′1(cid:174)𝑥 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧( (cid:174)𝐷′, (cid:174)𝐷) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) . + 𝑉 (cid:12) (cid:12) 1(cid:174)𝑦 (cid:174)Δ1(cid:174)𝑧 (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 + 𝑉 ′(cid:17) , 𝑃′′(cid:17) Compare the simplifications of contributions from each side. One one hand, 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥 (cid:174)𝑦((cid:174)Δ′′( (cid:174)𝐷′′), (cid:174)Δ′( (cid:174)𝐷′) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:17) · 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑦(cid:174)𝑧(((cid:174)Δ′′ • (cid:174)Δ′)( (cid:174)𝐷′( (cid:174)𝐷′′)), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:17) is equal to 𝜆 (cid:16) (cid:12) (cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦(cid:174)𝑧((cid:174)Δ′( (cid:174)𝐷′), (cid:174)Δ( (cid:174)𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:17) · 𝜆 (cid:12) (cid:16) (cid:12) 𝑊 (cid:174)𝑤 (cid:174)𝑥(cid:174)𝑧((cid:174)Δ′′( (cid:174)𝐷′′), ((cid:174)Δ′ • (cid:174)Δ)( (cid:174)𝐷( (cid:174)𝐷′))) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉 ′′ + 𝑉 ′ + 𝑉 (cid:17) by Lemma 4.2.1. On the other hand, careful observation reveals that, via bilinearity of 𝜆 alone, the two collections of terms apart from these, and the terms marked 𝜁 and 𝜂, are equivalent. The conclusion is that 𝜂 × (Left of (5.1.1)) = 𝜁 × (Right of (5.1.1)). This completes the proof. □ 77 As we proceed, for simplicity of exposition (and because it is the only situation which matters in our application) we will only consider multipaths which end in a single multimorphism; we have shown in the previous arguments how the situation is generalized without problem. 5.2 Generalities on shifting systems for grading multicategories We conclude this discussion by detailing the generalities of C -shifting systems. These are results of [NP20] which lift to the setting of grading multicategories. Throughout, let C be a grading multicategory with associator 𝛼, and 𝑆 = {I , {𝜑𝑖}𝑖∈I } a C -shifting system compatible with 𝛼 through compatibility maps 𝛽. Just as in the non-multi setting, we define for each 𝑖 ∈ I a grading shift functor 𝜑𝑖 : ModC → ModC by putting 𝜑𝑖(𝑀) = (cid:202) 𝜑𝑖(𝑀)𝜑𝑖(𝑔) 𝑔∈D𝑖 for 𝜑𝑖(𝑀)𝜑𝑖(𝑔) := 𝑀𝑔; that is, 𝜑𝑖 sends elements in degree 𝑔 ∈ D𝑖 to elements in degree 𝜑𝑖(𝑔), and elements whose degree does not belong to D𝑖 to zero. Sometimes we call 𝜑𝑖 a C -grading shift or just a grading shift. Now, if 𝑀, 𝑀1, . . . , 𝑀𝑘 are C -graded modules, there is a canonical isomorphism 𝛽( 𝑗1,..., 𝑗𝑘),𝑖 : (cid:0)𝜑 𝑗1(𝑀1), . . . , 𝜑 𝑗𝑘 (𝑀𝑘 )(cid:1) ⊗ 𝜑𝑖(𝑀) ∼ (cid:12) (cid:174)𝑚(cid:12) The compatibility requirement, equation (5.1.1), ensures that this isomorphism is compatible with by (𝑚1, . . . , 𝑚𝑘 ) ⊗ 𝑚 ↦→ 𝛽( 𝑗1,..., 𝑗𝑘),𝑖((cid:12) −→ 𝜑( 𝑗1,..., 𝑗𝑘)•𝑖 (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) (cid:12) ,|𝑚|)(𝑚1, . . . , 𝑚𝑘 ) ⊗ 𝑚. (5.2.1) the coherence isomorphism given by 𝛼. Moreover, since grading shift functors do not have effect on graded maps, the compatibility maps 𝛽 (cid:174)𝑗,𝑖 define natural isomorphisms (denoted by the same symbol) of multifunctors 𝛽 (cid:174)𝑗,𝑖 : (cid:0)𝜑 𝑗1(−), . . . , 𝜑 𝑗𝑘 (−)(cid:1) ⊗ 𝜑𝑖(−) ∼ −→ 𝜑 (cid:174)𝑗,𝑖 (cid:0)(−, . . . , −), −(cid:1) (5.2.2) for all 𝑗1, . . . , 𝑗𝑘 , 𝑖 ∈ I . We define the identity shift functor 𝜑Id as (cid:201) 𝑖∈IId 𝜑𝑖; thus, 𝜑id(𝑀) (cid:27) 𝑀. In general, the identity shift and the neutral shift are not the same (see, for example, [NP20], Remark 4.10). We’ll consider the set (cid:102)I , defined to be I ⊔ {Id}. We do not think of (cid:102)I as a multimonoid—writing it this way 78 means (cid:201) just helps to simplify notation. For example, we will write 𝜑 𝑗•Id to mean (cid:201) 𝜑 𝑗•𝑖. Similarly, (cid:174)𝑗 ∈𝐽 𝜑 (cid:174)𝑗•𝑖 where 𝐽 = {( 𝑗1, . . . , 𝑗𝑘 ) : 𝑗ℓ ∈ IId for all ℓ = 1, . . . , 𝑘 }. To extend the (𝑔′, 𝑔) = 𝛽 (cid:174)𝑗,𝑖(𝑔′, 𝑔) where 𝑔′ ∈ D (cid:174)𝑗 and (cid:174)𝑗 ∈ IId; similarly 𝜑 (cid:174)Id•𝑖 compatibility maps 𝛽 to (cid:102)I , define 𝛽 (cid:174)Id,𝑖 𝛽 𝑗,Id(𝑔′, 𝑔) = 𝛽 𝑗,𝑖(𝑔′, 𝑔) where 𝑔 ∈ D𝑖, 𝑖 ∈ IId. Lastly, we fix 𝛽 (cid:174)Id,Id = 1. 𝑖∈IId 5.2.1 Shifting multimodules To continue in the general setting, we must make the following assumption. Assumption: Hereafter, all C -graded algebras 𝐴 are supported only in Σ; that is, 𝐴𝑔 = 0 whenever 𝑔 ∉ HomΣ Thus, for C -algebras 𝐴 which satisfy this assumption, we have that 𝜑𝑒(𝐴) (cid:27) 𝐴 (really, 𝜑𝑒(𝐴) = 𝐴, since 𝜑𝑒 acts as the identity wherever defined). Recall that, since 𝐻𝑛 = F (1𝑛) = (cid:202) 𝑎,𝑏∈𝐵𝑛 F (𝑎1𝑛𝑏) any 𝑚 ∈ 𝐻𝑛 has degree degG (𝑚) = (1𝑛, deg𝑅(𝑚)) : 𝑎 → 𝑏; that is, arc algebras are G -graded algebras supported only in Σ. If 𝑀 is a C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule, and 𝜑𝑖 is a C -grading shifting functor, then we can view 𝜑𝑖(𝑀) as a C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule by defining left- and right-acitons and 𝜑𝑖 𝜌𝐿 : (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑖(𝑀) → 𝜑𝑖(𝑀) (cid:12) (cid:174)𝑎(cid:12) by 𝜑𝑖 𝜌𝐿( (cid:174)𝑎, 𝜑𝑖(𝑚)) = 𝛽(𝑒,...,𝑒),𝑖((cid:12) (cid:12) ,|𝑚|)𝜑𝑖(𝜌𝐿( (cid:174)𝑎, 𝑚)) 𝜑𝑖 𝜌𝑅 : 𝜑𝑖(𝑀) ⊗ 𝐵 → 𝜑𝑖(𝑀) by 𝜑𝑖 𝜌𝑅(𝜑𝑖(𝑚), 𝑏) = 𝛽𝑖,𝑒(|𝑚| ,|𝑏|)𝜑𝑖(𝜌𝑅(𝑚, 𝑏)). In other words, 𝜑𝑖 𝜌𝐿 and 𝜑𝑖 𝜌𝑅 are defined as the composites (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑖(𝑀) 𝜑𝑖(𝑀) ★ (cid:0)𝜑𝑒(𝐴1), . . . , 𝜑𝑒(𝐴𝑘 )(cid:1) ⊗ 𝜑𝑖(𝑀) 𝛽 (cid:174)𝑒,𝑖 𝜑(𝑒,...,𝑒)•𝑖 (cid:0)(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝑀(cid:1) 𝜌𝐿 𝜑(𝑒,...,𝑒)•𝑖(𝑀) 79 and 𝜑𝑖(𝑀) 𝜑𝑖(𝑀) ⊗ 𝐵 ★ 𝜑𝑖(𝑀) ⊗ 𝜑𝑒(𝐵) 𝛽𝑖,𝑒 𝜑𝑖•𝑒 (𝑀 ⊗ 𝐵) 𝜌𝑅 𝜑𝑖•𝑒(𝑀) where the maps labeled ★ are isomorphisms thanks to the assumption from the start of the section. We’ll breifly describe why 𝜑𝑖(𝑀) is indeed a C -graded multimodule. First, notice that 𝜑𝑖 𝜌𝐿 and 𝜑𝑖 𝜌𝑅 are both graded maps. To illustrate for the left action, if (𝑎1, . . . , 𝑎𝑘 ) ⊗ 𝑚 has grading 𝑔 ◦ (𝑔1, . . . , 𝑔𝑘 ) in (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝑀, it has grading 𝜑𝑖(𝑔) ◦ (𝑔1, . . . , 𝑔𝑘 ) in (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑖(𝑀). Thanks to the assumption from the start of the section, 𝑔𝑖 = 𝜑𝑒(𝑔𝑖) since all algebras in sight are supported only in Σ, and thus 𝜑𝑒 acts as the identity map. Applying the natural isomorphism (5.2.2) provides the desired result. To see that requirements (i)-(iv) of the definition of C -graded multimodules holds, one must simply apply equation (5.1.1) and 𝛽(𝑒,...,𝑒),(𝑒,...,𝑒) = 1 in each of the scenarios. Thus, grading shift functors are also functors for categories of multimodules. In conclusion, we have the following. Proposition 5.2.1. Let 𝑀 ∈ MultiModC (𝐵1, . . . , 𝐵𝑘 ; 𝐶) and 𝑀𝑖 ∈ MultiModC (𝐴𝑖1, . . . , 𝐴𝑖𝛼𝑖 ; 𝐵𝑖) for each 𝑖 = 1, . . . , 𝑘. Then, for each 𝑖, 𝑗1, . . . , 𝑗𝑘 ∈ I , there is an isomorphism of C -graded (𝐴11, . . . , 𝐴𝑘𝛼𝑘 )-multimodules 𝛽( 𝑗1,..., 𝑗𝑘),𝑖 : (cid:0)𝜑 𝑗1(𝑀1), . . . , 𝜑 𝑗𝑘 (𝑀𝑘 )(cid:1) ⊗(𝐵1,...,𝐵𝑘) 𝜑𝑖(𝑀) ∼ −→ 𝜑( 𝑗1,..., 𝑗𝑘)•𝑖 (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗(𝐵1,...,𝐵𝑘) 𝑀(cid:1) induced by the canonical isomorphism (5.2.1). Proof. We direct the reader to [NP20] Proposition 4.18 for a complete proof; the one here is completely analogous. 5.3 Homogeneous maps □ One of the goals of this thesis is to prove an adjunction for unified Khovanov homology, generalizing Theorem 2.31 of [Hog19]. This means we must define HOM-complexes which, in 80 our case, necessitates defining what is meant by maps of homogeneous G -degree. This opens a whole can of worms, which most of the rest of this section is devoted to describing. We proceed with the same assumptions as before: (C , 𝛼) is a grading multicategory, and 𝑆 = (I , {𝜑𝑖}𝑖∈I ) is a shifting system compatible with 𝛼 through maps 𝛽. Moreover, all C -graded algebras are assumed to be supported entirely in Σ so that previous results hold. Definition 5.3.1. Suppose 𝑀 and 𝑁 are C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodules. A K-linear map 𝑓 : 𝑀 → 𝑁 is called purely homogeneous of degree 𝑖 (for 𝑖 ∈ I ⊔ {Id}) if, for all 𝑚 ∈ 𝑀, (i) 𝑓 (𝑚) = 0 if |𝑚| ∉ D𝑖, (ii) (cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) = 𝜑𝑖(|𝑚|) if |𝑚| ∈ D𝑖, (cid:12) (cid:174)𝑎(cid:12) (iii) 𝜌𝐿( (cid:174)𝑎, 𝑓 (𝑚)) = 𝛽(𝑒,...,𝑒),𝑖((cid:12) (cid:12) ,|𝑚|) 𝑓 (𝜌𝐿( (cid:174)𝑎, 𝑚) for all (cid:174)𝑎 ∈ (𝐴1, . . . , 𝐴𝑘 ), and (iv) 𝜌𝑅( 𝑓 (𝑚), 𝑏) = 𝛽𝑖,𝑒(|𝑚| ,|𝑏|) 𝑓 (𝜌𝑅(𝑚, 𝑏)) for all 𝑏 ∈ 𝐵. A map 𝑓 : 𝑀 → 𝑀 is called homogeneous if it is a finite sum of purely homogeneous maps, written 𝑓 = (cid:205) 𝑗 𝑓 𝑗 . We’ll write(cid:12) (cid:12) 𝑓 (cid:12) (cid:12) = 𝑖 if 𝑓 is a purely homogeneous map of degree 𝑖. Importantly, we do not require that a purely homogeneous map preserve C -degree; however, every purely homogeneous map of degree 𝑖, 𝑓 : 𝑀 → 𝑁, induces a graded one, (cid:101)𝑓 : 𝜑𝑖(𝑀) → 𝑁, by setting (cid:101)𝑓 (𝜑𝑖(𝑚)) = 𝑓 (𝑚). Using the shifting system and compatibility maps, we can define the tensor product of ho- mogeneous maps. Let 𝑓𝑖 : 𝑀𝑖 → 𝑁𝑖 for 𝑖 = 1, . . . , 𝑘 and 𝑓 : 𝑀 → 𝑁 be (not necessarily purely) homogeneous maps of (𝐴𝑖1, . . . , 𝐴𝑖𝛼𝑖 ; 𝐵𝑖)-multimodules and (𝐵1, . . . , 𝐵𝑘 ; 𝐶)-multimodules respectively. Then, define ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 : (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀 → (𝑁1, . . . , 𝑁𝑘 ) ⊗ 𝑁 by setting ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 = (cid:205) 𝑗 [( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 ] 𝑗 where [( 𝑓1, . . . , 𝑓𝑘 )⊗ 𝑓 ] 𝑗 ((𝑚1, . . . , 𝑚𝑘 )⊗𝑚) = ∑︁ (𝑖1,...,𝑖𝑘)•𝑖= 𝑗 𝛽(cid:12) (cid:12) (cid:12) (cid:12) (cid:174)𝑓 (cid:12) (cid:12) ,| 𝑓 | 81 (cid:12) (cid:174)𝑚(cid:12) ((cid:12) (cid:12) ,|𝑚|)−1 Ä 𝑓 𝑖1 1 (𝑚1), . . . , 𝑓 𝑖𝑘 𝑘 (𝑚𝑘 )ä ⊗ 𝑓 𝑖(𝑚) for all homogeneous elements (cid:174)𝑚 ∈ (𝑀1, . . . , 𝑀𝑘 ), 𝑚 ∈ 𝑀. First, notice that homogeneous maps behave well with respect to this tensor product (or, horizontal composition). Proposition 5.3.2. If 𝑓1, . . . , 𝑓𝑘 , 𝑓 are purely homogeneous maps of degrees 𝑖1, . . . , 𝑖𝑘 and 𝑖 re- spectively, then ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 is purely homogeneous of degree (𝑖1, . . . , 𝑖𝑘 ) • 𝑖. Proof. For requirement (i), recall that |(𝑚1, . . . , 𝑚𝑘 ) ⊗ 𝑚| = 𝑔 ◦ (𝑔1, . . . , 𝑔𝑘 ). The assumption that (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12) (cid:12) ∉ D(cid:174)𝑖•𝑖 implies that either 𝑔 ∉ D𝑖, hence 𝑓 (𝑚) = 0 since 𝑓 is homogeneous of degree 𝑖, or 𝑔ℓ ∉ D𝑖ℓ for some ℓ, in which case 𝑓 (𝑚ℓ) = 0 for the same reason. Thus (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) ( (cid:174)𝑚 ⊗𝑚) = 0. For (ii), we compute (cid:12) (cid:12) (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) ( (cid:174)𝑚 ⊗ 𝑚)(cid:12) (cid:12) = 𝛽(cid:174)𝑖,𝑖 = 𝛽(cid:174)𝑖,𝑖 = 𝜑(cid:174)𝑖•𝑖 as desired. For (iii), (cid:12) ,|𝑚| Ä(cid:12) (cid:12) (cid:174)𝑚(cid:12) Ä(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚| Ä(cid:12) (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12) ä−1(cid:12) (cid:12) (cid:0) 𝑓1(𝑚1), . . . , 𝑓𝑘 (𝑚𝑘 )(cid:1) ⊗ 𝑓 (𝑚)(cid:12) (cid:12) ä−1 (cid:0)𝜑𝑖1(|𝑚1|), . . . , 𝜑𝑖𝑘 (|𝑚𝑘 |)(cid:1) ◦ 𝜑𝑖(|𝑚|) ä Ä 𝜌𝐿 (cid:174)𝑎, (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) ( (cid:174)𝑚 ⊗ 𝑚)ä Ä(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛼((cid:12) (cid:12) , (cid:12) ,|𝑚| ä−1 = 𝛽(cid:174)𝑖,𝑖 Ä(cid:12) (cid:12) (cid:174)𝑚(cid:12) )−1 Ä𝜌1 𝐿 = 𝛽(cid:174)𝑖,𝑖 ä−1 Ä 𝜌𝐿 (cid:12) ,|𝑚| (cid:174)𝑎, (cid:0)( 𝑓1(𝑚1), . . . , 𝑓𝑘 (𝑚𝑘 )(cid:1) ⊗ 𝑓 (𝑚)ä (cid:0) (cid:174)𝑎𝑘 , 𝑓𝑘 (𝑚𝑘 )(cid:1)ä ⊗ 𝑓 (𝑚) (cid:0) (cid:174)𝑎1, 𝑓1(𝑚1)(cid:1) , . . . , 𝜌𝑘 𝐿 (cid:12) , 𝜑(cid:174)𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛼((cid:12) (cid:12)), 𝜑𝑖(|𝑚|))−1𝛽 (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:174)𝑒,(cid:174)𝑖((cid:12) (cid:12)) (cid:16) 𝑓1 Ä𝜌1 𝐿( (cid:174)𝑎1, 𝑚1)ä , . . . , 𝑓𝑘 Ä𝜌𝑘 𝐿( (cid:174)𝑎𝑘 , 𝑚𝑘 )ä(cid:17) (cid:12) 𝑓 (𝑚)(cid:12) ,(cid:12) (cid:12) (cid:32)(cid:32) (cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) =𝜑𝑖(|𝑚|) (cid:12) (cid:12) (cid:174)𝑓 ( (cid:174)𝑚) (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32) (cid:32)(cid:32) (cid:124) (cid:125) (cid:123)(cid:122) =𝜑(cid:174)𝑖(| (cid:174)𝑚|) (cid:12) (cid:174)𝑚(cid:12) = 𝛽(cid:174)𝑖,𝑖 Ä(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚| ä−1 ⊗ 𝑓 (𝑚) = 𝛽 (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:174)𝑒,(cid:174)𝑖•𝑖((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ◦|𝑚|)𝛼((cid:12) (cid:12) ,|𝑚|)−1𝛽 (cid:12) ◦(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:174)𝑒•(cid:174)𝑖,𝑖((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚|)−1(cid:16) 𝑓1(cid:0)𝜌1 𝐿( (cid:174)𝑎1, 𝑚1)(cid:1), . . . , 𝑓𝑘 (cid:0)𝜌𝑘 𝐿( (cid:174)𝑎𝑘 , 𝑚𝑘 )(cid:1)(cid:17) ⊗ 𝑓 (𝑚) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ◦|𝑚|)𝛼((cid:12) (cid:12) ,|𝑚|)−1 (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) (cid:16)Ä𝜌1 𝐿( (cid:174)𝑎1, 𝑚1), . . . , 𝜌𝑘 𝐿( (cid:174)𝑎𝑘 , 𝑚𝑘 )ä (cid:17) ⊗ 𝑚 = 𝛽 = 𝛽 (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:174)𝑒,(cid:174)𝑖•𝑖((cid:12) (cid:174)𝑒,(cid:174)𝑖•𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ◦|𝑚|) (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) (cid:0)𝜌𝐿( (cid:174)𝑎, (cid:174)𝑚 ⊗ 𝑚)(cid:1) 82 The first equality is by definition and K-linearity of the left action. The second equality is by the (cid:12) 𝑓ℓ definition of the C -graded multimodule left-action on (𝑀1, . . . , 𝑀𝑘 )⊗ 𝑀. The third equality follows from the assumption that (cid:12) (cid:12) (cid:12) = 𝑖ℓ. The fourth equality follows from equation (5.1.1). Finally, the fifth and sixth equalities follow from unraveling definitions; in particular, the fifth follows since (cid:12) ◦(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) 𝐿( (cid:174)𝑎1, 𝑚1), . . . , 𝜌𝑘 Finally, this gives us the desired result since(cid:12) and the sixth invokes the K-linearity of ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 . 𝐿( (cid:174)𝑎𝑘 , 𝑚𝑘 Ä𝜌1 (cid:12) (cid:174)𝑚(cid:12) ä(cid:12) (cid:12) (cid:12) (cid:12) = (cid:12) (cid:12) (cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ◦|𝑚| = (cid:12) (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12): we have 𝜌𝐿 Ä (cid:174)𝑎, (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) ( (cid:174)𝑚 ⊗ 𝑚)ä = 𝛽 (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:174)𝑒,(cid:174)𝑖•𝑖((cid:12) (cid:12)) (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) (cid:0)𝜌𝐿( (cid:174)𝑎, (cid:174)𝑚 ⊗ 𝑚)(cid:1) . Showing that (iv) holds is completely analogous (and easier)—we leave it to the reader. □ On the other hand, we do not yet have a method for composing grading shifts vertically, so that we cannot define the composition of homogeneous maps. We introduce the fix in the following section. 5.3.1 Extension to a shifting 2-system As before, we will consider the G -graded situation and then present generalities. Thankfully, the extension of a C -shifting system to a C -shifting 2-system is almost exactly like the categorically- graded situation. Suppose Δ1 : 𝐷1 → 𝐷′ 2 are cobordisms of planar arc diagrams, so that (Δ1, 𝑣1) and (Δ2, 𝑣2) induce grading shift functors for any 𝑣1, 𝑣2 ∈ Z ⊕ Z; that is, they belong to the 1 and Δ2 : 𝐷2 → 𝐷′ multimonoid I of G . Define a binary operation, which we call vertical composition, by stacking: set ◦ : I × I → I (Δ2, 𝑣2) ◦ (Δ1, 𝑣1) =   (Δ2 ◦ Δ1, 𝑣2 + 𝑣1)  0 if 𝐷′ 1 = 𝐷2, and otherwise. In our multivariable setting, vertical composition must be extended to a family of vertical compo- sitions for each 𝑘 ≥ 1, ◦ : I 𝑘 × I 𝑘 → I 𝑘 . 83 So, if (cid:174)Δ𝑖 = (Δ𝑖1, . . . , Δ𝑖𝑘 ) for 𝑖 = 1, 2 and 𝐷1 𝑗 Δ1 𝑗 −−→ 𝐷′ 1 𝑗 and 𝐷2 𝑗 Δ2 𝑗 −−→ 𝐷′ 2 𝑗 for 𝑗 = 1, . . . , 𝑘, we set ((cid:174)Δ2, (cid:174)𝑣2) ◦ ((cid:174)Δ1, (cid:174)𝑣1) =    ((cid:174)Δ2 ◦ (cid:174)Δ1, (cid:174)𝑣2 + (cid:174)𝑣1) if 𝐷′ 1 𝑗 = 𝐷2 𝑗 for all 𝑗, and 0 otherwise. The nonzero term on the right is given a chronology as follows. 𝑣2𝑘 𝑣1𝑘 Δ2𝑘 · · · (cid:174)𝑣2 + (cid:174)𝑣1 · · · Δ1𝑘 (cid:174)Δ2 ◦ (cid:174)Δ1 𝑣22 𝑣12 Δ22 Δ12 𝑣21 𝑣11 Δ21 Δ11 Again, we choose this particular chronology so that the arguments of [NP20] lift to our setting. In general, if 𝑆 = {I , {𝜑𝑖}} is already a C -shifting system, equipping I with a vertical composition map I 𝑘 × I 𝑘 → I 𝑘 of this form constitutes what is called a C -shifting 2-system, granted it satisfies the following requirements: (i) 𝑒 ◦ 𝑒 = 𝑒, (ii) D 𝑗◦𝑖 = D𝑖 ∩ 𝜑−1 𝑖 (D 𝑗 ), (iii) 𝜑 𝑗◦𝑖 = 𝜑 𝑗 |𝜑𝑖(D𝑖)∩D 𝑗 ◦ 𝜑𝑖 |D 𝑗◦𝑖 , and (iv) 𝜑(( 𝑗1,..., 𝑗𝑘)◦(𝑖1,...,𝑖𝑘))•( 𝑗◦𝑖) = 𝜑(( 𝑗1,..., 𝑗𝑘)• 𝑗)◦((𝑖1,...,𝑖𝑘)•𝑖) for all 𝑗1, . . . , 𝑗𝑘 , 𝑗, 𝑖1, . . . , 𝑖𝑘 , 𝑖 ∈ I . The first three requirements are written in the single-input case to ignore burdensome notation and should be extended to the 𝑘-input cases. To elucidate the above requirements notice that (in particular, if 𝑗 ◦ 𝑖 is nonzero) 𝜑 𝑗 and 𝜑𝑖 must be defined on (frequently distinct) subsets of the same hom-set. In the G -graded case, this causes no confusion: D 𝑗◦𝑖 = D𝑖 since cobordisms which start 84 at 𝐷1 and factor through 𝐷′ 1 = 𝐷2 still start at 𝐷1. In general, we should be a little more careful: 𝜑𝑖 D𝑖 HomC (𝑋1, . . . , 𝑋𝑘 ; 𝑌 ) ⊃ D 𝑗 𝜑 𝑗 HomC (𝑋1, . . . , 𝑋𝑘 ; 𝑌 ) so, in general, 𝜑 𝑗◦𝑖 is defined only on the subset D𝑖 ∩ 𝜑−1 𝑖 (D 𝑗 ), as in (ii) and (iii). Condition (iv) just ensures that vertical composition and horizontal composition play nicely together—(iv) obviously holds in the G -setting for weighted cobordisms of planar arc diagrams. For completeness, we include a description of compatibility maps. We say that a C -shifting 2-system 𝑆 = {I , {𝜑𝑖}𝑖∈I } is compatible with the associator 𝛼 of C if there are (𝛽, 𝛾, Ξ) such that the underlying C -shifting system is compatible with 𝛼 through 𝛽, and 𝛾 and Ξ are as follows. First, 𝛾 stands for a collection of maps 𝛾 (cid:174)𝑋→𝑌 𝑖, 𝑗 : D (cid:174)𝑋→𝑌 𝑖 → K× for all 𝑖, 𝑗 ∈ I and (cid:174)𝑋, 𝑌 ∈ C satisfying 𝛾𝑖, 𝑗 = 1 whenever 𝑖 ∈ Iid, 𝑗 ∈ Iid, or 𝑖 = 𝑗 = 𝑒. More generally, we construct multivariable functions 𝛾 (cid:174)𝑋→ (cid:174)𝑌 (cid:174)𝑖, (cid:174)𝑗 : 𝑘 (cid:214) ℓ=1 (cid:174)𝑋𝑖→𝑌𝑖 D 𝑖ℓ → K× with analogous requirements (𝛾(cid:174)𝑖, (cid:174)𝑗 = 1 whenever each entry of (cid:174)𝑖 belongs to IId, each entry of (cid:174)𝑗 belongs to IId, or (cid:174)𝑖 = (cid:174)𝑗 = (cid:174)𝑒). We do not require that 𝛾 (cid:174)𝑋→ (cid:174)𝑌 . For example, this = 𝛾 (cid:174)𝑋1→𝑌1 𝑖1, 𝑗1 · · · 𝛾 (cid:174)𝑋𝑘→𝑌𝑘 𝑖𝑘, 𝑗𝑘 (cid:174)𝑖, (cid:174)𝑗 is not the case for the G -graded setting, at least the way we’ve set things up. Second, Ξ stands for a collection of invertible scalars Ξ (cid:174)𝑋→ (cid:174)𝑌 →𝑍 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 ∈ K× satisfying (i) Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 = 1 whenever ( (cid:174)𝑗 ◦ (cid:174)𝑖) • ( 𝑗 ◦ 𝑖) = ( (cid:174)𝑗 • 𝑗) ◦ ((cid:174)𝑖 • 𝑖) and (ii) Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 exchanging elements of Iid out with other elements of Iid. We often write Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 is invariant when (𝑔′𝑔) for Ξ (cid:174)𝑋→ (cid:174)𝑌 →𝑍 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 (𝑔) for Ξ (cid:174)𝑋→𝑌 when (cid:174)𝑋 𝑔 −→ 𝑌 . Finally, we say that the shifting 2-system when (cid:174)𝑋 𝑔′ −→ (cid:174)𝑌 𝑔 −→ 𝑍, or Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 is compatible with 𝛼 through (𝛽, 𝛾, Ξ) if, in addition, the two following equations hold. The first reads 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 (𝑔′𝑔)𝛽𝑖,(cid:174)𝑖(𝑔′, 𝑔)𝛽 𝑗, (cid:174)𝑗 (𝜑(cid:174)𝑖(𝑔′), 𝜑𝑖(𝑔)) = Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 (𝑔′𝑔)𝛽 𝑗◦𝑖, (cid:174)𝑗◦(cid:174)𝑖(𝑔′, 𝑔)𝛾(cid:174)𝑖, (cid:174)𝑗 (𝑔′)𝛾𝑖, 𝑗 (𝑔), (5.3.1) 85 (cid:174)𝑍→ (cid:174)𝑌 for all 𝑔′ ∈ D (cid:174)𝑖 (cid:174)𝑋→𝑌 and 𝑔 ∈ D 𝑖 . Again, this looks burdensome, but it is just to say that 𝛾 and Ξ are chosen so that the following diagram commutes. (cid:174)𝑗 (cid:174)𝑖 𝑔′ 𝑗 𝑖 𝑔 𝛾(cid:174)𝑖, (cid:174)𝑗 (𝑔′) 𝛾𝑖, 𝑗 (𝑔) (cid:174)𝑗 ◦ (cid:174)𝑖 𝑔′ 𝑗 ◦ 𝑖 𝑔 𝛽 𝑗, (cid:174)𝑗 (𝜑(cid:174)𝑖(𝑔′), 𝜑𝑖(𝑔)) 𝛽𝑖, (cid:174)𝑖(𝑔′, 𝑔) (cid:174)𝑗 • 𝑗 (cid:174)𝑖 𝑔′ 𝑖 𝑔 (cid:174)𝑗 • 𝑗 (cid:174)𝑖 • 𝑖 𝑔′ 𝑔 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 (𝑔′𝑔) 𝛽 𝑗◦𝑖, (cid:174)𝑗◦(cid:174)𝑖(𝑔′, 𝑔) ( (cid:174)𝑗 ◦ (cid:174)𝑖) • ( 𝑗 ◦ 𝑖) (𝑔′𝑔) Ξ 𝑖, (cid:174)𝑖 𝑗, (cid:174)𝑗 ( (cid:174)𝑗 • 𝑗) ◦ ((cid:174)𝑖 • 𝑖) 𝑔′ 𝑔 𝑔′ 𝑔 The second requirement reads 𝛾𝑖,𝑘◦ 𝑗 (𝑔)𝛾 𝑗,𝑘 (𝜑𝑖(𝑔)) = 𝛾 𝑗◦𝑖,𝑘 (𝑔)𝛾𝑖, 𝑗 (𝑔) (5.3.2) (cid:174)𝑋→𝑌 for all 𝑔 ∈ D 𝑖 which, a little more obviously, is to say the following diagram commutes. 𝑘 𝑗 𝑖 𝑔 𝛾𝑖, 𝑗 (𝑔) 𝑘 𝑖 ◦ 𝑗 𝑔 𝛾 𝑗, 𝑘(𝜑𝑖(𝑔)) 𝛾 𝑗◦𝑖, 𝑘(𝑔) 𝑘 ◦ 𝑗 𝛾𝑖, 𝑘◦ 𝑗 (𝑔) 𝑖 𝑔 𝑘 ◦ 𝑗 ◦ 𝑖 𝑔 86 So, to extend the G -shifting system we have to a G -shifting 2-system, we choose compatibility maps 𝛾(𝑥1,...,𝑥𝑘)→𝑦 (Δ1,𝑣1),(Δ2,𝑣2) (𝐷, 𝑝) = 𝜆 Ä(cid:12) (cid:12)1(cid:174)𝑥Δ21𝑦 (cid:12) (cid:12) , 𝑣1 ä or, in general, 𝛾 (cid:174)𝑥→(cid:174)𝑦 ((cid:174)Δ1,(cid:174)𝑣1),((cid:174)Δ2,(cid:174)𝑣2) ( (cid:174)𝐷, (cid:174)𝑝) = 𝜆 (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:17) , 𝑉1 (cid:12) (cid:12) (cid:12) where 𝑉1 is the sum of entries in (cid:174)𝑣1, and Ξ(cid:174)𝑥→(cid:174)𝑦→𝑧 (Δ1,𝑣1),((cid:174)Δ1,(cid:174)𝑣1) (Δ2,𝑣2),((cid:174)Δ2,(cid:174)𝑣2) = 𝜄((cid:174)𝑥 𝐻𝑧)𝜆(𝑉1, 𝑣2) where 𝐻 : ((cid:174)Δ2 ◦ (cid:174)Δ1) • (Δ2 ◦ Δ1) ⇒ ((cid:174)Δ2 • Δ2) ◦ ((cid:174)Δ1 • Δ1) and 𝑉1, as before, means the sum of the entries of (cid:174)𝑣1. We refer to the first factor of Ξ by Ξ1 and the second factor by Ξ2. Of course, the definitions above only hold if the cobordisms involved are vertically composable with respect to the chosen order; otherwise, these maps are zero. To understand where these choices come from, notice that (Δ2, 𝑣2) ◦ (Δ1, 𝑣1) can be rewritten schematically as 𝑣2 Δ2 𝑣1 Δ1 𝑔 = 𝜆((cid:12) (cid:12)1(cid:174)𝑥Δ21𝑦 (cid:12) (cid:12) , 𝑣1) 𝑣2 𝑣1 Δ2 Δ1 𝑔 for 𝑔 : (cid:174)𝑥 → 𝑦. That is, (Δ2, 𝑣2) ◦ (Δ1, 𝑣1) = 𝜆((cid:12) (cid:12) (cid:12) , 𝑣1)(Δ2 ◦ Δ1, 𝑣2 + 𝑣1), so we hope 𝛾 has the (cid:12)1(cid:174)𝑥Δ21𝑦 form above. For Ξ, we can start by recognizing that, schematically (and thanks to our chronology conventions), ((cid:174)Δ2 ◦ (cid:174)Δ1) • (Δ2 ◦ Δ1) looks like (cid:174)Δ2 • 1 (cid:174)Δ1 • 1 (cid:174)1 • Δ2 (cid:174)1 • Δ1 𝑔 Where “1” and “(cid:174)1” just stand for the identity cobordism on their respective components (in partic- 87 ular, an element of IId).On the other hand, ((cid:174)Δ2 • Δ2) ◦ ((cid:174)Δ1 • Δ1) looks like (cid:174)Δ2 • 1 (cid:174)1 • Δ2 (cid:174)Δ1 • 1 (cid:174)1 • Δ1 𝑔 So, we have that ((cid:174)Δ2 ◦ (cid:174)Δ1) • (Δ2 ◦ Δ1) = 𝜄((cid:174)𝑥 𝐻𝑧)((cid:174)Δ2 • Δ2) ◦ ((cid:174)Δ1 • Δ1) where 𝐻 : ((cid:174)Δ2 ◦ (cid:174)Δ2) • (Δ2 ◦ Δ1) ⇒ ((cid:174)Δ2 • Δ2) ◦ ((cid:174)Δ1 • Δ1) is the locally vertical change of chronology which simply pushes Δ2 past the cobordisms involved in (cid:174)Δ1. So that Ξ satisfies equation (5.3.1), we must also multiply by 𝜆(𝑉1, 𝑣2). Proposition 5.3.3. The G -shifting 2-system 𝑆 defined above is compatible with 𝛼 through (𝛽, 𝛾, Ξ). Proof. We know that the underlying shifting system is compatible with 𝛼 through 𝛽 by Proposition 5.1.5. Since Iid = {(1𝐷∧, (0, 0)) : 𝐷 is a planar arc diagram}, it is clear that 𝛾 and Ξ as chosen satisfy preliminary requirements; all we need to do is verify equations (5.3.1) and (5.3.2). Verifying (5.3.2) is easy: computing both sides yields 𝜆((cid:12) (cid:12)1(cid:174)𝑥(Δ3 ◦ Δ2)1𝑦 (cid:12) , 𝑣1)𝜆((cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ31𝑦 (cid:12) , 𝑣2) = 𝜆((cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ31𝑦 (cid:12) , 𝑣1 + 𝑣2)𝜆((cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ21𝑦 (cid:12) (cid:12) , 𝑣1) which is true since(cid:12) are equal to 𝜆((cid:12) (cid:12)1(cid:174)𝑥Δ31𝑦 (cid:12)1(cid:174)𝑥(Δ3 ◦ Δ2)1𝑦 (cid:12) , 𝑣1)𝜆((cid:12) (cid:12) (cid:12) = (cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ31𝑦 (cid:12) , 𝑣2)𝜆((cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ31𝑦 (cid:12) +(cid:12) (cid:12) (cid:12)1(cid:174)𝑥Δ21𝑦 (cid:12)1(cid:174)𝑥Δ21𝑦 (cid:12) (cid:12) , 𝑣1). (cid:12) (cid:12); applying bilinearity shows that both sides Verifying equation (5.3.1) looks a lot like the proofs of Propositions 4.2.3 and 5.1.5. Again, start by considering the contributions of 𝛽1 and Ξ1 only. To do this, one can consider the two 88 sequences of changes of chronology encoded by the diagram below. (cid:174)2 (cid:174)1 2 1 𝜁 (cid:174)2 (cid:174)1 2 1 𝛽1 (cid:174)𝑗 • 𝑗 (cid:174)𝑖 • 𝑖 𝛽1 (cid:174)𝑗 • 𝑗 (cid:174)𝑖 𝑖 𝛽1 ( (cid:174)𝑗 ◦ (cid:174)𝑖) • ( 𝑗 ◦ 𝑖) ( (cid:174)𝑗 • 𝑗) ◦ ((cid:174)𝑖 • 𝑖) Ξ1 (cid:174)𝑗 ◦ (cid:174)𝑖 𝑗 ◦ 𝑖 where 𝜁 = 𝜆 (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 (cid:12) (cid:12) (cid:12) , (cid:12) 1(cid:174)𝑦Δ21𝑧 (cid:12) (cid:12) (cid:12) (cid:17) (cid:12) (cid:12) . Thus, by Proposition 3.1.3, we see that the contributions of 𝛽1 and Ξ1 in equation (5.3.1) is 𝜁 × (Left of (5.3.1)) = (Right of (5.3.1)). The remainder of the proof is computing the contributions of 𝛽2, 𝛽3, 𝛽4, 𝛾, and Ξ2. The contributions of these on the left-hand side of (5.3.1) are 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧(((cid:174)Δ2 ◦ (cid:174)Δ1)( (cid:174)𝐷), (Δ2 ◦ Δ1)(𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) (cid:17) 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 (cid:12) (cid:12) · 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) + 𝑃 + 𝑉1, (cid:123)(cid:122) (∗) 1(cid:174)𝑦Δ21𝑧 + 𝑣2 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉2 + 𝑣2 (cid:17) · 𝜆 (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:17) , 𝑣2 (cid:12) (cid:12) (cid:12) (𝛽2,3,4) 𝑗, (cid:174)𝑗 (𝜑(cid:174)𝑖(𝑔′), 𝜑𝑖(𝑔)) 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((cid:174)Δ1( (cid:174)𝐷), Δ1(𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17) 𝑃, 1(cid:174)𝑦Δ11𝑧 + 𝑣1 · 𝜆 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) , 𝑉1 + 𝑣1 (cid:17) · 𝜆 (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 (cid:17) , 𝑣1 (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 1(cid:174)𝑥((cid:174)Δ2 • Δ2)1𝑧 (cid:12) (cid:12) 𝜆 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) , 𝑉1 + 𝑣1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:17) (cid:12) (cid:12) (cid:12) (cid:123)(cid:122) (∗∗) 89 (𝛽2,3,4)𝑖,(cid:174)𝑖(𝑔′, 𝑔) 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 (𝑔′𝑔) We rewrite this product by expanding (∗) via bilinearity, expanding (∗∗) via Lemma 5.1.4 and bilinearity, and then performing the obvious cancellations; the result is the following. 𝜆 (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧(((cid:174)Δ2 ◦ (cid:174)Δ1)( (cid:174)𝐷), (Δ2 ◦ Δ1)(𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) (cid:17) (cid:16) 1(cid:174)𝑦Δ21𝑧 , (cid:12) (cid:12) (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:12) 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 · 𝜆 (cid:12) (cid:12) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:124) 𝜁 1(cid:174)𝑦Δ21𝑧 , 𝑉2 + 𝑣2 + 𝑣2 (cid:16)(cid:12) (cid:12) (cid:12) 𝑃, (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) · 𝜆 (cid:17) , 𝑣2 (cid:12) (cid:12) (cid:12) (cid:17) · 𝜆 ·𝜆 (cid:16)(cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 (cid:12) (cid:12) (cid:17) · 𝜆(𝑉1, 𝑣2) , 𝑣2 𝜆 𝜆 (cid:17) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16) (cid:17) 𝑃, · 𝜆 , 𝑣1 + 𝑣1 1(cid:174)𝑦Δ11𝑧 1(cid:174)𝑥 (cid:174)Δ11(cid:174)𝑦 (cid:16)(cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧(((cid:174)Δ2 ◦ (cid:174)Δ1)( (cid:174)𝐷), (Δ2 ◦ Δ1)(𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:12) (cid:12) , 𝑣1 (cid:17) (cid:12) (cid:12) (cid:12) · 𝜆 , 𝑉1 + 𝑣1 (cid:17) · 𝜆 (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑦Δ21𝑧 (cid:17) , 𝑣1 · 𝜆 (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) (cid:12) (cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:17) , 𝑉1 (cid:12) (cid:12) (cid:12) On the other hand, the contributions of 𝛽2, 𝛽3, 𝛽4, 𝛾 and Ξ2 on the right-hand side of (5.3.1) are 𝜆 𝜆 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:17) (cid:17) · 𝜆 , 𝑣1 , 𝑉1 1(cid:174)𝑦Δ21𝑧 (cid:16)(cid:12) (cid:16)(cid:12) 1(cid:174)𝑥 (cid:174)Δ21(cid:174)𝑦 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧(((cid:174)Δ2 ◦ (cid:174)Δ1)( (cid:174)𝐷), (Δ2 ◦ Δ1)(𝐷)) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:16)(cid:12) 1(cid:174)𝑦(Δ2◦Δ1)1𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) 1(cid:174)𝑥((cid:174)Δ2 ◦ (cid:174)Δ1)1(cid:174)𝑦 , 𝑣2 + 𝑣1 · 𝜆 𝑃, (cid:17) (cid:16) · 𝜆 , 𝑉1 + 𝑉2 + 𝑣1 + 𝑣2 (cid:17) + 𝑣2 + 𝑣2 (cid:12) (cid:12) (cid:12) (cid:17) 𝛾(cid:174)𝑖, (cid:174)𝑗 (𝑔′) · 𝛾𝑖, 𝑗 (𝑔) (𝛽2,3,4) 𝑗◦𝑖, (cid:174)𝑗◦(cid:174)𝑖(𝑔′, 𝑔) 𝜆 (𝑉1, 𝑣2) Ξ2 Comparing the updated form of the left-hand side with this, we see that everything cancels except for the 𝜁 term present in the former. Thus, we conclude that the contributions of 𝛽2, 𝛽3, 𝛽4, 𝛾 and Ξ2 in equation (5.3.1) is (Left of (5.3.1)) = 𝜁 × (Right of (5.3.1)), which completes the proof. □ 5.3.2 C -graded vertical composition As before, we construct natural isomorphisms 𝜑 𝑗 ◦ 𝜑𝑖 ⇒ 𝜑 𝑗◦𝑖 or 𝜑 (cid:174)𝑗 ◦ 𝜑(cid:174)𝑖 ⇒ 𝜑 (cid:174)𝑗◦(cid:174)𝑖 given by (𝜑 𝑗 ◦ 𝜑𝑖)(𝑀) → 𝜑 𝑗◦𝑖(𝑀) 𝑚 ↦→ 𝛾𝑖, 𝑗 (|𝑚|)𝑚 or, in general, 90 (𝜑 (cid:174)𝑗 ◦ 𝜑(cid:174)𝑖)( (cid:174)𝑀) → 𝜑 (cid:174)𝑗◦(cid:174)𝑖( (cid:174)𝑀) (cid:12) (cid:174)𝑚(cid:12) (cid:174)𝑚 ↦→ 𝛾(cid:174)𝑖, (cid:174)𝑗 ((cid:12) (cid:12)) (cid:174)𝑚 respectively, and 𝜑( (cid:174)𝑗◦(cid:174)𝑖)•( 𝑗◦𝑖) ⇒ 𝜑( (cid:174)𝑗• 𝑗)◦((cid:174)𝑖•𝑖) by (𝜑( (cid:174)𝑗◦(cid:174)𝑖)•( 𝑗◦𝑖))(𝑀) → 𝜑( (cid:174)𝑗• 𝑗)◦((cid:174)𝑖•𝑖)(𝑀) 𝑚 ↦→ Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 (|𝑚|)𝑚 for all homogeneous 𝑚 ∈ 𝑀. In terms of these natural isomorphisms, equations (5.3.1) and (5.3.2) translate to mean that the diagram 𝜑 (cid:174)𝑗• 𝑗 Ä𝜑(cid:174)𝑖(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝜑𝑖(𝑀)ä 𝛽 𝑗, (cid:174)𝑗 𝛽𝑖,(cid:174)𝑖 𝜑 (cid:174)𝑗 ◦ 𝜑(cid:174)𝑖(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝜑 𝑗 ◦ 𝜑𝑖(𝑀) 𝛾(cid:174)𝑖, (cid:174)𝑗 ⊗𝛾𝑖, 𝑗 𝜑 (cid:174)𝑗◦(cid:174)𝑖(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝜑 𝑗◦𝑖(𝑀) 𝜑 (cid:174)𝑗• 𝑗 ◦ 𝜑(cid:174)𝑖•𝑖 (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 𝜑( (cid:174)𝑗• 𝑗)◦((cid:174)𝑖•𝑖) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) 𝛽 𝑗◦𝑖, (cid:174)𝑗◦(cid:174)𝑖 𝜑( (cid:174)𝑗◦(cid:174)𝑖)•( 𝑗◦𝑖) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 commutes for all C -graded multimodules 𝑀1, . . . , 𝑀𝑘 , 𝑀, and 𝜑𝑘 ◦ 𝜑 𝑗 ◦ 𝜑𝑖(𝑀) 𝛾 𝑗,𝑘 𝜑𝑘◦ 𝑗,𝑖(𝑀) 𝛾𝑖, 𝑗 𝛾𝑖,𝑘◦ 𝑗 𝜑𝑘 ◦ 𝜑 𝑗◦𝑖(𝑀) 𝛾 𝑗◦𝑖,𝑘 𝜑𝑘◦ 𝑗◦𝑖(𝑀) commutes for each C -graded multimodule 𝑀. Before moving on, we note that a shifting 2-system may be extended to (cid:102)I . In particular, since 𝜑Id acts as the identity, we can extend vertical composition itself by declaring Id ◦ 𝑖 = 𝑖 = 𝑖 ◦ Id. If Id appears in the subscript of Ξ, it can be replaced by an compatible element in IId. Finally, we can properly define a vertical composition of homogeneous maps. Suppose 𝑓 : 𝑀 → 𝑁 is homogeneous of degree 𝑖, and 𝑔 : 𝑁 → 𝐿 is homogeneous of degree 𝑗. Define their C -graded composition as (cid:0)𝑔 ◦C 𝑓 (cid:1) (𝑚) = 𝛾𝑖, 𝑗 (|𝑚|)−1 (cid:0)𝑔 ◦ 𝑓 (cid:1) (𝑚). 91 Proposition 5.3.4. With the assumptions above, 𝑔 ◦C 𝑓 is purely homogeneous of degree 𝑗 ◦ 𝑖. Proof. Requirement (i) of Definition 5.3.1 follows easily since D 𝑗◦𝑖 = D𝑖 ∩ 𝜑−1 𝑖 (cid:12) = 𝑖 so(cid:12) (cid:12) 𝑓 (cid:12) (cid:12) (cid:12) = 𝑗 so(cid:12) (cid:12)𝑔(cid:12) (cid:12) = 𝜑𝑖(|𝑚|), and(cid:12) (cid:12) = 𝜑 𝑗 ◦ 𝜑𝑖(|𝑚|). Thus, (cid:12)𝑔( 𝑓 (𝑚))(cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (Dj). Additionally, (cid:12) (cid:12) (cid:0)𝑔 ◦C 𝑓 (cid:1) (𝑚)(cid:12) (cid:12) = 𝛾𝑖, 𝑗 (|𝑚|)−1𝜑 𝑗 ◦ 𝜑𝑖(|𝑚|) = 𝜑 𝑗◦𝑖(|𝑚|), so (ii) is satisfied. For (iii) we claim that, for any (cid:174)𝑎 = (𝑎1, . . . , 𝑎𝑘 ) ∈ (𝐴1, . . . , 𝐴𝑘 ), (cid:12) (cid:174)𝑎(cid:12) (cid:12) , 𝜑𝑖(|𝑚|)) = 𝛽 (cid:174)𝑒, 𝑗◦𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑚|)𝛽 (cid:174)𝑒, 𝑗 ((cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛾𝑖, 𝑗 (|𝑚|)−1𝛽 (cid:174)𝑒,𝑖((cid:12) (cid:12) ,|𝑚|)𝛾𝑖, 𝑗 ä−1 Ä(cid:12) (cid:12)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:12) (cid:12) where (cid:174)𝑒 = (𝑒, . . . , 𝑒) as usual. The desired result follows easily from here, since 𝜌𝐿 (cid:0) (cid:174)𝑎, (𝑔 ◦C 𝑓 )(𝑚)(cid:1) = 𝜌𝐿 Ä (cid:174)𝑎, 𝛾𝑖, 𝑗 (|𝑚|)−1𝑔( 𝑓 (𝑚))ä = 𝛾𝑖, 𝑗 (|𝑚|)−1𝜌𝐿 (cid:0) (cid:174)𝑎, 𝑔( 𝑓 (𝑚))(cid:1) )𝑔 (cid:0)𝜌𝐿( (cid:174)𝑎, 𝑓 (𝑚))(cid:1) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) = 𝛾𝑖, 𝑗 (|𝑚|)−1𝛽 (cid:174)𝑒, 𝑗 ((cid:12) (cid:12) (cid:32)(cid:32) (cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) =𝜑𝑖(|𝑚|) (cid:12) ,|𝑚|)𝑔 Ä 𝑓 (cid:0)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:1)ä (cid:12) (cid:174)𝑎(cid:12) (cid:12) , 𝜑𝑖(|𝑚|))𝛽 (cid:174)𝑒,𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) = 𝛾𝑖, 𝑗 (|𝑚|)−1𝛽 (cid:174)𝑒, 𝑗 ((cid:12) ä−1 Ä(cid:12) = 𝛽 (cid:174)𝑒, 𝑗◦𝑖((cid:12) (cid:12)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:12) (cid:12) (cid:174)𝑎(cid:12) (𝑔 ◦ 𝑓 ) (cid:0)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:1) (cid:12) ,|𝑚|)𝛾𝑖, 𝑗 (cid:12) (cid:12) (cid:174)𝑎(cid:12) = 𝛽 (cid:174)𝑒, 𝑗◦𝑖((cid:12) (cid:12) ,|𝑚|) (cid:0)𝑔 ◦C 𝑓 (cid:1) (𝜌𝐿( (cid:174)𝑎, 𝑚)) by definition, by K-linearity of 𝜌𝐿, (cid:12)𝑔(cid:12) since (cid:12) (cid:12) = 𝑗, since (cid:12) (cid:12) 𝑓 (cid:12) (cid:12) = 𝑖 & 𝑔 is K-linear, by the claim, and by definition. (cid:12) (cid:174)𝑎(cid:12) To prove the claim, we apply equation (5.3.1) when (cid:174)𝑗 = (cid:174)𝑖 = (cid:174)𝑒 and 𝑔′ = (cid:12) (cid:12) and 𝑔 = |𝑚|; it reads (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑚|)𝛾 (cid:174)𝑒, (cid:174)𝑒((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ◦|𝑚|)𝛽( (cid:174)𝑒◦ (cid:174)𝑒), 𝑗◦𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) ((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑚|)𝛽 (cid:174)𝑒, 𝑗 (𝜑 (cid:174)𝑒((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ◦|𝑚|)𝛽 (cid:174)𝑒,𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛾 (cid:174)𝑒•𝑖, (cid:174)𝑒• 𝑗 ((cid:12) (cid:12))𝛾𝑖, 𝑗 (|𝑚|). (cid:12)), 𝜑𝑖(|𝑚|)) = Ξ𝑖, (cid:174)𝑒 𝑗, (cid:174)𝑒 = 1 since ( (cid:174)𝑒 ◦ (cid:174)𝑒) • ( 𝑗 ◦ 𝑖) = (cid:174)𝑒 • ( 𝑗 ◦ 𝑖) = 𝑗 ◦ 𝑖 = ( (cid:174)𝑒 • 𝑗) ◦ ( (cid:174)𝑒 • 𝑖). Moreover, by our Now, Ξ𝑖, (cid:174)𝑒 𝑗, (cid:174)𝑒 (cid:12) (cid:174)𝑎(cid:12) (cid:12)) = (cid:12) (cid:12) (cid:174)𝑎(cid:12) working assumption that all C -graded algebras are supported entirely in Σ, 𝜑 (cid:174)𝑒((cid:12) (cid:12). Then, noting 𝛾 (cid:174)𝑒, (cid:174)𝑒 = 1, the equation above may be rewritten (cid:12) (cid:174)𝑎(cid:12) (cid:12) , 𝜑𝑖(|𝑚|)) = 𝛽 (cid:174)𝑒, 𝑗◦𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑚|)𝛽 (cid:174)𝑒, 𝑗 ((cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ◦|𝑚|)𝛽 (cid:174)𝑒,𝑖((cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛾𝑖, 𝑗 ((cid:12) (cid:12) ,|𝑚|)𝛾𝑖, 𝑗 (|𝑚|). 92 Note that (cid:12) (cid:12)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) = (cid:12) (cid:12) ◦ |𝑚|, since the action maps of multimodules are C -graded—thus, rear- ranging provides the desired result. Requirement (iv) is proven in exactly the same manner, noting that Ξ 𝑒,𝑖 𝑒, 𝑗 = 1. □ In general, suppose 𝑀ℓ 𝑓ℓ −→ 𝑁ℓ 𝑔ℓ −→ 𝐿ℓ is a composition of purely homogeneous maps of degree 𝑖ℓ and 𝑗ℓ respectively, for ℓ = 1, . . . , 𝑘. We say that (cid:174)𝑓 is purely homogeneous of degree (cid:174)𝑖, and (cid:174)𝑔 is purely homogeneous of degree (cid:174)𝑗. Then, for (cid:174)𝑚 ∈ (𝑀1, . . . , 𝑀𝑘 ), we define ((cid:174)𝑔 ◦C (cid:174)𝑓 )( (cid:174)𝑚) = 𝛾(cid:174)𝑖, (cid:174)𝑗 ((cid:12) = 𝛾(cid:174)𝑖, (cid:174)𝑗 ((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12))−1((cid:174)𝑔 ◦ (cid:174)𝑓 )( (cid:174)𝑚) (cid:12))−1(𝑔1( 𝑓1(𝑚1)), . . . , 𝑔𝑘 ( 𝑓𝑘 (𝑚𝑘 ))). The proof above extends to this situation without trouble, so ((cid:174)𝑔 ◦C (cid:174)𝑓 ) is purely homogeneous of degree (cid:174)𝑗 ◦ (cid:174)𝑖. Proposition 5.3.5. C -graded vertical composition is associative. Proof. Suppose 𝑀 𝑓 −→ 𝑁 𝑔 −→ 𝐿 On one hand, ℎ −→ 𝐾 are purely homogeneous of degrees(cid:12) (cid:12) 𝑓 (cid:12) (cid:12)𝑔(cid:12) (cid:12) = 𝑖,(cid:12) (cid:12) = 𝑗, and|ℎ| = 𝑘. (cid:0)ℎ ◦C (𝑔 ◦C 𝑓 )(cid:1) (𝑚) = 𝛾𝑖, 𝑗 (|𝑚|)−1 (cid:0)ℎ ◦C 𝑔 𝑓 (cid:1) (𝑚) = 𝛾𝑖, 𝑗 (|𝑚|)−1𝛾 𝑗◦𝑖,𝑘 (|𝑚|)−1ℎ𝑔 𝑓 (𝑚). On the other, Ä(cid:0)ℎ ◦C 𝑔(cid:1) ◦C 𝑓 ä (𝑚) = 𝛾 𝑗,𝑘 Ä(cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) ä−1 (cid:0)ℎ𝑔 ◦C 𝑓 (cid:1) (𝑚) = 𝛾 𝑗,𝑘 Ä(cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) ä−1 𝛾𝑖,𝑘◦ 𝑗 (|𝑚|)−1ℎ𝑔 𝑓 (𝑚). Since(cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) = 𝜑𝑖(|𝑚|), associativity follows from equation (5.3.2). □ Propositions 5.3.2 and 5.3.4 imply that the C -graded composition and tensor product of homo- geneous maps is again a homogeneous map. The last thing we must do is check the compatibility of ⊗ and ◦C . Proposition 5.3.6. Suppose 𝑓 : 𝑀 → 𝑁 and { 𝑓𝛼 : 𝑀𝛼 → 𝑁𝛼}𝛼=1,...,𝑘 are purely homogeneous maps of degree 𝑖 and 𝑖𝛼 respectively, and similarly 𝑔 : 𝑁 → 𝐿 and {𝑔𝛽 : 𝑁𝛽 → 𝐿 𝛽}𝛽=1,...,𝑘 are 93 purely homogeneous maps of degree 𝑗 and 𝑗𝛽 respectively. Then (cid:0)(𝑔1 ◦C 𝑓1), . . . , (𝑔𝑘 ◦C 𝑓𝑘 )(cid:1) ⊗ 𝑔 ◦C 𝑓 = Ξ 𝑖,(cid:174)𝑖 (cid:0)(𝑔1, . . . , 𝑔𝑘 ) ⊗ 𝑔(cid:1) ◦C (cid:0)( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 (cid:1) . 𝑗, (cid:174)𝑗 Proof. We will just unravel both sides of the equation above. The equality will follow from equation (5.3.1). On one hand, (((𝑔1 ◦C 𝑓1), . . . , (𝑔𝑘 ◦C 𝑓𝑘 )) ⊗ 𝑔 ◦C 𝑓 )( (cid:174)𝑚 ⊗ 𝑚) = 𝛽 (cid:174)𝑗◦(cid:174)𝑖, 𝑗◦𝑖((cid:12) = 𝛽 (cid:174)𝑗◦(cid:174)𝑖, 𝑗◦𝑖((cid:12) (cid:12) ,|𝑚|)−1 (cid:0)(𝑔1 ◦C 𝑓1)(𝑚1), . . . , (𝑔𝑘 ◦C 𝑓𝑘 )(𝑚𝑘 )(cid:1) ⊗ (𝑔 ◦C 𝑓 )(𝑚) (cid:12) ,|𝑚|)−1 Ä𝛾𝑖1, 𝑗1(|𝑚1|)−1(𝑔1 ◦ 𝑓1)(𝑚1), . . . , 𝛾𝑖𝑘, 𝑗𝑘 (|𝑚1|)−1(𝑔𝑘 ◦ 𝑓𝑘 )(𝑚𝑘 )ä (cid:12) (cid:174)𝑚(cid:12) (cid:12) (cid:174)𝑚(cid:12) ⊗ 𝛾𝑖, 𝑗 (|𝑚1|)−1(𝑔 ◦ 𝑓 )(𝑚) = 𝛽 (cid:174)𝑗◦(cid:174)𝑖, 𝑗◦𝑖((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚|)−1𝛾(cid:174)𝑖, (cid:174)𝑗 ((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12))−1𝛾𝑖, 𝑗 (|𝑚|)−1 (cid:0)(𝑔1 ◦ 𝑓1)(𝑚1), . . . , (𝑔𝑘 ◦ 𝑓𝑘 )(𝑚𝑘 )(cid:1) ⊗ (𝑔 ◦ 𝑓 )(𝑚). The first equality follows from Proposition 5.3.4 since each 𝑔ℓ ◦ 𝑓ℓ is purely homogeneous of degree 𝑗ℓ ◦ 𝑖ℓ, so (cid:0)(𝑔1 ◦C 𝑓1), . . . , 𝑔𝑘 ◦C 𝑓𝑘 (cid:1) is purely homogeneous of degree (cid:174)𝑗 ◦ (cid:174)𝑖. The second equality follows from the definition of ◦C , while the third is just a rewriting step. On the other hand, (((𝑔1, . . ., 𝑔𝑘 ) ⊗ 𝑔) ◦C ( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 ))( (cid:174)𝑚 ⊗ 𝑚) = 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 ((cid:12) = 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 ((cid:12) = 𝛾(cid:174)𝑖•𝑖, (cid:174)𝑗• 𝑗 ((cid:12) (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12) (cid:174)𝑚 ⊗ 𝑚(cid:12) (cid:12))−1 (cid:0)((𝑔1, . . . , 𝑔𝑘 ) ⊗ 𝑔) ◦ (( 𝑓1, . . . , 𝑓𝑘 ) ⊗ 𝑓 )(cid:1) ( (cid:174)𝑚 ⊗ 𝑚) (cid:12) ,|𝑚|)−1( 𝑓1(𝑚1), . . . , 𝑓𝑘 (𝑚𝑘 )) ⊗ 𝑓 (𝑚)ä (cid:12))−1((𝑔1, . . . , 𝑔𝑘 ) ⊗ 𝑔) Ä𝛽(cid:174)𝑖,𝑖((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) (cid:12) 𝑓 (𝑚)(cid:12) ,(cid:12) (cid:12))−1𝛽(cid:174)𝑖,𝑖((cid:12) (cid:12))−1 (cid:0)𝑔1( 𝑓1(𝑚)), . . . , 𝑔𝑘 ( 𝑓𝑘 (𝑚𝑘 ))(cid:1) (cid:174)𝑓 ( (cid:174)𝑚) (cid:12) (cid:12) (cid:12) (cid:12) ,|𝑚|)−1𝛽 (cid:174)𝑗, 𝑗 ( (cid:12) (cid:12) (cid:12) (cid:174)𝑚(cid:12) ⊗ 𝑔( 𝑓 (𝑚)). The first equality follows from Proposition 5.3.2, and the second and third follow from the definition of the tensor product of homogeneous maps. As suggested, the equality follows from equation (5.3.1), taking 𝑔′ = (cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) and 𝑔 = |𝑚|—we must only compensate by Ξ 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 . □ 5.4 Changes of chronology An important feature of G -shifting systems in particular is that changes of chronology induce natural transformations of grading shift functors. Recall that we have a few different notions of composition for changes of chronolology: 94 • to a sequence of chronological cobordisms 𝐴 𝑊 −→ 𝐵 𝑊 ′ −−→ 𝐶 and changes of chronology 𝐻 on 𝑊 and 𝐻′ on 𝑊 ′, there is a change of chronology 𝐻′ ◦ 𝐻 on 𝑊 ′ ◦ 𝑊; • A sequence of changes of chronology 𝑊1 𝐻1 ==⇒ 𝑊2 𝐻2 ==⇒ 𝑊3 is itself a change of chronology, denoted 𝐻2 ★ 𝐻1. The compositions ◦ and ★ extend to chronological cobordisms with corners Δ in the obvious way. In this setting we obtain another way of composing changes of chronology. Suppose Δ, Δ1, . . . , Δ𝑘 are chronological cobordisms with corners so that (Δ1, . . . , Δ𝑘 ) • Δ is nonzero, and suppose 𝐻, 𝐻1, . . . , 𝐻𝑘 are changes of chronology on Δ, Δ1, . . . , Δ𝑘 . Then we denote by (𝐻1, . . . , 𝐻𝑘 ) • 𝐻 the change of chronology on (Δ1, . . . , Δ𝑘 ) • Δ defined by applying the 𝐻𝑖 and the 𝐻 in order according to the chronology. Indeed, we could define the • operation in terms of successive applications of the ◦ operation after extending each change of chronology to be trivial outside of its original component. Now, each change of chronology 𝐻 : Δ → Δ′ of chronological cobordisms with corners extends to a change of chronology without corners given appropriate crossingless matchings 𝑥1, . . . , 𝑥𝑘 , 𝑦. The latter is denoted by (cid:174)𝑥 𝐻𝑦 : 1(cid:174)𝑥Δ1𝑦 → 1(cid:174)𝑥Δ′1𝑦. We claim that this observation induces a natural transformation of grading shift functors 𝜑𝐻 : 𝜑Δ ⇒ 𝜑Δ′ defined on each 𝑀 ∈ Ob(MultiModG ) by 𝜑𝐻(𝑀) : 𝜑Δ(𝑀) → 𝜑Δ′(𝑀) 𝜑Δ(𝑚) ↦→ 𝜄(𝐻(|𝑚|))−1𝜑Δ′(𝑚) where 𝐻(|𝑚|) means (cid:174)𝑥 𝐻𝑦 for |𝑚| : (cid:174)𝑥 → 𝑦. In general, 𝜑(𝐻1,...,𝐻𝑘) : 𝜑(Δ1,...,Δ𝑘) ⇒ 𝜑(Δ′ 1,...,Δ′ 𝑘 ) 95 where is given by 𝜑(𝐻1,...,𝐻𝑘) : (𝜑Δ1(𝑀1), . . . , 𝜑Δ𝑘 (𝑀𝑘 )) → (𝜑Δ′ 1 (𝑀1), . . . , 𝜑Δ′ 𝑘 (𝑀𝑘 )) 𝑘 (cid:214) 𝜑 (cid:174)Δ ( (cid:174)𝑚) ↦→ 𝜄(𝐻𝑖(|𝑚𝑖 |))−1𝜑 (cid:174)Δ′( (cid:174)𝑚). We abbreviate (cid:206)𝑘 𝑖=1 𝜄(𝐻𝑖(|𝑚𝑖 |))−1 to 𝜄( (cid:174)𝐻((cid:12) (cid:12) (cid:174)𝑚(cid:12) 𝑖=1 (cid:12)))−1. Sometimes, we write 𝜑𝐻 when we mean 𝜑𝐻(𝑀). Proposition 5.4.1. The diagram (cid:0)𝜑Δ1(𝑀1), . . . , 𝜑Δ𝑘 (𝑀𝑘 )(cid:1) ⊗ 𝜑Δ(𝑀) 𝛽(Δ1,...,Δ𝑘 ),Δ 𝜑(Δ1,...,Δ𝑘)•Δ (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) (𝜑𝐻1 , ..., 𝜑𝐻𝑘 )⊗𝜑𝐻 𝜑(𝐻1, ..., 𝐻𝑘 )•𝐻 Ä𝜑Δ′ 1 (𝑀1), . . . , 𝜑Δ′ 𝑘 (𝑀𝑘 )ä ⊗ 𝜑Δ′(𝑀) 𝛽(Δ′ 1 ,...,Δ′ 𝑘 ),Δ′ 𝜑(Δ′ 1,...,Δ′ 𝑘 )•Δ′ (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) commutes. Thus, 𝜑𝐻(𝑀) is a map of G -graded multimodules and, in particular, 𝜑𝐻 is a natural transformation of MultiModG (𝐴1, . . . , 𝐴𝑘 ; 𝐵) functors. Proof. Assume that the gradings of (𝑚1, . . . , 𝑚𝑘 ) ∈ (𝑀1, . . . , 𝑀𝑘 ) and 𝑚 ∈ 𝑀 are compatible in the sense that |𝑚𝑖 | ∈ Hom(𝑥𝑖1, . . . , 𝑥𝑖𝛼𝑖 ; 𝑦𝑖) for each 𝑖 = 1, . . . , 𝑘 and |𝑚| ∈ Hom(𝑦1, . . . , 𝑦𝑘 ; 𝑧). Recall that 𝛽 is defined as the composite 𝛽1𝛽2𝛽3𝛽4 and notice that (𝛽𝑖)(Δ1,...,Δ𝑘),Δ = (𝛽𝑖)(Δ′ 1,...,Δ′ 𝑘 ),Δ′ for 𝑖 = 2, 3, and 4. Denote by 𝐻𝛽 and 𝐻𝛽′ the changes of chronology used to define (𝛽1)(Δ1,...,Δ𝑘),Δ and (𝛽1)(Δ′ 1,...,Δ′ 𝑘 ),Δ′ respectively. Also, consider the changes of chronology (cid:0)(𝐻1, . . . , 𝐻𝑘 ) • 𝐻(cid:1) (cid:174)𝑥 : 1(cid:174)𝑥((cid:174)Δ • Δ)1𝑧 ⇒ 1(cid:174)𝑥((cid:174)Δ′ • Δ′)1𝑧, 𝑧 which we abbreviate to 𝐻•, and (cid:0) (𝐻1)𝑦1, . . . , (cid:174)𝑥𝑘 (𝐻𝑘 )𝑦𝑘 (cid:174)𝑥1 (cid:1) ⊔ (cid:174)𝑦𝐻𝑧 : 1(cid:174)𝑥 (cid:174)Δ1(cid:174)𝑦 ⊔ 1(cid:174)𝑦Δ1𝑧 ⇒ 1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 ⊔ 1(cid:174)𝑦Δ′1𝑧, which we abbreviate to 𝐻⊔. Then we have the following sequences of changes of chronology. Ä1(cid:174)𝑥((cid:174)Δ • Δ)1𝑧 ä ◦ 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧( (cid:174)𝐷, 𝐷) 𝐻•◦Id Ä1(cid:174)𝑥((cid:174)Δ′ • Δ′)1𝑧 ä ◦ 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧( (cid:174)𝐷, 𝐷) 𝐻𝛽1 𝐻𝛽1 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((cid:174)Δ( (cid:174)𝐷), Δ(𝐷)) ◦ Ä1(cid:174)𝑥 (cid:174)Δ1(cid:174)𝑦 ⊔ 1(cid:174)𝑦Δ1𝑧 ä 𝑊(cid:174)𝑥 (cid:174)𝑦𝑧((cid:174)Δ′( (cid:174)𝐷), Δ′(𝐷)) ◦ Id◦𝐻⊔ Ä1(cid:174)𝑥 (cid:174)Δ′1(cid:174)𝑦 ⊔ 1(cid:174)𝑦Δ′1𝑧 ä 96 Then, proposition 3.1.3 implies that 𝜄(𝐻⊔)𝛽 (cid:174)Δ,Δ On the other hand, (cid:12) (cid:174)𝑚(cid:12) ((cid:12) (cid:12) ,|𝑚|) = 𝛽 (cid:174)Δ′,Δ′((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,|𝑚|)𝜄(𝐻•). (cid:0)(𝜑𝐻1, . . . , 𝜑𝐻𝑘 ) ⊗ 𝜑𝐻(cid:1) ((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ⊗ |𝑚|) = 𝜑 (cid:174)𝐻((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12))𝜑𝐻(|𝑚|) = 𝜄( (cid:174)𝐻((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12)))−1𝜄(𝐻(|𝑚|))−1 = 𝜄(𝐻⊔)−1 □ □ and which concludes the proof. Proposition 5.4.2. We have that 𝜑(𝐻1,...,𝐻𝑘)•𝐻(|𝑚| ◦(cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12)) = 𝜄(𝐻•)−1, 𝜑𝐻′ ◦ 𝜑𝐻 (cid:27) 𝜑𝐻′★𝐻 Proof. This is immediate, since 𝜄((cid:174)𝑥 𝐻′ 𝑦 ◦ (cid:174)𝑥 𝐻𝑦) = 𝜄((cid:174)𝑥 𝐻′ 𝑦)𝜄((cid:174)𝑥 𝐻𝑦) = 𝜄((cid:174)𝑥 𝐻′ 𝑦 ★ (cid:174)𝑥 𝐻𝑦). 97 CHAPTER 6 TANGLES, DG-MULTIMODULES, AND MULTIGLUING In this chapter, we finally prove multigluing in generality upon detailing our method for associating to a diskular tangle a G -graded dg-multimodule; see §6.2. This is preceded by §6.1 wherein we define C -graded dg-multimodules, whose differential preserves C -degree. We also define the HOM-complex associated to C -graded dg-multimodules, important to Chapter 8. Finally, in §6.3, we also discuss graded commutativity and analogues of Naisse-Putyra’s “dg-C-graded” bimodules, whose differential is C -homogeneous. We do this mostly for completeness, and cite §6.3 very sparingly in successive sections. 6.1 C -graded dg-multimodules and related concepts We remark that we only consider the situation of C -graded dg-multimodules over C -graded algebras, rather than over C -graded dg-algebras. Definition 6.1.1. If 𝐴1, . . . , 𝐴𝑘 , 𝐵 are C -graded algebras, we define a C -graded dg-(𝐴1, . . . , 𝐴𝑘 ; 𝐵)- multimodule (𝑀, 𝑑𝑀) as a Z × C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule 𝑀 = (cid:201) 𝑛∈Z,𝑔∈Mor(C ) 𝑀 𝑛 𝑔 together with a K-linear map 𝑑𝑀 : 𝑀 → 𝑀 satisfying (i) 𝑑𝑀(𝑀 𝑛 𝑔 ) ⊂ 𝑀 𝑛+1 𝑔 , (ii) 𝑑𝑀(𝜌𝐿( (cid:174)𝑎, 𝑚)) = 𝜌𝐿( (cid:174)𝑎, 𝑑𝑀(𝑚)), (iii) 𝑑𝑀(𝜌𝑅(𝑚, 𝑏)) = 𝜌𝑅(𝑑𝑀(𝑚), 𝑏), and (iv) 𝑑𝑀 ◦ 𝑑𝑀 = 0. for all (cid:174)𝑎 ∈ (𝐴1, . . . , 𝐴𝑘 ), 𝑏 ∈ 𝐵, and 𝑚 ∈ 𝑀. The Z-grading is called the homological grading; the homological grading of 𝑚 ∈ 𝑀 is denoted |𝑚|ℎ. We assume the left and right action on a multimodule preserves homological grading; i.e., (cid:12) (cid:12)𝜌𝐿( (cid:174)𝑎, 𝑚)(cid:12) (cid:12)ℎ = |𝑚|ℎ. A map of C -graded dg- : 𝑀 → 𝑁 will always mean a K-linear chain map (i.e., it commutes with the bimodules 𝑓 differentials) which preserves both homological and C -grading. 98 Given C -graded dg-(𝐴𝑖1, . . . , 𝐴𝑖𝛼𝑖 ; 𝐵𝑖)-multimodules (𝑀𝑖, 𝑑𝑀𝑖 ) for each 𝑖 = 1, . . . , 𝑘 and a C - graded dg-(𝐵1, . . . , 𝐵𝑘 ; 𝐶)-multimodules (𝑀, 𝑑𝑀), we define the C -graded dg-(𝐴11, . . . , 𝐴𝑘𝛼𝑘 ; 𝐶)- multimodule (cid:0)(𝑀1, 𝑑𝑀1), . . . , (𝑀𝑘 , 𝑑𝑀𝑘 )(cid:1) ⊗(𝐵1,...,𝐵𝑘) (𝑀, 𝑑𝑀) = Ä(𝑀1, . . . , 𝑀𝑘 ) ⊗(𝐵1,...,𝐵𝑘) 𝑀, 𝑑 (cid:174)𝑀 ⊗𝑀 ä where 𝑑 (cid:174)𝑀 ⊗𝑀( (cid:174)𝑚 ⊗ 𝑚) = 𝑘 ∑︁ 𝑖=1 (−1)(cid:205)𝑖−1 𝑗=1|𝑚 𝑗 | ℎ(𝑚1, . . . , 𝑑𝑀𝑖 (𝑚𝑖), . . . , 𝑚𝑘 ) ⊗ 𝑚 + (−1)(cid:205)𝑘 𝑖=1|𝑚𝑖 |ℎ (cid:174)𝑚 ⊗ 𝑑𝑀(𝑚). We will sometimes denote the first large summation, perhaps confusingly, by simply 𝑑 (cid:174)𝑀( (cid:174)𝑚). Proposition 6.1.2. The tensor product of C -graded dg-multimodules, as defined above, is a C - graded dg-multimodule. Proof. The requirement (i) is obvious. Also, it is routine (but tedious) to check requirement (iv), that 𝑑2 (cid:174)𝑀 ⊗𝑀 = 0. To see requirements (ii) and (iii), note that 𝑑𝑀𝑖 preserves C -grading, so for any 𝑖, (cid:12)(𝑚1, . . . , 𝑑𝑀𝑖 (𝑚𝑖), . . . , 𝑚𝑘 )(cid:12) (cid:12) (cid:12) = |(𝑚1, . . . , 𝑚𝑖, . . . , 𝑚𝑘 )| thus, in particular, (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) 𝛼((cid:12) (cid:12) (cid:174)𝑚(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑚|) = 𝛼((cid:12) (cid:12)(𝑚1, . . . , 𝑑𝑀𝑖 (𝑚𝑖), . . . , 𝑚𝑘 )(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) (cid:12) ,|𝑑𝑀(𝑚)|) = 𝛼((cid:12) (cid:12) ,|𝑚|). We leave the rest of the proof to the reader. □ The homology of a C -graded dg-multimodule (𝑀, 𝑑𝑀) is the C × Z-graded multimodule 𝐻(𝑀, 𝑑𝑀) = ker(𝑑𝑀)(cid:14)im(𝑑𝑀). We call a map of C -graded dg-multimodules 𝑓 : (𝑀, 𝑑𝑀) → (𝑁, 𝑑𝑁 ) a quasi-isomorphism if the induced map 𝑓∗ : 𝐻(𝑀, 𝑑𝑀) → 𝐻(𝑁, 𝑑𝑁 ) is an isomorphism. We define the mapping cone of a map of C -graded dg-multimodules as follows. First, recall the homological shifting functor [𝑘] which sends the dg-multimodule (𝑀, 𝑑𝑀) to (𝑀[𝑘], 𝑑𝑀[𝑘]) where 𝑀[𝑘]𝑛 𝑔 = 𝑀 𝑛−𝑘 𝑔 , 𝑑𝑀[𝑘] = (−1)𝑘 𝑑𝑀, and 𝑀[𝑘] inherits the left and right actions of 𝑀. Then the mapping cone of 𝑓 : (𝑀, 𝑑𝑀) → (𝑁, 𝑑𝑁 ) is the C -graded dg-multimodule Ö Cone( 𝑓 ) = (𝑀[−1] ⊕ 𝑁, 𝑑Cone( 𝑓 )) where 𝑑Cone( 𝑓 ) = 99 −𝑑𝑀 𝑓 è . 0 𝑑𝑁 We also define the HOM complex of C -graded dg-multimodules. Suppose 𝑀 and 𝑁 are two C -graded dg-(𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodules. Let HOM(𝑀, 𝑁) denote the chain complex of bihomogeneous (that is, homogeneous in homological degree and purely homogeneous in (cid:102)I - degree) maps 𝑓 of arbitrary (Z × (cid:102)I )-degree, with differential 𝐷( 𝑓 ) = 𝑑𝑁 ◦ 𝑓 − (−1)| 𝑓 | ℎ 𝑓 ◦ 𝑑𝑀 . Thus, 𝐷 preserves the (cid:102)I -degree of a bihomogeneous map, but increases the homological degree by one. For example, if 𝑓 has degree (𝑘, 𝑖) ∈ Z × (cid:102)I , then the differential of HOM(𝑀, 𝑁) simply takes the difference of the following paths. 𝑓 𝑓 𝑀 𝑛 𝑔 𝑑𝑀 𝑀 𝑛+1 𝑔 𝑁 𝑛+𝑘 𝜑𝑖(𝑔) 𝑑 𝑁 𝑁 𝑛+𝑘+1 𝜑𝑖(𝑔) Recall that each purely homogeneous map of degree 𝑖 induces a graded map (cid:101)𝑓 Moreover, C -grading preserving maps can be viewed as purely homogeneous of degree Id ∈ (cid:102)I ; : 𝜑𝑖(𝑀) → 𝑀. indeed, purely homogeneous maps of degree Id induce maps graded maps 𝜑Id(𝑀) → 𝑁, but, 𝜑Id(𝑀) = 𝑀. This (tautological) correspondence allows us to view the HOM complex as a bigraded abelian group HOM(𝑀, 𝑁)𝑘 𝑖 (cid:27) (cid:214) HomMultiModC (𝜑𝑖(𝑀 𝑛), 𝑁 𝑛+𝑘 ) 𝑛∈Z with differential of bidegree (1, 𝑒). 6.2 Resolution of diskular tangles A diskular (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-tangle is a tangle diagram 𝑇 in D2 − ( ˚𝐷1 ∪ · · · ∪ ˚𝐷 𝑘 ), where each of the 𝐷𝑖 are disjoint disks lying within the interior of D2, each of the form {𝑧 ∈ D2 : |𝑧 − 𝑧𝑖 | ≤ 𝑟𝑖} for some 𝑧𝑖 ∈ ˚D2 and 𝑟𝑖 > 0, so that • Each 𝐷𝑖 has 2𝑚𝑖 marked points on its boundary, all disjoint from a fixed basepoint in 𝜕𝐷𝑖, and 100 • D2 itself has 2𝑛 marked points on its boundary, all disjoint from a fixed basepoint on 𝜕D2. By “𝑇 is a tangle diagram in D2 − ( ˚𝐷1 ∪ · · · ∪ ˚𝐷 𝑘 ),” we mean that the interval components of 𝑇 all have endpoints lying on the marked points of D2 − ( ˚𝐷1 ∪ · · · ∪ ˚𝐷 𝑘 ). We view the disks 𝐷1, . . . , 𝐷 𝑘 as ordered. As with planar arc diagrams, if 𝑆𝑖 is a diskular (ℓ𝑖1, . . . , ℓ𝑖𝛼𝑖 ; 𝑚𝑖)-tangle for each 𝑖 = 1, . . . , 𝑘, we denote by 𝑇(𝑆1, . . . , 𝑆𝑘 ) the diskular (ℓ11, . . . , ℓ𝑘𝛼𝑘 ; 𝑛)-tangle obtained by filling the 𝑖th removed disk with 𝑆𝑖, identifying distinguished points and basepoints appropriately. Again, there is also a pairwise composition, which we write as 𝑇 ◦𝑖 𝑆𝑖, and the two are related by 𝑇(𝑆1, . . . , 𝑆𝑘 ) = (· · · ((𝑇 ◦𝑘 𝑆𝑘 ) ◦𝑘−1 · · · ) ◦1 𝑆1. A diskular (; 𝑛)-tangle is referred to as a diskular 𝑛-tangle. Let 𝑐(𝑇) denote the number of crossings in 𝑇 and take an ordering 𝜒(𝑇) = { 𝜒1, . . . , 𝜒𝑐(𝑇)} of the crossings of 𝑇. Let 𝑣 = (𝑣1, . . . , 𝑣𝑐(𝑇)) : 𝜒(𝑇) → {0, 1}𝑐(𝑇) be an assignment of 0 or 1 to each crossing of 𝑇. To each 𝑣, thought of as the coordinates of the vertices of the hypercube [0, 1]𝑐(𝑇), we associate a planar arc diagram 𝑇𝑣 of type (𝑚1, . . . , 𝑚𝑘 ; 𝑛) by resolving each crossing according to the following rule. 𝜒𝑖 𝑣𝑖=0 𝑣𝑖=1 We call 𝑇𝑣 a resolution of 𝑇. As this procedure associates planar arc diagrams to each vertex of the cube [0, 1]𝑐(𝑇), we can associate to each edge a cobordism of planar arc diagrams. First, to ensure this cobordism comes with a chronology, we require that 𝑇 come labeled with one of or at each crossing. For each 𝑣𝑖 = 0 in some vertex 𝑣, we write 𝑣 + 𝑖 to denote the vertex which is identical to 𝑣 except that (𝑣 +𝑖)𝑖 = 1. Introduce a direction on the edges of the cube so that 𝑣 → 𝑣 +𝑖. 101 Finally, to each of these edges, we associate the chronological cobordism 𝑊𝑣,𝑖 : 𝑇𝑣 → 𝑇𝑣+𝑖 obtained by putting a saddle in a small cylinder above the 0-resolution of the 𝑖th crossing with chronology determined by the labeling, and taking the identity everywhere outside of this cylinder. Our goal is to assign a G -graded dg-(𝐻𝑚1, . . . , 𝐻𝑚𝑘 ; 𝐻𝑛)-multimodule F (𝑇) to each diskular (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-tangle 𝑇. We have already seen that F (𝑇𝑣) is a G -graded (𝐻𝑚1, . . . , 𝐻𝑚𝑘 ; 𝐻𝑛)- multimodule for each resolution 𝑇𝑣 of 𝑇. Also, to each edge cobordism 𝑊𝑣,𝑖 : 𝑇𝑣 → 𝑇𝑣+𝑖, we can associate a G -graded map F (𝑊𝑣,𝑖) : 𝜑𝑊𝑣,𝑖 (F (𝑇𝑣)) → F (𝑇𝑣+𝑖). We will need a slightly different graded map, achieved by constructing another family of chronolog- ical cobordisms for each 𝑣. Denote by 1 the “all one” vertex (1, . . . , 1). Recursively, set 𝑊1 = 1𝑇1, the identity cobordism of 𝑇1. For 𝑣 ≠ 1, let ℓ denote the lowest integer so that 𝑣ℓ = 0. Then, define which has path 𝑇𝑣 𝑊𝑣,ℓ −−−→ 𝑇𝑣+ℓ 𝑊𝑣+ℓ−−−→ 𝑇1. Additionally, notice that for each 𝑣 𝑗 = 0, there is a locally 𝑊𝑣 := 𝑊𝑣+ℓ ◦ 𝑊𝑣,ℓ vertical change of chronology 𝐻𝑣, 𝑗 : 𝑊𝑣 ⇒ 𝑊𝑣+ 𝑗 ◦ 𝑊𝑣, 𝑗 obtained by pushing the saddle over the 𝑗th crossing to the beginning of the sequence of saddles. Now, set 𝐶(𝑇)𝑟 = (cid:202) |𝑣|=𝑟 𝐶(𝑇)𝑣[𝑟] where 𝐶(𝑇)𝑣 = 𝜑𝑊𝑣 (F (𝑇𝑣)). Here, 𝑟 is the homological index of the dg-bimodule we are building. The first step in defining the differential is to associate to each edge 𝑣 → 𝑣 + 𝑗 the G -graded map 𝑑𝑣, 𝑗 = F (𝑊𝑣, 𝑗 ) ◦ 𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) : 𝐶(𝑇)𝑣 → 𝐶(𝑇)𝑣+ 𝑗 . Perhaps this doesn’t seem to make sense. Indeed, there should be an intermediary 𝛾𝑊𝑣+ 𝑗 ,𝑊𝑣, 𝑗 for the composition to parse: 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) −−−−−−−−−−→ 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) 𝛾𝑊𝑣+ 𝑗 ,𝑊𝑣, 𝑗 −−−−−−−−→ 𝜑𝑊𝑣+ 𝑗 (𝜑𝑊𝑣, 𝑗 (F (𝑇𝑣))) F (𝑊𝑣, 𝑗 ) −−−−−−→ 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )). 102 However, by our definition of these compatibility maps, 𝛾𝑊𝑣+ 𝑗 ,𝑊𝑣, 𝑗 = 1 since both cobordisms in- volved are unweighted. Actually, the grading shifting system imposed on this grading multicategory implies that 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 = 𝜑𝑊𝑣+ 𝑗 ◦ 𝜑𝑊𝑣, 𝑗 . Lemma 6.2.1 ([NP20], Lemma 6.7). The diagram 𝐶(𝑇)𝑣 𝑑𝑣,𝑖 𝑑𝑣, 𝑗 𝐶(𝑇)𝑣+𝑖 𝑑𝑣+𝑖, 𝑗 𝐶(𝑇)𝑣+𝑖+ 𝑗 𝑑𝑣+ 𝑗,𝑖 𝐶(𝑇)𝑣+ 𝑗 commutes for all 𝑣 and 𝑖, 𝑗 for which 𝑣𝑖 = 𝑣 𝑗 = 0. The proof of this Lemma is exactly as Naisse-Putyra. Indeed, the validity of this Lemma, without sign assignments, is the first meaningful benefit of working with grading (multi)categories. Finally, define 𝑑𝑟 : 𝐶(𝑇)𝑟 → 𝐶(𝑇)𝑟+1 by setting 𝑑𝑟 |𝐶(𝑇)𝑣 = ∑︁ (−1)𝑝(𝑣, 𝑗)𝑑𝑣, 𝑗 { 𝑗:𝑣 𝑗 =0} for all 𝑣 with |𝑣| = 𝑟, where 𝑝(𝑣, 𝑗) = {ℓ : 𝑗 < ℓ ≤ 𝑐(𝑇) and 𝑣ℓ = 1} counts the number of 1-resolutions occurring after the 𝑗th entry of 𝑣. In conclusion, we set F (𝑇) = (cid:32) (cid:202) 𝑟 𝐶(𝑇)𝑟, 𝑑 = (cid:33) 𝑑𝑟 . ∑︁ 𝑟 The following is apparent, but we write it as a proposition for future reference. Proposition 6.2.2. Suppose 𝑇 is a diskular tangle. Given a specified crossing of 𝑇, write 𝑇𝑖, for 𝑖 = 0, 1, to denote the diskular tangles resulting from taking the 𝑖th resolution of this crossing. Write 𝜎 to denote the saddle from 𝑇0 to 𝑇1. Then, F (𝑇) (cid:27) Cone Å Equivalently, we have an exact triangle 𝜑𝜎F (𝑇0) F (𝜎) −−−−→ F (𝑇1) ã . F (𝑇1) → F (𝑇) → 𝜑𝜎F (𝑇0)[1]. 103 Less apparent is the fact that F (𝑇) is actually a C -graded dg-multimodule. Proposition 6.2.3. If 𝑇 is a diskular (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-tangle, F (𝑇) has the structure of a G -graded dg-(𝐻𝑚1, . . . , 𝐻𝑚𝑘 ; 𝐻𝑛)-multimodule. Proof. It is clear that 𝑑(F (𝑇)ℓ 𝑔) ⊂ F (𝑇)ℓ+1 𝑔 and 𝑑2 = 0 by definition. We will show that 𝑑(𝜌𝐿( (cid:174)𝑎, 𝑢)) = 𝜌𝐿( (cid:174)𝑎, 𝑑(𝑢)); the requirement for the right action follows by a similar argument. By linearity, it suffices to show that the diagram (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣 (F (𝑇𝑣)) 1⊗𝑑𝑣, 𝑗 (𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) 𝜑𝑊𝑣 𝜌𝑣 𝐿 𝜑𝑊𝑣+ 𝑗 𝜌𝑣+ 𝑗 𝐿 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝑑𝑣, 𝑗 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) commutes, where 𝜌𝑣 𝐿 denotes 𝜇[(1𝑚1, . . . , 1𝑚𝑘 ); 𝑇𝑣]. By definition of 𝑑𝑣, 𝑗 and left actions on shifted multimodules, this diagram factors as follows (we’ve refrained from labeling arrows to avoid clutter. 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝜑𝑊𝑣 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) 1 3 2 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) 4 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) 𝜑𝑊𝑣+ 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣+ 𝑗 )) 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) Here, we are using the fact that (𝐴1, . . . , 𝐴𝑘 ) = 𝜑 (cid:174)𝑒(𝐴1, . . . 𝐴𝑘 ). We will show that the original diagram commutes by showing that squares 1 – 4 commute up to constants which cancel with one another. Square 1 , 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝛽 (cid:174)𝑒,𝑊𝑣 𝜑𝑊𝑣 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 1⊗𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) 𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) 𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 104 commutes on the nose by Proposition 5.4.1, taking (cid:174)Δ = (cid:174)𝑒 and Δ = 𝑊𝑣. Technically, if ℎ is the “do nothing” change of chronology, we are acting by 𝜑(cid:174)ℎ to 1. Similarly, the vertical arrow on the right should be 𝜑(cid:174)ℎ•𝐻𝑣, 𝑗 . ( (cid:174)𝐴) on the left terms, but this is clearly equal Square 2 , 𝜑𝑊𝑣 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 𝜌𝑣 𝐿 𝜌𝑣 𝐿 𝜑𝑊𝑣 (F (𝑇𝑣)) 𝜑𝐻𝑣, 𝑗 (F (𝑇𝑣)) 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) commutes by the naturality of 𝜑𝐻𝑣, 𝑗 ; again, see Proposition 5.4.1. Square 3 , 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) 𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 ◦𝑤𝑣, 𝑗 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 1⊗F (𝑊𝑣, 𝑗 ) 1⊗F (𝑊𝑣, 𝑗 ) 𝜑 (cid:174)𝑒(𝐴1, . . . , 𝐴𝑘 ) ⊗ 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) 𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 𝜑𝑊𝑣+ 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣+ 𝑗 )) commutes up to a factor of 𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 (cid:12) (cid:174)𝑎(cid:12) ((cid:12) (cid:12) ,|𝑢|), where we’ve fixed (cid:174)𝑎 ∈ (𝐴1, . . . , 𝐴𝑘 ) and 𝑢 ∈ F (𝑇𝑣). To see this, recall that 𝛽 decomposes into 4 terms, 𝛽1–𝛽4, and that here 𝛽2 = 𝛽3 = 1 for both compatibility maps since all cobordisms involved are unweighted. Otherwise, suppose|𝑎𝑖 | : 𝑥𝑖 → 𝑦𝑖 and |𝑢| : (𝑦1, . . . , 𝑦𝑘 ) → 𝑧. Note that if |𝑢| : (cid:174)𝑦 → 𝑥 then (cid:12) (cid:12) : (cid:174)𝑦 → 𝑥 and |F (Δ)(𝑢)| (cid:174)𝑦 → 𝑥, as (cid:12)𝜑𝑊 (𝑢)(cid:12) long as the values are nonzero. Then and Ä𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (cid:12) ,|𝑢|)ä (cid:12) (cid:174)𝑎(cid:12) ((cid:12) 4 = 𝜆(𝑃′, (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦(𝑊𝑣+ 𝑗 ◦ 𝑊𝑣, 𝑗 )1𝑧 (cid:12) ) (cid:12) (cid:12) Ä𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 (cid:12))ä (cid:12)F (𝑊𝑣, 𝑗 )(𝑢)(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) ((cid:12) 4 = 𝜆(𝑃′, (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦(𝑊𝑣+ 𝑗 )1𝑧 ) (cid:12) (cid:12) (cid:12) where 𝑃′ is the sum of the second coordinaters of (cid:174)𝑎𝑖 for 𝑖 = 1, . . . , 𝑘. Bilinearity of 𝜆 implies that the contribution from the 𝛽4 terms is 𝜆(𝑃′, (cid:12) 1(cid:174)𝑦(𝑊𝑣, 𝑗 )1𝑧 (cid:12) (cid:12) (cid:12) (cid:12) (cid:12) ) × (down, then right) = (right, then down). 105 On the other hand, the 𝛽1 terms are computed via changes of chronology. Similar to before, we have that Ä𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (cid:12) ,|𝑢|)ä (cid:12) (cid:174)𝑎(cid:12) ((cid:12) 1 Ä𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 (cid:12) ,|𝑢|)ä (cid:12) (cid:174)𝑎(cid:12) ((cid:12) = Ä𝛽 (cid:174)𝑒,𝑊𝑣+ 𝑗 (cid:12))ä (cid:12)F (𝑊𝑣, 𝑗 )(𝑢)(cid:12) (cid:12) ,(cid:12) (cid:12) (cid:174)𝑎(cid:12) ((cid:12) . 1 × 1 The easiest way to see this is by noticing that the change of chronology on the left factors into changes of chronologies corresponding to the right terms. 𝑊𝑣+ 𝑗 𝑊𝑣, 𝑗 (1𝑚1, . . . , 1𝑚𝑘 ) 𝑇𝑣 𝑊𝑣+ 𝑗 𝑊𝑣, 𝑗 𝑇𝑣+ 𝑗 (1𝑚1, . . . , 1𝑚𝑘 ) 𝑇𝑣 𝑊𝑣+ 𝑗 𝑇𝑣+ 𝑗 𝑊𝑣, 𝑗 (1𝑚1, . . . , 1𝑚𝑘 ) 𝑇𝑣 Together, this means that Ä𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 (cid:12) ,|𝑢|)ä (cid:12) (cid:174)𝑎(cid:12) ((cid:12) × (down, then right) = (right, then down). Finally, square 4 , 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣)) 1⊗F (𝑊𝑣, 𝑗 ) 𝜑𝑊𝑣+ 𝑗 ((𝐴1, . . . , 𝐴𝑘 ) ⊗ F (𝑇𝑣+ 𝑗 )) 𝜌𝑣 𝐿 𝜌𝑣+ 𝑗 𝐿 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 (F (𝑇𝑣)) F (𝑊𝑣, 𝑗 ) 𝜑𝑊𝑣+ 𝑗 (F (𝑇𝑣+ 𝑗 )) commutes up to a factor of 𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 (cid:12) (cid:174)𝑎(cid:12) ((cid:12) (cid:12) ,|𝑢|). To see this, recall that F (𝑊𝑖, 𝑗 ) is homogeneous of degree 𝑊𝑖, 𝑗 , hence 𝜌𝑣+ 𝑗 𝐿 ( (cid:174)𝑎, F (𝑊𝑣, 𝑗 )(𝑢)) = 𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 (cid:12) (cid:174)𝑎(cid:12) ((cid:12) (cid:12) ,|𝑢|)F (𝑊𝑣, 𝑗 )(𝜌𝑣 𝐿( (cid:174)𝑎, 𝑢)). These two contributions of 𝛽 (cid:174)𝑒,𝑊𝑣, 𝑗 cancel each other out, which concludes the proof. □ 106 6.2.1 Multigluing Finally, we prove that F behaves as we hope with respect to composition of tangles; this isomorphism is referred to as multigluing. Theorem 6.2.4. Suppose 𝑇 is a diskular (𝑚1, . . . , 𝑚𝑘 ; 𝑛)-tangle and 𝑇𝑖 is a diskular (ℓ𝑖1, . . . , ℓ𝑖𝛼𝑖 ; 𝑚𝑖) tangle for each 𝑖 = 1, . . . , 𝑘. Then there is an isomorphism (cid:0)F (𝑇1), . . . , F (𝑇𝑘 )(cid:1) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) F (𝑇) (cid:27) F (𝑇(𝑇1, . . . , 𝑇𝑘 )) induced by 𝜇[((𝑇1)𝑣1, . . . , (𝑇𝑘 )𝑣 𝑘 ); 𝑇𝑣]. Proof. Recall that 𝐶(𝑇)𝑣 = 𝜑𝑊𝑣 F (𝑇𝑣) and 𝐶(𝑇𝑖)𝑣𝑖 = 𝜑𝑊𝑣𝑖 F (𝑇𝑖)𝑣𝑖 . We’ll write ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑘 ) as ⊗ (cid:174)𝑚. First, notice that (𝜑𝑊𝑣1 F (𝑇1)𝑣1, . . . , 𝜑𝑊𝑣𝑘 F (𝑇𝑘)𝑣𝑘 ) ⊗ (cid:174)𝑚 𝜑𝑊𝑣 F (𝑇𝑣) )•𝑊𝑣 ((F (𝑇1)𝑣1, . . . , F (𝑇𝑘)𝑣𝑘 ) ⊗ (cid:174)𝑚 F (𝑇𝑣)) 𝛽(𝑊𝑣1 ,...,𝑊𝑣𝑘 ),𝑊𝑣 −−−−−−−−−−−−−→ 𝜑(𝑊𝑣1 𝜇[((𝑇1)𝑣1 ,...,(𝑇𝑘 )𝑣𝑘 −−−−−−−−−−−−−−−−−−→ 𝜑(𝑊𝑣1 );𝑇𝑣] ,...,𝑊𝑣𝑘 ,...,𝑊𝑣𝑘 )•𝑊𝑣 F (𝑇𝑣((𝑇1)𝑣1, . . . , (𝑇𝑘)𝑣𝑘 )) is an isomorphism thanks to Proposition 4.4.6. This composition is what we mean by “the map induced by 𝜇[((𝑇1)𝑣1, . . . , (𝑇𝑘 )𝑣 𝑘 ); 𝑇𝑣]”; we will denote it by 𝜇★ when there is no confusion. Notice that the target of this composition can be rewritten 𝜑𝑊(𝑣,𝑣1,...,𝑣𝑘 ) F (𝑇(𝑇1, . . . , 𝑇𝑘 )(𝑣,𝑣1,...,𝑣 𝑘)) = 𝐶(𝑇(𝑇1, . . . , 𝑇𝑘 ))(𝑣,𝑣1,...,𝑣 𝑘) where we’ve ordered the crossings of 𝑇(𝑇1, . . . , 𝑇𝑘 ) by the crossings of 𝑇 first, and then the crossings of 𝑇1, 𝑇2, and so on. So, to conclude the proof, we need only show the diagrams (cid:0)𝐶(𝑇1)𝑣1, . . . , 𝐶(𝑇𝑖)𝑣𝑖 , . . . , 𝐶(𝑇𝑘 )𝑣 𝑘 (1,...,𝑑𝑣𝑖 , 𝑗 ,...,1)⊗1 (cid:0)𝐶(𝑇1)𝑣1, . . . , 𝐶(𝑇𝑖)𝑣𝑖+ 𝑗 , . . . , 𝐶(𝑇𝑘 )𝑣 𝑘 (cid:1) ⊗ (cid:174)𝑚 𝐶(𝑇)𝑣 (cid:1) ⊗ (cid:174)𝑚 𝐶(𝑇)𝑣 𝜇★ 𝜇★ 𝐶(𝑇(𝑇1, . . . , 𝑇𝑘 ))(𝑣,𝑣1,...,𝑣𝑖,...,𝑣 𝑘) 𝑑(𝑣,𝑣1,...,𝑣𝑘 ),𝑐+𝑐1+...+𝑐𝑖−1+ 𝑗 𝐶(𝑇(𝑇1, . . . , 𝑇𝑘 ))(𝑣,𝑣1,...,𝑣𝑖+ 𝑗,...,𝑣 𝑘) (where 𝑐 is the number of crossings of 𝑇 and 𝑐𝑖 is the number of crossings of 𝑇𝑖) and (cid:0)𝐶(𝑇1)𝑣1, . . . , 𝐶(𝑇𝑘 )𝑣 𝑘 (cid:1) ⊗ (cid:174)𝑚 𝐶(𝑇)𝑣 (1,...,1)⊗𝑑𝑣, 𝑗 (cid:0)𝐶(𝑇1)𝑣1, . . . , 𝐶(𝑇𝑘 )𝑣 𝑘 (cid:1) ⊗ (cid:174)𝑚 𝐶(𝑇)𝑣+ 𝑗 𝜇★ 𝜇★ 𝐶(𝑇(𝑇1, . . . , 𝑇𝑘 ))(𝑣,𝑣1,...,𝑣 𝑘) 𝑑(𝑣,𝑣1,...,𝑣𝑘 ), 𝑗 𝐶(𝑇(𝑇1, . . . , 𝑇𝑘 ))(𝑣+ 𝑗,𝑣1,...,𝑣 𝑘) 107 commute. As in the proof of Proposition 6.2.3, we will show that each square factors into squares which commute up to values which cancel. We introduce the following notation: we’ll write • 𝜑𝑊𝑣𝑖 = 𝜑𝑖 and F (𝑇𝑖)𝑣𝑖 = 𝐶𝑖, so that 𝐶(𝑇𝑖)𝑣𝑖 = 𝜑𝑊𝑣𝑖 F (𝑇𝑖)𝑣𝑖 can be written 𝜑𝑖𝐶𝑖; • 𝜑𝑊𝑣 = 𝜑0 and F (𝑇)𝑣 = 𝐶0, so that 𝐶(𝑇)𝑣 = 𝜑𝑊𝑣 F (𝑇)𝑣 can be written 𝜑0𝐶0; • 𝜑𝑊𝑣𝑖 + 𝑗 ◦𝑊𝑣𝑖 , 𝑗 = 𝜑𝑖′ and 𝜑𝑊𝑣𝑖 + 𝑗 = 𝜑𝑖′′. Similarly, 𝜑𝑊𝑣+ 𝑗 ◦𝑊𝑣, 𝑗 = 𝜑0′ and 𝜑𝑊𝑣+ 𝑗 = 𝜑0′′. • 𝐶 = F (𝑇𝑣((𝑇1)𝑣1, . . . , (𝑇𝑖)𝑣𝑖 , . . . , (𝑇𝑘 )𝑣 𝑘 )), 𝐶′ = F (𝑇𝑣((𝑇1)𝑣1, . . . , (𝑇𝑖)𝑣𝑖+ 𝑗 , . . . , (𝑇𝑘 )𝑣 𝑘 )), and 𝐶′′ = F (𝑇𝑣+ 𝑗 ((𝑇1)𝑣1, . . . , (𝑇𝑘 )𝑣 𝑘 )). Other notation is defined accordingly; for example, 𝜑(𝑊𝑣1 and so on. The maps involved also adapt, including the writing of 𝜑𝐻𝑖 for 𝜑𝐻𝑣𝑖 , 𝑗 , and F𝑖 and F ′ F (𝑊𝑣𝑖, 𝑗 ) and F (𝑊(𝑣,𝑣1,...,𝑣 𝑘),𝑐+𝑐1+···+𝑐𝑖−1+ 𝑗 ). )•𝑊𝑣 is rewritten 𝜑(1,...,𝑖,...,𝑘)•0, ,...,𝑊𝑣𝑖 ,...,𝑊𝑣𝑘 𝑖 for With this new notation, the first diagram factorizes as follows. (cid:0)𝜑1𝐶1, . . . , 𝜑𝑖𝐶𝑖, . . . , 𝜑𝑘𝐶𝑘 (cid:1) ⊗ (cid:174)𝑚 𝜑0𝐶0 𝛽(1,...,𝑖,...,𝑘),0 𝜑(1,...,𝑖,...,𝑘)•0 (𝐶0, . . . , 𝐶𝑖, . . . , 𝐶𝑘 ) ⊗ (cid:174)𝑚 𝐶0 (1,...,𝜑𝐻𝑖 ,...,1)⊗1 1 𝜑(1,...,𝐻𝑖 ,...,1)•1 2 (cid:0)𝜑1𝐶1, . . . , 𝜑𝑖′𝐶𝑖, . . . , 𝜑𝑘𝐶𝑘 (cid:1) ⊗ (cid:174)𝑚 𝜑0𝐶0 𝛽(1,...,𝑖′ ,...,𝑘),0 𝜑(1,...,𝑖′,...,𝑘)•0 (𝐶0, . . . , 𝐶𝑖, . . . , 𝐶𝑘 ) ⊗ (cid:174)𝑚 𝐶0 (1,...,F𝑖,...,1)⊗1 3 (1,...,F𝑖,...,1)⊗1 4 Ä𝜑1𝐶1, . . . , 𝜑𝑖′′𝐶′ 𝑖 , . . . , 𝜑𝑘𝐶𝑘 ä ⊗ (cid:174)𝑚 𝜑0𝐶0 𝛽(1,...,𝑖′′ ,...,𝑘),0 𝜑(1,...,𝑖′′,...,𝑘)•0 Ä𝐶0, . . . , 𝐶′ 𝑖 , . . . , 𝐶𝑘 ä ⊗ (cid:174)𝑚 𝐶0 𝜇 𝜇 𝜇 𝜑(1,...,𝑖,...,𝑘)•0𝐶 𝜑(1,...,𝐻𝑖 ,...,1)•1 𝜑(1,...,𝑖′,...,𝑘)•0𝐶 F ′ 𝑖 𝜑(1,...,𝑖′′,...,𝑘)•0𝐶′ Squares 1 and 2 both commute by Proposition 5.4.1. Comparing the horizontal arrows, square 3 commutes up to a factor of 𝛽(1,...,𝑊𝑣𝑖 , 𝑗 ,...,1),1, in the sense that 𝛽(1,...,𝑊𝑣𝑖 , 𝑗 ,...,1),1 × (down, then right) = (right, then down). Notice that the 𝛽2 and 𝛽3 terms are both equal to 1, since all cobordisms involved are unweighted. Moreover, the 𝛽4 term is equal to 1 since (cid:12) (cid:12) (cid:12) = (0, 0) given any closures (cid:174)𝑥, 𝑦. Thus, the two (cid:12)1(cid:174)𝑥11𝑦 108 sides differ by a value given by a single change of chronology 𝛽(1,...,𝑊𝑣𝑖 , 𝑗 ,...,1),1 = 𝜄 á 𝑊𝑣𝑖, 𝑗 ë ⇒ 𝑊𝑣𝑖, 𝑗 as in the definition of the compatibility maps 𝛽. Of course, we should view the 𝑊𝑣𝑖, 𝑗 on the left as (1, . . . , 𝑊𝑣𝑖, 𝑗 , . . . , 1) • 1, and the 𝑊𝑣𝑖, 𝑗 on the right as (1, . . . , 𝑊𝑣𝑖, 𝑗 , . . . , 1). On the other hand, square 4 commutes up to the value á 𝜄 𝑊𝑣𝑖, 𝑗 ë 𝑊𝑣𝑖, 𝑗 ⇒ = (𝛽(1,...,𝑊𝑣𝑖 , 𝑗 ,...,1),1)−1 in the sense that (𝛽(1,...,𝑊𝑣𝑖 , 𝑗 ,...,1),1)−1 × (down, then right) = (right, then down). Thus, the former diagram commutes. There are subtle differences in validating the commutativity of the latter diagram—in particular, the 𝛽4 term in the analogue to square 3 is nontrivial. Anyway, the diagram in question factorizes as follows. (cid:0)𝜑1𝐶1, . . . , 𝜑𝑘𝐶𝑘 (cid:1) ⊗ (cid:174)𝑚 𝜑0𝐶0 𝛽(1,...,𝑘),0 𝜑(1,...,𝑘)•0 (𝐶0, . . . , 𝐶𝑘 ) ⊗ (cid:174)𝑚 𝐶0 (1,...,1)⊗𝜑𝐻 1′ 𝜑(1,...,1)•𝐻 2′ (cid:0)𝜑1𝐶1, . . . , 𝜑𝑘𝐶𝑘 (cid:1) ⊗ (cid:174)𝑚 𝜑0′𝐶0 𝛽(1,...,𝑘),0′ 𝜑(1,...,𝑘)•0′ (𝐶0, . . . , 𝐶𝑘 ) ⊗ (cid:174)𝑚 𝐶0 (1,...,1)⊗F0 3′ (1,...,1)⊗F0 4′ (cid:0)𝜑1𝐶1, . . . , 𝜑𝑘𝐶𝑘 (cid:1) ⊗ (cid:174)𝑚 𝜑0′′𝐶′ 0 𝛽(1,...,𝑘),0′′ 𝜑(1,...,𝑘)•0′′ (𝐶0, . . . , 𝐶𝑘 ) ⊗ (cid:174)𝑚 𝐶′ 0 𝜇 𝜇 𝜇 𝜑(1,...,𝑘)•0𝐶 𝜑(1,...,1)•𝐻 𝜑(1,...,𝑘)•0′𝐶 F ′ 0 𝜑(1,...,𝑘)•0′′𝐶′′ Again, squares 1′ and 2′ commute thanks to Proposition 5.4.1, and square 3′ commutes up to a factor of 𝛽 (cid:174)1,𝑊𝑣, 𝑗 in the sense that 𝛽 (cid:174)1,𝑊𝑣, 𝑗 × (down, then right) = (right, then down). 109 Indeed, the 𝛽2 and 𝛽3 terms are trivial, but otherwise we have á 𝑊𝑣, 𝑗 ë 𝛽 (cid:174)1,𝑊𝑣, 𝑗 = 𝜄 ⇒ 𝑊𝑣, 𝑗 × 𝜆 (cid:16) 𝑃′, (cid:12) (cid:12) (cid:12) 1(cid:174)𝑦𝑊𝑣, 𝑗 1𝑧 (cid:12) (cid:17) (cid:12) (cid:12) . Then again, the coherence isomorphisms specify that 𝑊𝑣, 𝑗 = 𝛽 (cid:174)1,𝑊𝑣, 𝑗 × 𝑊𝑣, 𝑗 which is precisely (down, then right) = 𝛽 (cid:174)1,𝑊𝑣, 𝑗 × (right, then down) for square 4′ . Thus the latter diagram commutes, concluding the proof. □ 6.3 dg-C -graded multimodules In [NP20], Naisse and Putrya provide a second notion of C -graded dg-multimodules with differential which is C -homogeneous rather than C -grading preserving: they are distinguished from the former notion by calling them dg-C -graded multimodules. The only difference lies in the differential. This section is devoted to showing that the analogous objects exists in the multicategorically graded setting. However, along the way, we develop the notion of C -commutative diagrams (see §6.3.2). While almost all succeeding work in this thesis does not rely on anything proven in this section, we will use C -commutative diagrams (especially Proposition 6.3.3) briefly in the discussion of duality (§8.2) and very minimally in the proof of properties of unified projectors (§8.3). The author suggests proceeding to Chapter 7 and referring back to this section as necessary. Definition 6.3.1. Assume 𝐴1, . . . , 𝐴𝑘 , 𝐵 are C -graded algebras. A dg-C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)- multimodule (𝑀, 𝑑𝑀) is a Z × C -graded (𝐴1, . . . , 𝐴𝑘 ; 𝐵)-multimodule 𝑀 = (cid:201) with a homogeneous differential 𝑑𝑀, written (cid:205) 𝑗 𝑑 𝑗 𝑀, satisfying 𝑑𝑀 ◦ 𝑑𝑀 = 0.1 Again, we assume 𝑛∈Z,𝑔∈C 𝑀 𝑛 𝑔 along 1Really, we could have written 𝑑𝑀 ◦C 𝑑𝑀 = 0, but clearly this is the case if and only if 𝑑𝑀 ◦ 𝑑𝑀 = 0, since 𝛾 does not take 0 for a value. 110 that the left and right action preserves homological grading. A map of dg-C -graded multimodules 𝑓 : 𝑀 → 𝑁 means a K-linear map which preserves homological grading and C -graded commutes with the differentials. To be explicit, since 𝑑𝑀 = (cid:205) 𝑗 𝑑 𝑗 𝑀 is homogeneous, we know that it is K-linear, the sum is finite, and for all 𝑚 ∈ 𝑀 it satisfies (i) 𝑑 𝑗 𝑀(𝑀 𝑛 𝑔 ) ⊂ 𝑀 𝑛+1 𝜑 𝑗 (𝑔) if 𝑔 ∈ D 𝑗 and 𝑑 𝑗 𝑀(𝑀 𝑛 𝑔 ) = 0 otherwise, (ii) 𝑑 𝑗 (cid:12) (cid:174)𝑎(cid:12) 𝑀(𝜌𝐿( (cid:174)𝑎, 𝑚) = 𝛽 (cid:174)𝑒, 𝑗 ((cid:12) (cid:12) ,|𝑚|)−1𝜌𝐿( (cid:174)𝑎, 𝑑 𝑗 𝑀(𝑚)) for all (cid:174)𝑎 ∈ (𝐴1, . . . , 𝐴𝑘 ), and (iii) 𝑑 𝑗 𝑀(𝜌𝑅(𝑚, 𝑏)) = 𝛽 𝑗,𝑒(|𝑚| ,|𝑏|)𝜌𝑅(𝑑 𝑗 𝑀(𝑚), 𝑏) for all 𝑏 ∈ 𝐵. Note that we do not require maps of dg-C -graded multimodules to preserve C -grading. The rest of this section is devoted to understanding what we mean by C -graded commutativity; see [NP20] for more details. A commutativity system on {I , Φ} is a collection (cid:110)(cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ (cid:0)I 𝑚(cid:1)2 × (cid:0)I 𝑚(cid:1)2(cid:111) T = (that is, each of 𝑖, 𝑗, 𝑖′, 𝑗 ′ may be 𝑚-vectors) such that • if (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T, then (a) 𝜑 𝑗◦𝑖 = 𝜑 𝑗 ′◦𝑖′, and (b) (cid:0)(𝑖′, 𝑗 ′), (𝑖, 𝑗)(cid:1) ∈ T and • for any 𝑘 ≥ 1, if Ä(𝑖1, 𝑗1), (𝑖′ 1, 𝑗 ′ 1)ä , . . . , Ä(𝑖𝑘 , 𝑗𝑘 ), (𝑖′ 𝑘 , 𝑗 ′ 𝑘 )ä , (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T, then (cid:0)((𝑖1, . . . , 𝑖𝑘 ) • 𝑖, ( 𝑗1, . . . , 𝑗𝑘 ) • 𝑗), ((𝑖′ 1, . . . , 𝑖′ 𝑘 ) • 𝑖′, ( 𝑗 ′ 1, . . . , 𝑗 ′ 𝑘 ) • 𝑗 ′)(cid:1) ∈ T. We abbreviate the last requirement to Ä((cid:174)𝑖, (cid:174)𝑗), ((cid:174)𝑖′, (cid:174)𝑗 ′)ä , (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T =⇒ ((cid:174)𝑖 • 𝑖, (cid:174)𝑗 • 𝑗). 111 For simplicity of exposition, assume 𝑖, 𝑗, 𝑖′, 𝑗 ′ are single-entry. To witness the commutativity system, we introduce a collection of scalars for each 𝑋1, . . . , 𝑋𝑘 , 𝑌 ∈ Ob(C ) and (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T, satisfying 𝜏 (cid:174)𝑋→𝑌 𝑖,𝑖′ 𝑗, 𝑗 ′ ∈ K× (i) 𝜏 (cid:174)𝑋→𝑌 𝑖,𝑖′ 𝑗, 𝑗 ′ = 1 whenever 𝑗 ◦ 𝑖 = 𝑗 ′ ◦ 𝑖′, and Ñ é−1 (ii) 𝜏 (cid:174)𝑋→𝑌 𝑖,𝑖′ 𝑗, 𝑗 ′ = 𝜏 (cid:174)𝑋→𝑌 𝑖′,𝑖 𝑗 ′, 𝑗 for each (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T. If (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∉ T, then we declare 𝜏 𝑖,𝑖′ 𝑗, 𝑗 ′ to be zero. We will write 𝜏 (cid:174)𝑋→ (cid:174)𝑌 for the scalar witness when 𝑚 ≠ 1—the above definition of 𝜏 extends to the case where 𝑖, 𝑗, 𝑖′, 𝑗 ′ are vectors, requiring (cid:174)𝑖,(cid:174)𝑖′ (cid:174)𝑗, (cid:174)𝑗 ′ = 1 whenever (cid:174)𝑗 ◦ (cid:174)𝑖 = (cid:174)𝑗 ′ ◦ (cid:174)𝑖′, interpreted correctly. As earlier, we write 𝜏 𝑖,𝑖′ 𝑗, 𝑗 ′ 𝜏 (cid:174)𝑋→ (cid:174)𝑌 (cid:174)𝑖,(cid:174)𝑖′ (cid:174)𝑗, (cid:174)𝑗 ′ whenever 𝑔 : (cid:174)𝑋 → 𝑌 . Finally, we say that a commutativity system T is compatible with a shifting (𝑔) to mean 𝜏 (cid:174)𝑋→𝑌 𝑖,𝑖′ 𝑗, 𝑗 ′ 2-system through 𝜏 if two equations are satisfied. The first is 𝜏 (cid:174)𝑖1•𝑖1,(cid:174)𝑖2•𝑖2 (cid:174)𝑗1• 𝑗1, (cid:174)𝑗2• 𝑗2 (𝑔′𝑔)Ξ 𝑖1,(cid:174)𝑖1 𝑗1, (cid:174)𝑗1 which translates to the following diagram. (𝑔′𝑔)𝛽 (cid:174)𝑗1◦(cid:174)𝑖1, 𝑗1◦𝑖1 (𝑔′, 𝑔) = Ξ 𝑖2,(cid:174)𝑖2 𝑗2, (cid:174)𝑗2 (𝑔′𝑔)𝛽 (cid:174)𝑗2◦(cid:174)𝑖2, 𝑗2◦𝑖2 (𝑔′, 𝑔)𝜏(cid:174)𝑖1,(cid:174)𝑖2 (cid:174)𝑗1, (cid:174)𝑗2 (𝑔′)𝜏𝑖1,𝑖2 𝑗1, 𝑗2 (𝑔) (6.3.1) 𝛽 (cid:174)𝑗1◦(cid:174)𝑖1, 𝑗1◦𝑖1 (𝑔′, 𝑔) (cid:174)𝑗1•1 (cid:174)𝑖1•1 (cid:174)1• 𝑗1 (cid:174)1•𝑖1 (𝑔′𝑔) Ξ𝑖1, (cid:174)𝑖1 𝑗1, (cid:174)𝑗1 (cid:174)𝑗1•1 (cid:174)1• 𝑗1 (cid:174)𝑖1•1 (cid:174)1•𝑖1 (cid:174)𝑗1 (cid:174)𝑖1 𝑔′ 𝑗1 𝑖1 𝑔 (𝑔′) 𝜏𝑖1, 𝑖2 𝑗1, 𝑗2 (𝑔) 𝜏 (cid:174)𝑖1, (cid:174)𝑖2 (cid:174)𝑗1, (cid:174)𝑗2 𝑔′ 𝑔 𝑔′ 𝑔 (𝑔′𝑔) 𝜏 (cid:174)𝑖1•𝑖1, (cid:174)𝑖2•𝑖2 (cid:174)𝑗1• 𝑗1, (cid:174)𝑗2• 𝑗2 (cid:174)𝑗2 (cid:174)𝑖2 𝑔′ 𝑗2 𝑖2 𝑔 𝛽 (cid:174)𝑗2◦(cid:174)𝑖2, 𝑗2◦𝑖2 (𝑔′, 𝑔) (cid:174)𝑗2•1 (cid:174)𝑖2•1 (cid:174)1• 𝑗2 (cid:174)1•𝑖2 (𝑔′𝑔) Ξ 𝑖2,(cid:174)𝑖2 𝑗2, (cid:174)𝑗2 (cid:174)𝑗2•1 (cid:174)1• 𝑗2 (cid:174)𝑖2•1 (cid:174)1•𝑖2 𝑔′ 𝑔 𝑔′ 𝑔 112 The second equation establishes consistency between 𝜏 and Ξ: we require that 𝜏 (cid:174)Id•𝑖, (cid:174)𝑗•Id (cid:174)𝑗•Id, (cid:174)Id•𝑖 = ΞId, (cid:174)𝑗 𝑖, (cid:174)Id . Ξ−1 𝑖, (cid:174)Id Id, (cid:174)𝑗 (6.3.2) In particular, notice that in order to conclude that Ä( (cid:174)Id • 𝑖, (cid:174)𝑗 • Id), ( (cid:174)𝑗 • Id, (cid:174)Id • 𝑖)ä ∈ T, where any Id may be replaced by any element of IId, it is sufficient if ((𝑖′, Id), (Id, 𝑖′)) ∈ T for any 𝑖′ ∈ I —this will clearly be the case in the G -graded setting. This equation translates to the following diagram. Ä (cid:174)𝑗 • Idä ◦ Ä (cid:174)Id • 𝑖ä 𝜏 (cid:174)Id•𝑖, (cid:174)𝑗•Id (cid:174)𝑗•Id, (cid:174)Id•𝑖 Ä (cid:174)Id • 𝑖ä Ä (cid:174)𝑗 • Idä ◦ Ξ−1 𝑖, (cid:174)Id Id, (cid:174)𝑗 Ä (cid:174)𝑗 ◦ (cid:174)Idä • (cid:0)Id ◦ 𝑖(cid:1) Ä (cid:174)Id ◦ (cid:174)𝑗ä • (cid:0)𝑖 ◦ Id(cid:1) ΞId, (cid:174)𝑗 𝑖, (cid:174)Id 6.3.1 G -graded commutativity As before, one last time, we will describe the G -graded setting before passing on to generalities. We will not consider dg-G -graded multimodules explicitly, but we can construct them using the information of this section. See [NP20] for more generalities of these objects. We’ll write Δ𝑣 for (Δ, 𝑣) ∈ I to reduce the number of nested ordered pairs. We’ll describe the non-vectorized setting first. Let T denote the collection of all pairs ß(cid:16) (Δ 𝑣1 1 , Δ 𝑣2 2 ), (Δ ′𝑣′ 1 , Δ 1 ′𝑣′ 2 ) 2 (cid:17)™ for which • there exists a locally vertical change of chronology 𝐻 : Δ2 ◦ Δ1 ⇒ Δ′ 2 ◦ Δ′ 1, and • 𝑣1 = 𝑣′ 1. 2 and 𝑣2 = 𝑣′ Similarly, in the vecotrized setting, changes of chronology 𝐻ℓ : Δ2,ℓ ◦ Δ1,ℓ ⇒ Δ′ 1. Notice that T satisfies the criteria of a commutativity system since cobordisms which differ only with respect to a 2 and (cid:174)𝑣2 = (cid:174)𝑣′ 2,ℓ ◦ Δ′ Å ((cid:174)Δ(cid:174)𝑣1 1 , (cid:174)Δ(cid:174)𝑣2 2 ), ((cid:174)Δ ′(cid:174)𝑣′ 1 , (cid:174)Δ 1 ã ′(cid:174)𝑣′ 2 ) 2 1,ℓ for all ℓ and (cid:174)𝑣1 = (cid:174)𝑣′ is in T if there are locally vertical locally vertical change of chronology induce the same G -grading shift, locally vertical changes of chronology are invertible, and locally vertical changes of chronology are well behaved with respect to horizontal composition of cobordisms. 113 Next, we set 𝜏 (cid:174)𝑥→𝑦 (Δ1,𝑣1),(Δ′ (Δ2,𝑣2),(Δ′ again, where 𝐻 is the locally vertical change of chronology 𝐻 : Δ2 ◦Δ1 ⇒ Δ′ = 𝜄((cid:174)𝑥 𝐻𝑦)𝜆(𝑣2, 𝑣1) 1) 1,𝑣′ 2) 2,𝑣′ 2 ◦Δ′ 1. In the vectorized setting, we set 𝜏 (cid:174)𝑥→(cid:174)𝑦 ((cid:174)Δ1,(cid:174)𝑣1),((cid:174)Δ′ ((cid:174)Δ2,(cid:174)𝑣2),((cid:174)Δ′ 1) 1,(cid:174)𝑣′ 2) 2,(cid:174)𝑣′ (cid:214) = ℓ 𝜄((cid:174)𝑥ℓ (𝐻ℓ)𝑦ℓ )𝜆(𝑉2, 𝑉1) where 𝑉2, 𝑉1 denote the sums of the entries of (cid:174)𝑣2 and (cid:174)𝑣1 respectively. We will write 𝜏 = 𝜏1𝜏2, for 𝜏1 the part coming from the change of chronology and 𝜏2 the other. Notice that if (Δ2, 𝑣2)◦(Δ1, 𝑣1) = (Δ′ 2)◦(Δ1,′ 𝑣′ 1 so 𝜆(𝑣2, 𝑣1) = 1, hence 𝜏 (cid:174)𝑥→𝑦 2, 𝑣′ 1), then 𝐻 is the identical change of chronology, 1 ⇒ Δ2 ◦ Δ1 is = 1. Also, if 𝐻 : Δ′ 2 ◦ Δ′ and 𝑣1 = 𝑣′ 2 = 𝑣2 = 𝑣′ (Δ1,𝑣1),(Δ′ (Δ2,𝑣2),(Δ′ 1) 1,𝑣′ 2) 2,𝑣′ also a locally vertical change of chronology (guaranteed to exist by the existence of 𝐻) then = 𝜄((cid:174)𝑥 𝐻 𝑦)𝜆(𝑣′ 2, 𝑣′ 1) = 𝜄((cid:174)𝑥 𝐻𝑦)−1𝜆(𝑣1, 𝑣2) = Ö è−1 𝜏 (cid:174)𝑥→𝑦 (Δ1,𝑣1),(Δ′ (Δ2,𝑣2),(Δ′ 1) 1,𝑣′ 2) 2,𝑣′ 𝜏 (cid:174)𝑥→𝑦 (Δ′ 1,𝑣′ (Δ′ 2,𝑣′ 1),(Δ1,𝑣1) 2),(Δ2,𝑣2) as desired. Proposition 6.3.2. This commutativity system T is compatible with the G -grading shifting 2-system defined previously, through the scalars 𝜏. Proof. The validity of (6.3.2) is simple: recall that IId consists of elements (1𝐷∧, (0, 0)) for any planar arc diagram 𝐷. Thus 𝜏 (cid:174)𝑥→𝑦 Id•(Δ,𝑣),((cid:174)Δ,(cid:174)𝑣•Id) ((cid:174)Δ,(cid:174)𝑣)•Id, (cid:174)Id•(Δ,𝑣) = 𝜄((cid:174)𝑥 𝐻𝑦)𝜆(𝑉, 𝑣) where 𝑉 is the sum of entries of (cid:174)𝑣 and 𝐻 : (((cid:174)Δ, (cid:174)𝑣) • Id) ◦ ( (cid:174)Id • (Δ, 𝑣)) ⇒ ( (cid:174)Id • (Δ, 𝑣)) ◦ (((cid:174)Δ, (cid:174)𝑣) • Id). On the other hand, ΞId,((cid:174)Δ,(cid:174)𝑣) (Δ,𝑣), (cid:174)Id Ξ−1 (Δ,𝑣), (cid:174)Id Id,((cid:174)Δ,(cid:174)𝑣) Ä𝜄((cid:174)𝑥 𝐻′′ 𝑦 )𝜆(𝑉, 𝑣)ä · Ä𝜄((cid:174)𝑥 𝐻′ 𝑦)𝜆((0, 0)(0, 0))−1ä = 114 where (((cid:174)Δ, (cid:174)𝑣) • Id) ◦ ( (cid:174)Id • (Δ, 𝑣)) 𝐻′ ==⇒ (((cid:174)Δ, (cid:174)𝑣) ◦ (cid:174)Id) • (Id ◦ (Δ, 𝑣)) = ( (cid:174)Id ◦ ((cid:174)Δ, (cid:174)𝑣)) • ((Δ, 𝑣) ◦ Id) 𝐻′′ ===⇒ ( (cid:174)Id • (Δ, 𝑣)) ◦ (((cid:174)Δ, (cid:174)𝑣) • Id). Since 𝐻 and 𝐻′′ ◦ 𝐻′ are locally vertical changes of chronology with the source and target, Proposition 3.1.3 implies that 𝜄((cid:174)𝑥 𝐻𝑦) = 𝜄((cid:174)𝑥(𝐻′′ ◦ 𝐻′)𝑦) = 𝜄((cid:174)𝑥 𝐻′′ 𝑦 )𝜄((cid:174)𝑥 𝐻′ 𝑦), so equation (6.3.2) is satisfied. To check equation (6.3.1), we apply familiar arguments. Actually, the computation is fairly simple compared to the previous proofs of this type. On one hand, ignoring Z × Z-degree to start, consider the diagram (cid:174)𝑗1 (cid:174)𝑖1 (cid:16) 𝛽 (cid:174)𝑗1◦(cid:174)𝑖1, 𝑗1◦𝑖1 (cid:17) 1 𝑗1 𝑖1 Ü ê 𝜏 (cid:174)𝑖1, (cid:174)𝑖2 (cid:174)𝑗1, (cid:174)𝑗2 1 Ñ é 𝜏𝑖1, 𝑖2 𝑗1, 𝑗2 1 (cid:174)𝑗2 (cid:174)𝑖2 (cid:16) 𝛽 (cid:174)𝑗2◦(cid:174)𝑖2, 𝑗2◦𝑖2 (cid:17) 1 𝑗2 𝑖2 (cid:174)𝑗1•1 (cid:174)𝑖1•1 (cid:174)1• 𝑗1 (cid:174)1•𝑖1 (cid:174)𝑗2•1 (cid:174)𝑖2•1 (cid:174)1• 𝑗2 (cid:174)1•𝑖2 Ü ê Ξ𝑖1, (cid:174)𝑖1 𝑗1, (cid:174)𝑗1 1 (cid:174)𝑗1•1 (cid:174)1• 𝑗1 (cid:174)𝑖1•1 (cid:174)1•𝑖1 Ü ê Ξ 𝑖2,(cid:174)𝑖2 𝑗2, (cid:174)𝑗2 1 Ü 𝜏 ê (cid:174)𝑖1•𝑖1, (cid:174)𝑖2•𝑖2 (cid:174)𝑗1• 𝑗1, (cid:174)𝑗2• 𝑗2 1 (cid:174)𝑗2•1 (cid:174)1• 𝑗2 (cid:174)𝑖2•1 (cid:174)1•𝑖2 where 𝑖1 = Δ1, 𝑗1 = Δ2, 𝑖2 = Δ′ 2, and so on. The two paths trace out changes of chronology with the same source and target, so we conclude that the contributions of 𝜏1, Ξ1, and 𝛽1 from 1, 𝑗2 = Δ′ equation (6.3.1) agree on the nose. On the other hand, since (cid:174)Δ1 ◦ (cid:174)Δ2 and (cid:174)Δ′ 1 ◦ (cid:174)Δ′ 2, as well as Δ2 ◦ Δ1 and (Δ′ a locally vertical changes of chronology, plus 𝑣1 + 𝑣2 = 𝑣′ 2 ◦ Δ′ 1 and 𝑉1 + 𝑉2 = 𝑉 ′ 1), differ only by 1, it is easy to 2 + 𝑉 ′ 2 + 𝑣′ find that Ä𝛽((cid:174)Δ2◦(cid:174)Δ1,(cid:174)𝑣1+(cid:174)𝑣2),(Δ2◦Δ1,𝑣1+𝑣2) ä 2,3,4 (cid:16) = 115 𝛽((cid:174)Δ′ 2◦(cid:174)Δ′ 1,(cid:174)𝑣′ 1+(cid:174)𝑣′ 2 ),(Δ′ 2◦Δ′ 1,𝑣′ 1+𝑣′ 2 (cid:17) ) . 2,3,4 If these conditions were not true, then the 𝜏 maps involved would be zero, and equation (6.3.1) would hold trivially. Moreover, we compute Ö and 𝜏((cid:174)Δ1•Δ1,𝑉1+𝑣1),((cid:174)Δ′ ((cid:174)Δ2•Δ2,𝑉2+𝑣2),((cid:174)Δ′ 1•Δ′ 2•Δ′ 1,𝑉 ′ 2,𝑉 ′ 1) 1+𝑣′ 2) 2+𝑣′ on one side, and è 2 = 𝜆(𝑉2 + 𝑣2, 𝑉1 + 𝑣1) = 𝜆(𝑉2, 𝑉1) · 𝜆(𝑉2, 𝑣1) · 𝜆(𝑣2, 𝑉1) · 𝜆(𝑣2, 𝑣1) Ñ é Ξ(Δ1,𝑣1),((cid:174)Δ1,(cid:174)𝑣1) (Δ2,𝑣2),((cid:174)Δ1,(cid:174)𝑣2) 2 Ö è 𝜏((cid:174)Δ1,(cid:174)𝑣1),((cid:174)Δ′ ((cid:174)Δ2,(cid:174)𝑣2),((cid:174)Δ′ 1) 1,(cid:174)𝑣′ 2) 2,(cid:174)𝑣′ Ñ 𝜏(Δ1,𝑣1),(Δ′ (Δ2,𝑣2),(Δ′ 1) 1,𝑣′ 2) 2,𝑣′ 2 é 2 = 𝜆(𝑉1, 𝑣2) = 𝜆(𝑉2, 𝑉1), = 𝜆(𝑣2, 𝑣1), and Ö è Ξ(Δ′ (Δ′ 1,𝑣′ 2,𝑣′ 1),((cid:174)Δ′ 2),((cid:174)Δ′ 1) 1,(cid:174)𝑣′ 2) 1,(cid:174)𝑣′ 2 = 𝜆(𝑉 ′ 1, 𝑣′ 2) = 𝜆(𝑉2, 𝑣1) on the other. Since 𝜆(𝑉1, 𝑣2) = 𝜆(𝑣2, 𝑉1)−1, these computations tell us that the contributions of 𝜏2, Ξ2, and 𝛽2,3,4 from equation (6.3.1) also agree on the nose, concluding the proof. □ There may be other choices of commutativity systems compatible with the G -grading shifting 2-system. However, this doesn’t matter so much: the existence of a commutativity system is more important than the commutativity system itself. 6.3.2 Generalities of commutativity systems As before, we obtain natural transformations 𝜑 𝑗◦𝑖 ⇒ 𝜑 𝑗 ′◦𝑖′ 𝜑 𝑗◦𝑖(𝑀) → 𝜑 𝑗 ′◦𝑖′(𝑀) 𝑚 ↦→ 𝜏 𝑖,𝑖′ 𝑗, 𝑗 ′ (|𝑚|)𝑚 116 or, more generally, 𝜑 (cid:174)𝑗◦(cid:174)𝑖 ⇒ 𝜑 (cid:174)𝑗 ′◦(cid:174)𝑖′ given by 𝜑 (cid:174)𝑗◦(cid:174)𝑖(𝑀1, . . . , 𝑀𝑘 ) → 𝜑 (cid:174)𝑗 ′◦(cid:174)𝑖′(𝑀1, . . . , 𝑀𝑘 ) (cid:174)𝑚 ↦→ 𝜏 (cid:174)𝑖,(cid:174)𝑖′ (cid:174)𝑗, (cid:174)𝑗 ′ (cid:12) (cid:174)𝑚(cid:12) ((cid:12) (cid:12)) (cid:174)𝑚 Then, the compatibility equations (6.3.1) and (6.3.2) imply the following commutative diagrams in categories of G -graded multimodules. 𝜑 (cid:174)𝑗1◦(cid:174)𝑖1 (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝜑 𝑗1◦𝑖1(𝑀) 𝛽 (cid:174)𝑗1◦(cid:174)𝑖1, 𝑗1◦𝑖1 𝜑( (cid:174)𝑗1◦(cid:174)𝑖1)•( 𝑗1◦𝑖1) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) 𝜏 (cid:174)𝑖1,(cid:174)𝑖2 (cid:174)𝑗1, (cid:174)𝑗2 ⊗𝜏𝑖1,𝑖2 𝑗1, 𝑗2 𝜑 (cid:174)𝑗2◦(cid:174)𝑖2 (𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝜑 𝑗2◦𝑖2(𝑀) 𝛽 (cid:174)𝑗2◦(cid:174)𝑖2, 𝑗2◦𝑖2 𝜑( (cid:174)𝑗2◦(cid:174)𝑖2)•( 𝑗2◦𝑖2) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) Ξ 𝑖1,(cid:174)𝑖1 𝑗1, (cid:174)𝑗1 Ξ 𝑖2,(cid:174)𝑖2 𝑗2, (cid:174)𝑗2 𝜑( (cid:174)𝑗1• 𝑗1)◦((cid:174)𝑖1•𝑖1) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) 𝜏 (cid:174)𝑖1•𝑖1,(cid:174)𝑖2•𝑖2 (cid:174)𝑗1• 𝑗1, (cid:174)𝑗2• 𝑗2 𝜑( (cid:174)𝑗2• 𝑗2)◦((cid:174)𝑖2•𝑖2) (cid:0)(𝑀1, . . . , 𝑀𝑘 ) ⊗ 𝑀(cid:1) 𝜏 (cid:174)Id•𝑖, (cid:174)𝑗•Id (cid:174)𝑗•Id, (cid:174)Id•𝑖 𝜑 ( (cid:174)𝑗•Id)◦( (cid:174)Id•𝑖) (𝑀) Ξ𝑖, (cid:174)Id Id, (cid:174)𝑗 𝜑 ( (cid:174)𝑗◦ (cid:174)Id)•(Id◦𝑖) (𝑀) 𝜑 (cid:174)𝑗•𝑖(𝑀) 𝜑 ( (cid:174)Id•𝑖)◦( (cid:174)𝑗•Id) (𝑀) ΞId, (cid:174)𝑗 𝑖,Id 𝜑 ( (cid:174)Id◦ (cid:174)𝑗)•(𝑖◦Id) Consider a diagram of purely homogeneous maps, with degrees pictured. 𝑀11 𝑓1∗ 𝑓∗1 𝑖′ 𝑖 𝑀12 𝑀21 𝑗 ′ 𝑗 𝑓∗2 𝑓2∗ 𝑀22 We say that the diagram is C -graded commutative if (cid:0)(𝑖, 𝑗), (𝑖′, 𝑗 ′)(cid:1) ∈ T, and (cid:0) 𝑓2∗ ◦C 𝑓∗1(cid:1) = 𝜏 𝑖,𝑖′ 𝑗, 𝑗 ′ (cid:0) 𝑓∗2 ◦C 𝑓1∗ (cid:1) . Note that (cid:0) 𝑓∗2 ◦C 𝑓1∗ (cid:1) has degree 𝑗 ′ ◦ 𝑖′ and (cid:0) 𝑓2∗ ◦C 𝑓∗1(cid:1) has degree 𝑗 ◦ 𝑖, so 𝜏 𝑖,𝑖′ 𝑗, 𝑗 ′ ensures their C - degrees agree. This situation is abbreviated by including an arrow⇒as in the following proposition. 117 Proposition 6.3.3. Given C -graded commutative diagrams 𝑀11 𝑓1∗ 𝑓∗1 𝑀12 𝑀21 𝑓∗2 𝑓2∗ 𝑀22 and (𝑀12)1 (𝑀12)𝑘 ( 𝑓1∗)1 ( 𝑓∗2)1 ( 𝑓1∗)𝑘 ( 𝑓∗2)𝑘 (𝑀11)1 (𝑀22)1 , . . . , (𝑀11)𝑘 (𝑀22)𝑘 ( 𝑓∗1)1 ( 𝑓2∗)1 (𝑀21)1 ( 𝑓∗1)𝑘 ( 𝑓2∗)𝑘 (𝑀21)𝑘 the diagram (cid:0)(𝑀12)1, . . . , (𝑀12)𝑘 (cid:1) ⊗ 𝑀12 (cid:174)𝑓1∗⊗ 𝑓1∗ (cid:174)𝑓∗2⊗ 𝑓∗2 (cid:0)(𝑀11)1, . . . , (𝑀11)𝑘 (cid:1) ⊗ 𝑀11 (cid:0)(𝑀22)1, . . . , (𝑀22)𝑘 (cid:1) ⊗ 𝑀22 (cid:174)𝑓∗1⊗ 𝑓∗1 (cid:174)𝑓2∗⊗ 𝑓2∗ (cid:0)(𝑀21)1, . . . , (𝑀21)𝑘 (cid:1) ⊗ 𝑀21 is C -graded commutative. Here, (cid:174)𝑓1∗ ⊗ 𝑓1∗ is shorthand for (cid:0)( 𝑓1∗)1, . . . , ( 𝑓1∗)𝑘 (cid:1) ⊗ 𝑓1∗, and so on. Proof. This is simple, given Proposition 5.3.6 and equation (6.3.1). We drop some notation in 118 what follows; hopefully it is clear: (( (cid:174)𝑓∗2 ⊗ 𝑓∗2) ◦C ( (cid:174)𝑓1∗ ⊗ 𝑓1∗))( (cid:174)𝑚 ⊗ 𝑚) = Ξ−1 𝑖′,(cid:174)𝑖′ 𝑗 ′, (cid:174)𝑗 ′ Ä( (cid:174)𝑓∗2 ◦C (cid:174)𝑓1∗) ⊗ ( 𝑓∗2 ◦C 𝑓1∗)ä ( (cid:174)𝑚 ⊗ 𝑚) = Ξ−1 𝑖′,(cid:174)𝑖′ 𝑗 ′, (cid:174)𝑗 ′ = Ξ−1 𝑖′,(cid:174)𝑖′ 𝑗 ′, (cid:174)𝑗 ′ 𝛽−1 (cid:174)𝑗 ′◦(cid:174)𝑖′, 𝑗 ′◦𝑖′ ( (cid:174)𝑓∗2 ◦C (cid:174)𝑓1∗)( (cid:174)𝑚) ⊗ ( 𝑓∗2 ◦C 𝑓1∗)(𝑚) 𝛽−1 (cid:174)𝑗 ′◦(cid:174)𝑖′, 𝑗 ′◦𝑖′ 𝜏−1 (cid:174)𝑖,(cid:174)𝑖′ (cid:174)𝑗, (cid:174)𝑗 ′ 𝜏−1 𝑖,𝑖′ 𝑗, 𝑗 ′ ( (cid:174)𝑓2∗ ◦C (cid:174)𝑓∗1)( (cid:174)𝑚) ⊗ ( 𝑓2∗ ◦C 𝑓∗1)(𝑚) = 𝜏−1 (cid:174)𝑖•𝑖,(cid:174)𝑖′•𝑖′ (cid:174)𝑗• 𝑗, (cid:174)𝑗 ′• 𝑗 ′ = 𝜏−1 (cid:174)𝑖•𝑖,(cid:174)𝑖′•𝑖′ (cid:174)𝑗• 𝑗, (cid:174)𝑗 ′• 𝑗 ′ = 𝜏−1 (cid:174)𝑖•𝑖,(cid:174)𝑖′•𝑖′ (cid:174)𝑗• 𝑗, (cid:174)𝑗 ′• 𝑗 ′ 𝛽−1 (cid:174)𝑗◦(cid:174)𝑖, 𝑗◦𝑖 ( (cid:174)𝑓2∗ ◦C (cid:174)𝑓∗1)( (cid:174)𝑚) ⊗ ( 𝑓2∗ ◦C 𝑓∗1)(𝑚) Ä( (cid:174)𝑓2∗ ◦C (cid:174)𝑓∗1) ⊗ ( 𝑓2∗ ◦C 𝑓∗1)ä ( (cid:174)𝑚 ⊗ 𝑚) Ξ−1 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 Ξ−1 𝑖,(cid:174)𝑖 𝑗, (cid:174)𝑗 (( (cid:174)𝑓2∗ ⊗ 𝑓2∗) ◦C ( (cid:174)𝑓∗1 ⊗ 𝑓∗1))( (cid:174)𝑚 ⊗ 𝑚). □ 119 CHAPTER 7 AN INVARIANT OF DISKULAR TANGLES In this chapter, we describe an invariant of diskular tangles. In §7.1, we describe useful compu- tational tools necessary to successive work, inspired by [BN07] but paying particular attention to the “simplification” of G -grading shifts. We remark that, as in [NP20], our G -grading system is a little too sensitive for the G -graded dg-multimodule we associate to a diskular tangle to be invariant under each Reidemeister move. However, we also describe a procedure (important to the results of chapter 8) which collapses G -grading to a 𝑞-grading, in which case we obtain an honest tangle invariant. The work here is motivated by and serves as a generalization of [NP20]. Recall that we write Kom(·) to indicate the category of complexes which are bounded below in homological degree and of finite rank in each quantum or G degree. 7.1 Quick computations in unified Khovanov homology To begin, we will describe a few tools which will allow for quick computations in the homotopy ä. In particular, we hope to use the methods category of G -graded 𝐻𝑛-modules, Kom Ä𝐻𝑛ModG 𝑅 introduced in [BN07], but must develop others to deal with problems posed by G -shifts. 7.1.1 Delooping As an internal check, we can derive a formula for delooping in the current setting. A birth (cid:17) : 𝜑 (𝑅) → 𝑉, since F (∅) = 𝑅 and F (⃝) = 𝑉. Notice that this G-grading shift functor has only the effect of adding (1, 0) in the second coordinate : ∅ → ⃝ induces a graded map F (cid:16) (free loops are ignored in the first coorinate): 𝜑 (cid:27) {1, 0}. So we have a graded map Similarly, (cid:16) F (cid:17) : 𝑅{1, 0} → 𝑉 . (cid:16) F • (cid:17) : 𝑅{0, −1} → 𝑉 is a graded map. The grading shift functors {𝑢, 𝑣} have clear inverses given by {−𝑢, −𝑣}. This fact, together with similar analysis on graded maps induced by deaths, yields the following array 120 of graded maps: • F (∅){1, 0} F (⃝) ⊕ F (⃝) (7.1.1) F (∅){0, −1} • It might seem pedantic, but we note that the arrows on the left-hand side of (7.1.1) should be precomposed with the isomorphisms coming from natural transformations of grading shift functors and Id ⇒ {1, 0} ◦ {−1, 0} Id ⇒ {0, −1} ◦ {0, 1} respectively, so that the maps on the left are graded with respect to our conventions. We will neglect writing these isomorphisms outside of special situations (e.g., the proof of Theorems 8.1.5 and 8.2.3). Proposition 7.1.1 (Delooping). F (⃝) (cid:27) F (∅){0, −1} ⊕ F (∅){1, 0}. Proof. This follows directly from the definition of F . For example, the composition shown in diagram 7.1.1 reads Ñ é Ñ F • + F • é Ñ é = F . One can verify this by checking that a dotted cylinder, followed by a positive death, and then a birth maps 𝑣+ to 𝑣+ and 𝑣− to zero, while a positive death, followed by a birth, and then a dotted cylinder maps 𝑣+ to zero and 𝑣− to 𝑣−. The other composition is also the identity: this amounts to showing that Ç å (cid:32) F = F (cid:33) • • = 0, and F Ç å • = 1. That is, the tube-cutting and sphere relations hold in the category 𝐻𝑛ModG 𝑅 . 121 We should expect the gradings as they are since deg𝑅(𝑣+) = (1, 0) and deg𝑅(𝑣−) = (0, −1), with 𝑉 = 𝑅⟨𝑣+, 𝑣−⟩. □ 7.1.2 Simplifying grading shift functors In the even setting, delooping and Gaussian elimination allowed us to perform quick compu- tations. To perform similar computations in the unified setting, we need to develop a system for simplifying G -shifts. In the best cases, this means that 𝑊 consists of no ambiguous saddles, and is equivalent to a grading shift supported entirely in the Z × Z component; for example, we previously used that 𝜑 (cid:27) {1, 0}. Usually this is not the case. Instead, given a cobordism 𝑊 : 𝑡 → 𝑡′, we’d like to equate 𝜑(𝑊,𝑣) with 𝜑( ˇ𝑊,𝑢) for some 𝑢 ∈ Z × Z where ˇ𝑊 is minimal. Recall that if 𝑊 is not minimal, it fails to be so up to some addition of tubes. Therefore, to approach the problem of simplify grading shift functors, it makes sense to ask how 𝜑(𝑊,𝑣) behaves under tube-cutting. Proposition 7.1.2. Let 𝑊 : 𝑡 → 𝑡′ be a cobordism. There is a minimal cobordism ˇ𝑊 : 𝑡 → 𝑡′ which is isotopic to 𝑊 outside of finitely many tubes. Denote the number of tubes in 𝑊 by 𝜏𝑊 . Then 𝜑(𝑊,𝑣) (cid:27) 𝜑( ˇ𝑊,𝑣+𝜏𝑊 (−1,−1)). Proof. Any tube in 𝑊 is either unambiguous (it is a split followed by a merge or vice versa) or it is ambiguous (it is impossible to determine the order of elementary cobordisms which constitute the tube without a given closure). Consider the (locally vertical) change of chronology 𝐻 : 𝑊 ⇒ 𝑊 ′ which changes all ambiguous tubes into unambiguous tubes, e.g., 𝐻 : ⇒ wherever ambiguous tubes are present. From our analysis earlier, there is an induced natural transformation 𝜑𝐻 : 𝜑𝑊 ⇒ 𝜑𝑊 ′. Note that deg(1𝑎𝑊1𝑏) = deg(1𝑎𝑊 ′1𝑏) since any tube in 𝑊 corresponds to the addition of (−1, −1) in degree on any closure, ambiguous or not. This implies that 𝜑(𝑊,𝑣) (cid:27) 𝜑(𝑊 ′,𝑣). Since each tube in 𝑊 ′ is unambiguous, we know that each tube in 𝑊 ′ acts as a degree (−1, −1) shift, so the result follows. □ 122 A consequence of this proposition is that all grading shift functors have inverses, not just {𝑢, 𝑣}. Corollary 7.1.3. For any pair (𝑊 : 𝑡 → 𝑡′, 𝑣), 𝜑(𝑊,𝑣) has a left inverse 𝜑−1 (𝑊,𝑣) = 𝜑(𝑊,−𝑣+𝜏𝑊◦𝑊 (1,1)) (where 𝑊 : 𝑡′ → 𝑡 is the mirror image of 𝑊) in the sense that 𝜑−1 (𝑊,𝑣) ◦ 𝜑(𝑊,𝑣) (cid:27) 𝜑1𝑡 . Proof. If the composition 𝑊 ◦ 𝑊 produces any tubes, the contribution by these tubes on the second coordinate are killed by the addition of the term 𝜏𝑊◦𝑊 (1, 1). □ Example. An elementary saddle cobordism (cid:16) F (cid:17) : 𝜑 F : (cid:16) → induces the graded map (cid:17) → F (cid:16) (cid:17) . Consider the isomorphism induced by change of chronology: 𝜑𝐻 : Id ⇒ 𝜑−1 ◦ 𝜑 . Then, precomposing with 𝜑𝐻, the saddle can be reinterpreted as the following graded map. (cid:16) F (cid:17) ◦ 𝜑𝐻 : F (cid:16) (cid:17) → 𝜑−1 F (cid:16) (cid:17) . We compute that 𝜑−1 = 𝜑Å , (1,1) 𝜑−1 Å ã since ◦ produces a tube. In general, ã = 𝜑Å , (𝑢,𝑣) ã. , (1−𝑢,1−𝑣) Remark 7.1.4. Returning to diagram (7.1.1), we see that the natural transformations of grading shifting functors actually take the forms and 𝜑𝐻 : Id ⇒ 𝜑−1 • ◦ 𝜑 • (cid:27) {1, 0} ◦ {−1, 0} 𝜑𝐻 : Id ⇒ 𝜑−1 ◦ 𝜑 (cid:27) {0, −1} ◦ {0, 1}. 123 The dismissal of free loops by the G-shifting system leads to another possibility for simplification of grading shift functors. We will frequently use the following simplification while cooking up projectors; see [NP20] for a proof. Proposition 7.1.5. Suppose 𝑊 : 𝑡 → 𝑡′ is a cobordism and 𝑡 contains a free loop ℓ. Then there is a natural isomorphism 𝜑(𝑊,(𝑢,𝑣)) (cid:27) 𝜑(𝑊 ′,(𝑢−1,𝑣)) by 𝑚 ↦→ 𝜆((1, 0), deg(1𝑎𝑊1𝑏))𝑚 where 𝑊 ′ : 𝑡 − ℓ → 𝑡′ is given by gluing a birth under the free loop in 𝑡, and 𝑚 ∈ 𝑀𝑔:𝑎→𝑏. If, on the other hand, 𝑡′ contains the free loop, the natural isomorphism is on the nose: 𝜑(𝑊,(𝑢,𝑣)) (cid:27) 𝜑(𝑊 ′′,(𝑢,𝑣−1)) and 𝑊 ′′ : 𝑡 → 𝑡′ − ℓ is given by gluing a death above 𝑊. Example. Here is a way we may use the preceding proposition. Consider the G -grading shifting map 𝜑 (the choice of chronology is unimportant). Then Proposition 7.1.5 says that this grading shift is isomorphic to the grading shift 𝜑(𝑊 ′,(−1,0)), where 𝑊 ′ is pictured below. Of course, 𝑊 ′ is isotopic to an elementary saddle , so Proposition 7.1.2 allows us to conclude that 𝜑 (cid:27) 𝜑Ä ,(−1,0)ä. 124 The examples of this subsection illustrate a peculiarity of computations in Kom(𝐻𝑛ModG 𝑅 )— that a G -grading shift has a few different representatives. A difficulty in coming work (cf. the proof of Lemma 7.2.6) is choosing the correct representative. 7.2 Tangle invariant In this section, we finally construct an invariant of diskular tangles, motivated by and general- izing Section 6.4 of [NP20]. Suppose 𝑇 is a diskular (𝑛1, . . . , 𝑛𝑘 ; 𝑚)-tangle with 𝑐-many crossings. We will continue under the assumption that 𝑇 carries an orientation. Then 𝑇 defines an oriented (2, . . . , 2 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) 𝑐 times , 𝑛1, . . . , 𝑛𝑘 ; 𝑚)- planar arc diagram 𝐷𝑇 by replacing each crossing of 𝑇 with a new diskular region with four endpoints; consult the schematic below. ⇝ × ⇝ Denote the crossings of 𝑇 by 𝑥1, . . . , 𝑥𝑐 and define the complex Kh(𝑇) := (cid:0)Kh(𝑥1), . . . , Kh(𝑥𝑐)(cid:1) ⊗(𝐻2,...,𝐻2) F (𝐷𝑇 ) where Ç Kh Ç Kh å å Ö := Cone 𝜑 F Ç å F Å ã Ç è å −−−−−−−→ F {−1, 0}, and Ö := Cone F Ç Å ã å F ◦𝜑𝐻 −−−−−−−−−−−→ 𝜑−1 F Ç è å {0, 1}, for 𝜑𝐻 : Id ⇒ 𝜑−1 ◦ 𝜑 . Recall that the underlined entry is in homological degree zero. The reader should compare this with the unoriented case, where we have F (𝑇) (cid:27) (cid:0)F (𝑥1), . . . , F (𝑥𝑐)(cid:1) ⊗(𝐻2,...,𝐻2) F (𝐷𝑇 ) by Theorem 6.2.4. So, we would expect the following lemma. Lemma 7.2.1. For any diskular tangle 𝑇, there exists a shifting functor 𝜑 and integer ℓ such that Kh(𝑇) (cid:27) 𝜑(F (𝑇))[ℓ] 125 Proof. Recall that the dg-multimodule associated to a single crossing is given by Ç F å Ö = Cone 𝜑 F Ç å F Å ã Ç −−−−−−−→ F è å . On one hand, it is obvious that Ç Kh Ç å å (cid:27) F {−1, 0}[−1]. On the other hand, the diagram Ç F å 𝜑𝐻 𝜑−1 ◦ 𝜑 F Ç å (cid:16) F (cid:17) ◦ 𝜑𝐻 Ç å 𝜑−1 F Ñ é F Ç å 𝜑−1 F commutes tautologically. The bottom line is exactly 𝜑−1 F Ç å , so we conclude that Ç Kh å (cid:27) 𝜑−1 F Ç å {0, 1}. Then the desired result follows from the definition of Kh and Theorem 6.2.4. □ Unfortunately, Kh is not an invariant of oriented tangles in the G -graded sense; rather, Kh will be an invariant of diskular tangles up to G -grading shift (Theorem 7.2.8). We break the computation up into three lemmas of increasing difficulty. Remark 7.2.2. Notice that invariance under planar isotopy is immediately apparent in the G -graded setting, in contrast to [NP20], since F (𝐷𝑇 ) (cid:27) F (𝐷𝑇 ′) if 𝑇 ′ is obtained from 𝑇 via planar isotopy. Moreover, in our setup, we no longer have to assume 𝑇 is presented in a generic position. Lemma 7.2.3. There are isomorphisms å Ç Kh Ç å Ç å (cid:27) Kh (cid:27) Kh in Kom(𝐻1ModG 𝑅 ) (here, the choice of orientation does not matter). 126 Proof. Picking an orientation for the right handed twist, we compute Ç Kh å (cid:27) Ü (cid:32) F (cid:33) ◦ 𝜑𝐻 (cid:32) 𝜑−1 F ê (cid:33) {0, 1}              (cid:27) (cid:27) Kh (cid:32) F (cid:33) ⊕ {0, −1} 𝑋 𝑍 𝜑𝐻 (cid:32) F (cid:33) {1, 0} F Ç (cid:32) (cid:33) 𝜑𝐻 {1, 0} å .              {0, 1} The second isomorphism is by delooping, noticing that 𝜑−1 is isomorphic to {1, 0} as shifting functors. Additionally, the maps are obtained from the former by precomposing with a birth or a dotted birth. The third isomorphism is by Gaussian elimination. The reader may verify that the computation for Kh Ç å is duplicate. Doing the same for the left handed twist, Ç Kh Ö å (cid:27) 𝜑 F (cid:32) (cid:33) (cid:32) F è (cid:33) {−1, 0}              (cid:27) (cid:32) F (cid:33) {0, −1} (cid:32) F 1 (cid:33) {0, −1} 𝑌 𝑍 (cid:32) F ⊕ (cid:33) {1, 0}              {−1, 0} Ç å (cid:27) Kh follows by the same reasoning, and the computation for Kh Ç å is its doppelgänger. □ 127 Lemma 7.2.4. There are isomoprhisms (cid:32) Kh (cid:33) (cid:32) (cid:27) Kh (cid:33) (cid:32) (cid:33) {−1, 1} (cid:27) Kh and (cid:32) Kh (cid:33) (cid:27) 𝜑Å ãKh , (0,1) (cid:32) (cid:33) (cid:32) (cid:27) Kh (cid:33) in Kom(𝐻2ModG 𝑅 ). We call the first pair of isomorphisms RII+ and the second pair RII−. Proof. By definition, Kh is the complex (cid:32) (cid:33) (cid:32) (cid:33) 𝜑 F (cid:32) F (cid:33) ◦ 𝜑𝐻2 ◦ 𝜑𝐻1 ⊕ 𝜑 ◦ 𝜑−1 F (cid:32) (cid:33) (cid:32) (cid:33) 𝜑−1 F with a global shift by {−1, 1}. However, up to isomorphism, we can rewrite the grading shifts on the 01 and 11 resolutions suggestively, so that the complex takes the form (cid:32) (cid:33) 𝜑 F (cid:32) F (cid:33) ◦ 𝜑𝐻2 ◦ 𝜑𝐻1 ⊕ (cid:32) (cid:33) 𝜑Å ã F , (0,1) 𝜑Å ã F , (1,1) (cid:32) (cid:33) again, with a global shift by {−1, 1}. Now, by delooping, (cid:32) (cid:33) (cid:32) (cid:33) 𝜑Å ã F , (0,1) (cid:27) 𝜑 Moreover, the maps F (cid:16) (cid:17) ◦ 𝜑𝐻1 and F F (cid:16) (cid:17) ⊕ 𝜑Å ã F , (1,1) (cid:32) (cid:33) . compose with the delooping isomorphism to yield invertible maps where desired, so that Gaussian elimination tells us that the entire complex is 128 homotopy equivalent to Kh {−1, 1}, as desired. Duplicate this work for the other side (cid:32) (cid:33) of RII+. We play the exact same game for RII−: Kh (cid:32) (cid:33) is (cid:32) (cid:33) ◦ 𝜑𝐻′ 1 𝜑 F 𝜑 ◦ 𝜑−1 F (cid:32) (cid:33) ⊕ (cid:32) F (cid:33) ◦ 𝜑𝐻′ 2 (cid:32) (cid:33) 𝜑−1 F with a global shift of {−1, 1}. By grading shift arethmetic, we know 𝜑 (cid:27) {0, −1}, 𝜑−1 (cid:27) 𝜑Å ã, and (1,1) 𝜑−1 (cid:27) {1, 0}, so that the complex may be rewritten (cid:32) F (cid:33) {0, −1} ◦ 𝜑𝐻′ 1 𝜑Å ãF (1,0) (cid:32) (cid:33) ⊕ (cid:32) F (cid:33) ◦ 𝜑𝐻′ 2 (cid:32) F (cid:33) {1, 0} Delooping the 01 entry and applying Gaussian elimination, we conclude that the entire complex (cid:32) (cid:33) (cid:32) (cid:33) is homotopy equivalent to 𝜑Å ãKh , (1,0) {−1, 1} ; i.e., 𝜑Å ãKh , (0,1) desired. Again, the other side of RII− is similar. , as □ Remark 7.2.5. Lemma 7.2.4 establishes that the grading shift coming from Reidemeister II moves is dependent on orientation. This, together with Lemma 7.2.3, implies that Reidemeister III moves must—at least, sometimes—come at the cost of a nontrivial grading shift. For example, if this was 129 not the case, the sequence of isomorphisms 𝜑Å ãKh , (0,1) (cid:32) (cid:33) RII−1 − (cid:32) Kh (cid:33) RI−1 (cid:32) Kh (cid:32) Kh (cid:33) {−1, 1} RI (cid:32) Kh (cid:33) {−1, 1} RII+ (cid:32) Kh (cid:33) (cid:33) ★ would yield a contradiction. Notice that the vertical arrow is an Reidemeister III move of type ⇝ . Lemma 7.2.6. We have the following isomorphisms in Kom(𝐻3ModG 𝑅 ): Ñ é Ñ é Ñ é Ñ é (cid:27) Kh Ñ é Ñ é Kh Kh (cid:27) Kh (cid:33) (cid:33) (cid:32) (cid:32) (cid:27) Kh (cid:27) Kh (cid:33) (cid:33) (cid:32) (cid:32) Kh Kh , , , , Kh Kh (cid:27) 𝜑 ◦ 𝜑−1 Kh Ñ é Ñ é (cid:27) 𝜑 ◦ 𝜑−1 Kh , , (cid:32) (cid:32) Kh Kh (cid:33) (cid:33) (cid:27) 𝜑 ◦ 𝜑−1 Kh (cid:27) 𝜑 ◦ 𝜑−1 Kh (cid:32) (cid:32) (cid:33) (cid:33) , . Proof. We will describe the proof by illustrating one of the isomorphisms on the left-hand side and its counterpart on the right-hand side. Each computation is slightly different, but we hope that this discussion sates the reader, or illuminates the procedure enough so that they might check the others on their own. The idea for any isomorphism on the left-hand side is to expand each complex and apply Gaussian elimination carefully. If Gaussian elimination is done properly, the two complexes are isotopic. If we do the same procedure for complexes appearing on the right-hand side, we will find that the entries of the complex are isotopic, but the grading shifts disagree. In this case, we will 130 argue that one is taken to the other by applying the grading shifts provided in the statement of the Lemma. Observe the complex associated to Kh (cid:32) (cid:33) . 𝜑 𝜑 ◦ 𝜑𝐻1 𝜑 (1,1) 𝜑 ◦ 𝜑𝐻2 𝜑 (1,1) ◦ 𝜑𝐻3 𝜑 (1,1) ◦ 𝜑𝐻4 𝜑 (1,1) Eyeing the boxed vertex, we have that 𝜑Ö è (cid:27) 𝜑Ö , (1,1) è , (0,1) 131 and, moreover, the delooping isomorphism provides that 𝜑Ö è , (0,1) (cid:27) 𝜑 ⊕ 𝜑Ö è , (1,1) . Now, we apply Gaussian elimination, so that the northwest and southeast vertices of the forward- facing face cancel with the northeast vertex which we just delooped. Here is the resulting complex. 𝜑 𝜑 ◦ 𝜑𝐻1 𝜑 (1,1) − ◦ 𝜑𝐻3 𝜑 (1,1) Now we do the same thing for Kh . We will refrain from writing out the initial cube (cid:32) (cid:33) this time. Mirroring the previous argument—delooping and then applying Gaussian elimination to toss three of the four terms appearing in the forward-facing face—this complex is homotopy equivalent to the following. 132 𝜑 𝜑 ◦ 𝜑𝐻′ 1 ◦ 𝜑𝐻′ 2 𝜑 (1,1) 𝜑 (1,1) − Finally, notice that and 𝜑Ö è (cid:27) 𝜑 , (1,1) 𝜑Ö è (cid:27) 𝜑 , (1,1) as grading shift functors. From here, it is straight forward to verify that the complexes are homotopy equivalent, showing that (cid:32) Kh (cid:33) (cid:32) (cid:27) Kh (cid:33) . On the other hand, working the same program for Kh and Kh we obtain (cid:32) (cid:33) (cid:32) (cid:33) 133 the following complexes. 𝜑 {0, −1} − ◦ 𝜑𝐻3 ◦ 𝜑𝐻1 ◦ 𝜑𝐻2 𝜑Ö è , (1,0) 𝜑Ö è , (1,1) 𝜑Ö è , (1,0) 𝜑 ◦ 𝜑𝐻′ 1 𝜑Ö è , (1,1) 𝜑Ö è , (1,0) {0, −1} ◦ 𝜑𝐻′ 3 𝜑Ö è , (1,0) − ◦ 𝜑𝐻′ 2 134 Again, we know these complexes are not homotopy equivalent by, for example, Remark 7.2.5. Instead, we will show that the latter is taken to the former by a global grading shift of 𝜑 ◦ 𝜑−1 . First, recall that 𝜑−1 may be written as 𝜑Ö è (up to equivalence of grading shift , (1,1) functors). On the other hand, 𝜑 and 𝜑−1 Ö è are isomorphic as grading shift , (−1,−1) functors. (i) Northwest vertex. As a warm-up, notice that 𝜑 has two isomorphic representatives important to understanding the intermediate complex. They are hardly different, but making a choice here is one way to describe two representatives of the G -grading shift obtained after the first global shift: 𝜑Ö è ◦ , (1,1) 𝜑Ö 𝜑Ö    è è    (cid:27) 𝜑Ö 𝜑Ö    , (0,−1) , (0,−1) è è    . , (1,0) , (1,0) Of course, yet another representative of this grading shift, encapsulating both of these representatives, is 𝜑Ö è. Anyway, applying the final global shift to the second , (1,1) representative above, we obtain the grading shift 𝜑Ö è = 𝜑Ö , (0,−1) è , (0,−1) which, similarly, is a representative of the grading shift 𝜑 . (ii) Southwest vertex. This is the trickiest since it is the vertex with one of its arrows altered by Gaussian elimination. On one hand, obviously if we apply 𝜑Ö è to {0, −1} , (1,1) 135 we are left with 𝜑Ö è. At first, this may not seem to square with the other arrow , (1,0) out of the vertex. To see that this questionable arrow is still a graded map, one may draw the original cube and trace it through the Gaussian elimination; we leave this as an exercise. Moving on, rewrite the grading shift as 𝜑Ö è and apply 𝜑 to , (1,0) obtain 𝜑Ö è. This is a representative of the grading shift 𝜑Ö è as we , (1,0) , (1,1) hoped. (iii) Northeast vertex. From 𝜑Ö è, we will consider the representatives 𝜑Ö , (1,1) and 𝜑Ö è. Then, , (1,0) è , (1,0) 𝜑−1 ◦ 𝜑Ö 𝜑Ö    è è    , (1,0) , (1,0) (cid:27) 𝜑Ö 𝜑Ö    è è    . , (1,0) , (1,0) The reader is invited to check that both representatives are used in the intermediary complex. Picking the latter and composing with 𝜑−1 Ö è, we obtain the grading shift {0, −1}. , (−1,−1) (iv) Southeast vertex. This is the most straightforward: applying the first global shift to 𝜑Ö è yields a shift by {1, 0}. Redrawing as , it is apparent that , (1,0) applying the second global shift provides 𝜑Ö è, as desired. , (1,0) □ 136 Remark 7.2.7. In light of Lemma 7.2.6, the sequence of Remark 7.2.5 is recitfied: notice that (cid:32) Kh (cid:33) (cid:27) 𝜑 ◦ 𝜑−1 Kh (cid:32) (cid:33) (cid:27) 𝜑Å ãKh , (1,0) (cid:32) (cid:33) . Composing with the grading shift by {−1, 1} and one last Reidemeister I move gives the desired grading shift by 𝜑Å ã. , (0,1) Theorem 7.2.8. If 𝑇 and 𝑆 are isotopic diskular tangles, then there exists a grading shifting functor 𝜑Δ𝑣 so that 𝜑Δ𝑣 Kh(𝑇) (cid:27) Kh(𝑆). Proof. In general, if one decomposes a diskular tangle 𝑇 into 𝑇𝐴(𝑇𝐵), as pictured below, then Theorem 6.2.4 tells us that F (𝑇) (cid:27) F (𝑇𝐵) ⊗𝐻𝑛 F (𝑇𝐴). By Lemma 7.2.1, there is a shifting functor 𝜑 and ℓ ∈ Z such that Kh(𝑇) (cid:27) 𝜑(F (𝑇))[ℓ]. By the coherence isomorphisms 𝛽, we have that 𝜑(F (𝑇𝐵) ⊗𝐻𝑛 F (𝑇𝐴)) (cid:27) 𝜑𝐵F (𝑇𝐵) ⊗𝐻𝑛 𝜑 𝐴F (𝑇𝐴) for 𝜑 𝐴 and 𝜑𝐵 restrictions of 𝜑 to the regions 𝐴 and 𝐵. Moreover, as described in the proof of Lemma 7.2.1, the G -grading and homological-grading shifts here are determined by local crossing information, so it follows similarly that 𝜑(F (𝑇))[ℓ] (cid:27) 𝜑𝐵F (𝑇𝐵)[ℓ𝐵] ⊗𝐻𝑛 𝜑 𝐴F (𝑇𝐴)[ℓ𝐴] for those particular ℓ𝐴, ℓ𝐵 ∈ Z satisfying ℓ𝐴 + ℓ𝐵 = ℓ. Indeed, since each 𝜑 𝐴, 𝜑𝐵, ℓ𝐴, ℓ𝐵 coming from 𝜑 and ℓ are the same as the shifts coming from the proof of Lemma 7.2.1, we have 𝜑𝐵F (𝑇𝐵)[ℓ𝐵] (cid:27) Kh(𝑇𝐵) and 𝜑 𝐴F (𝑇𝐴)[ℓ𝐴] (cid:27) Kh(𝑇𝐴). Summarizing, we have that Kh(𝑇) (cid:27) Kh(𝑇𝐵) ⊗𝐻𝑛 Kh(𝑇𝐴). 137 𝑇 𝑇𝐴 𝑇𝐵 If 𝑇 and 𝑆 are isotopic, then 𝑆 is obtained from 𝑇 by a finite sequence of Reidemeister moves. For each move in this sequence, apply the isomorphism above to the diskular region containing the Reidemeister move. Then, the theorem follows by applying this isomorphism, invoking one of Lemmas 7.2.3, 7.2.4, and 7.2.6, and repeating as needed. □ 7.2.1 Collapse to 𝑞-grading To obtain a genuine tangle invariant, we will perform the same trick as is in Section 6.5 of [NP20]. Define the degree collapsing map 𝜅 :HomG → Z (𝐷, (𝑝1, 𝑝2)) ↦→ 𝑝1 + 𝑝2 which forgets the planar arc diagram input of a G -grading and sums the entries of the second coordinate. We will use 𝜅 to notice that the G -grading of any G -graded object induces a coarser integral grading. First, by F𝑞(𝐷), we mean the multimodule F (𝐷) with an additional Z-grading determined by its G -grading: fix Ñ degZ×G (𝑢) := 𝜅(degG (𝑢)) + é 𝑚𝑖, degG (𝑢) . 𝑘 ∑︁ 𝑖=1 This additional Z-degree, determined by G -degree, is called the quantum degree, or 𝑞-degree; we denoted it by deg𝑞(𝑢). Notice that the composition maps 𝜇 preserve quantum degree. Furthermore, any cobordism Δ : 𝐷𝑑 → 𝐷′ induces a map F (Δ) : F𝑞(𝐷) → F𝑞(𝐷′) which is homogeneous of 𝑞-degree deg𝑞(F (Δ)) = #births + #deaths − #saddles. 138 Sometimes we just write deg𝑞(Δ) for deg𝑞(F (Δ)). Finally, we reinterpret a grading shift functor 𝜑Δ𝑣 in the Z × G -graded setting by deg𝑞 (cid:0)𝜑 Δ(𝑣1,𝑣2)(𝑚)(cid:1) := deg𝑞(𝑚) + deg𝑞(Δ) + 𝑣1 + 𝑣2. This is to say that any cobordism Δ induces a Z × G -graded map F (Δ) : 𝜑ΔF𝑞(𝐷) → F𝑞(𝐷′). In conclusion, all results in the G -graded setting extend to the Z × G -graded setting with no change to the compatibility maps: all isomorphisms involved are graded with respect to quantum degree. In particular, each 𝑑𝑣, 𝑗 preserves 𝑞-degree, so we can define F𝑞(𝑇) as a Z × G -graded dg-multimodule, using F𝑞(𝑇𝑣) in the place of F (𝑇𝑣); define Kh𝑞(𝑇) similarly. Suppressing notation, we let MultiMod𝑞 denote the category whose objects are the same as MultiModG except we record the quantum degree (that is, objects are Z × G -graded multimodules obtained from the regular G -graded ones) but now, maps are only required to be homogeneous with respect to G -degree, with the caveat that they must preserve quantum degree. By collapsing to 𝑞-degree, we just mean that we are working in the category MultiMod𝑞 rather than MultiModG . This is perhaps misleading, as the G -degree is still present—what we mean to relay is that we have relaxed the requirement of G -degree preservation to G -degree homogeneity up to 𝑞-degree preservation. We think of Kh𝑞(𝑇) as an object of Kom(MultiMod𝑞). In the final chapter, we are mostly interested in objects of Kom(𝐻𝑛ModG and also to Kom(𝐻𝑛Mod𝑞 𝑒 ) and Kom(𝐻𝑛Mod𝑞 𝑅 ), which we say descend to objects of Kom(𝐻𝑛Mod𝑞 𝑅), 𝑜), specializing 𝑋, 𝑌 , 𝑍 = 1 and 𝑋, 𝑍 = 1, 𝑌 = −1 respectively. We call these objects of Kom(𝐻𝑛Mod𝑞) the image of whatever object(s) of Kom(𝐻𝑛ModG ) which descends to it. Notice that a gluing property holds for F𝑞(𝑇) and Kh𝑞(𝑇), as before. Again, the benefit of working in Kom(MultiMod𝑞) is that Kh𝑞 becomes an honest tangle invariant. Theorem 7.2.9. If 𝑇 and 𝑆 are isotopic diskular tangles, then Kh𝑞(𝑇) (cid:27) Kh𝑞(𝑆). 139 Proof. This follows as long as the homotopy equivalences of Lemmas 7.2.3, 7.2.4, and 7.2.6 are graded with respect to quantum degree. For Reidemeister I moves, this is trivial, as the homotopy equivalence was already graded with respect to G -degree. For Reidemeister II moves, deg𝑞({−1, 1}) = 0 obviously, and Ñ é deg𝑞 𝜑Å ã , (0,1) = 1 + deg𝑞 (cid:16) (cid:17) = 1 + (−1) = 0. Similarly, it is clear that the 𝑞-degree of 𝜑 ◦ 𝜑−1 is zero. Therefore, the grading shift appearing in Theorem 7.2.8 has deg𝑞(𝜑𝑊 𝑣 ) = 0, and the result follows. □ Remark 7.2.10. If 𝑇 is a link, then the homology of Kh𝑞(𝑇) is isomorphic to the unified Khovanov homology of 𝑇, as constructed in [Put14]; see [NP20] for a proof. In particular, setting 𝑋 = 𝑍 = 1 and 𝑌 = −1 (before taking homology), we get a tangle invariant for odd Khovanov homology, as desired. 140 CHAPTER 8 UNIFIED AND ODD PROJECTORS Finally, we apply Theorem 6.2.4 (multigluing) to mimic the constructions of Cooper-Krushkal [CK12] and produce projectors living in Kom(𝐻𝑛ModG 𝑅 ). Our work in this chapter follows an outline similar to [SW24], since we exploit the flexibility provided by diskular tangles, as Stoffregen and Willis do in the spectral setting. More explicitly, in §8.1, we use multigluing to define the stacking ⊗, juxtaposing ⊔, and partial trace Tr operations, and the category Chom(𝑛)G (which we conjecture is the same as Kom(𝐻𝑛PModG ), as in [Kho02]). We also take this opportunity to prove an adjunction, generalizing a theorem of Hogancamp [Hog19]. The next section, §8.2, is mostly stand-alone: the main takeaway for this thesis is Corollary 8.2.4, which we use in the proof of Lemma 8.3.4, itself used in the proofs of Proposition 8.3.5 and Corollary 8.3.7. In §8.3, we define unified projectors as in [Hog19], though our proofs follow the methods outlined in [SW24], as their setting most resembles our own. We hope to illuminate preceding and successive work by computing the 2-stranded unified projector two different ways in §8.4. We also compute the homology of the closure of 𝑃2 (cf. Section 4.3.1 of [CK12]), which we will use to show that our categorification of the colored Jones polynomial is distinct from that of [CK12]. Finally, we prove the existence of unified projectors (using the same procedure as [SW24]) in §8.5, and the existence of a unified colored link homology (which collapses to the categorification of the colored Jones polynomial of [CK12] on one hand, and to a new categorification on the other) in §8.6. We establish some notation. Proceeding, for 𝐴, 𝐵 ∈ Kom(𝐻𝑛ModG ), we will denote the HOM- complex of 𝐴 and 𝐵 by Hom𝑛(𝐴, 𝐵). If 𝐴 and 𝐵 are (non-dg) G -graded 𝐻𝑛 modules, we’ll write Hom𝑛(𝐴, 𝐵) as shorthand for Hom𝐻𝑛 𝑀𝑜𝑑G (𝐴, 𝐵). 8.1 Operations defined via multigluing As far as the existence of projectors is concerned, the main payoff of multigluing in the unified setting is that we can develop a notion for stacking and juxtaposing complexes of G -graded modules. We can also use multigluing to define a partial trace for these complexes, allowing for an adjunction 141 statement. Given a diskular 𝑛-tangle 𝑇, we’ll view it as a tangle in a rectangle as follows: traveling counter- clockwise from the basepoint along the boundary, place the first 𝑛 endpoints along the top of the rectangle and the last 𝑛 endpoints along the bottom. For this reason, flat diskular 𝑛-tangles are also called a Temperley-Lieb 𝑛-diagrams, i.e., each resolution of 𝑇 is a Temperley-Lieb diagram. Definition 8.1.1 (Stacking). Vertical composition is the operation ⊗ : Kom(𝐻𝑛ModG 𝑅 ) × Kom(𝐻𝑛ModG 𝑅 ) → Kom(𝐻𝑛ModG 𝑅 ) defined as follows: given complexes 𝐴, 𝐵 ∈ Kom(𝐻𝑛ModG 𝑅 ), 𝐴 ⊗ 𝐵 is the complex (𝐴, 𝐵) ⊗(𝐻𝑛,𝐻𝑛) F (𝐷 ⊗ 𝑛 ) where 𝐷 ⊗ 𝑛 is the (𝑛, 𝑛; 𝑛)-planar arc diagram · · · 𝑛 · · · 1 · · · 𝑛 · · · 2 · · · 𝑛 · · · × × × with removed disks ordered as shown. In particular, if 𝑇1 and 𝑇2 are both diskular 𝑛-tangles, Theorem 6.2.4 says that F (𝑇1) ⊗ F (𝑇2) (cid:27) (F (𝑇1), F (𝑇2)) ⊗(𝐻𝑛,𝐻𝑛) F (𝐷 ⊗ 𝑛 ) (cid:27) F (𝐷 ⊗ 𝑛 (𝑇1, 𝑇2)). We say that this complex is the result of stacking F (𝑇1) and F (𝑇2). Definition 8.1.2. Consider the full subcategory Chom(𝑛)G of Kom(𝐻𝑛ModG ) consisting of (par- tially unbounded) G -graded dg-modules whose entries are all direct sums of G -graded modules associated to flat diskular 𝑛-tangles. In analogy with [Kho00], we expect that the subcategory Chom(𝑛)G is just the category Kom(𝐻𝑛PModG ) for 𝐻𝑛PModG the category of projective G -graded 𝐻𝑛-modules, although this 142 seems worthy of further study. Additionally, we expect that vertical composition ⊗ for this subcat- egory is a monoidal product with monoidal identity I𝑛 := · · · 𝑛 · · · × (that is, I𝑛 is the dg-module associated to the picture above), with monoidal structure provided by multigluing (Theorem 6.2.4). We let Chom(𝑛)𝑞 denote the image of Chom(𝑛)G in Kom(𝐻𝑛Mod𝑞) after collapsing to 𝑞-degree, §7.2.1. By definition, for 𝐾 𝑞 0 the Grothendieck group which records only the 𝑞-degree of G -graded objects, we have that 0 (Chom(𝑛)G ) (cid:27) 𝐾 𝑞 𝐾 𝑞 0 (Chom(𝑛)𝑞) (cid:27) 𝑇 𝐿𝑛. Just as stacking can be realized as a multigluing operation, the horizontal juxtaposition can as well. Definition 8.1.3 (Juxtaposing). Horizontal composition is the operation ⊔ : Kom(𝐻𝑛1ModG 𝑅 ) × Kom(𝐻𝑛2ModG 𝑅 ) → Kom(𝐻𝑛1+𝑛2ModG 𝑅 ) defined as follows: for complexes 𝐴 ∈ Kom(𝐻𝑛1ModG 𝑅 ) and 𝐵 ∈ Kom(𝐻𝑛2ModG 𝑅 ), 𝐴 ⊔ 𝐵 is the complex (F (𝑇1), F (𝑇2)) ⊗(𝐻𝑛1 ,𝐻𝑛2 ) F (𝐷⊔ (𝑛1,𝑛2)) where 𝐷⊔ (𝑛1,𝑛2) is the (𝑛1, 𝑛2; 𝑛1 + 𝑛2)-planar arc diagram 𝑛1 𝑛2 × × × 𝑛1 𝑛2 If 𝑇𝑖 a diskular 𝑛𝑖-tangle, we’ll write F (𝑇1) ⊔ F (𝑇2) to denote the tensor product (F (𝑇1), F (𝑇2)) ⊗(𝐻𝑛1 ,𝐻𝑛2 ) F (𝐷⊔ (𝑛1,𝑛2)) (cid:27) F (𝐷⊔ (𝑛1,𝑛2)(𝑇1, 𝑇2)). We say that this complex is the result of juxtaposing F (𝑇1) and F (𝑇2). 143 8.1.1 Adjunction First, consider the following operation on complexes in Kom(𝐻𝑛ModG 𝑅 ). Definition 8.1.4. (Trace) The trace is an operation Tr : Kom(𝐻𝑛ModG 𝑅 ) → Kom(𝐻𝑛−1ModG 𝑅 ) defined as follows: for 𝐴 ∈ Kom(𝐻𝑛ModG 𝑅 ), Tr(𝐴) is the complex where 𝐷Tr 𝑛 is the (𝑛; 𝑛 − 1)-planar arc diagram 𝐴 ⊗𝐻𝑛 F (𝐷Tr 𝑛 ) 𝑛 − 1 𝑛 − 1 × × If 𝑇 is a diskular 𝑛-tangle, we’ll write Tr(F (𝑇)) to denote the complex F (𝑇) ⊗𝐻𝑛 F (𝐷Tr 𝑛 ) (cid:27) F (𝐷Tr 𝑛 (𝑇)). By the 𝑘th partial trace of 𝐴, we mean the complex obtained from applying the partial trace 𝑘 times to obtain Tr𝑘 (𝐴) ∈ Kom(𝐻𝑛−𝑘 ModG 𝑅 ). The 𝑛th partial trace of 𝐴 is known simply as the trace or closure of 𝐴. In [Hog19], we saw that the operations − ⊔ 1 and Tr(−) were adjoint. Impressively, we can prove that a generalization of this adjunction exists in the G -graded setting! Theorem 8.1.5. Suppose 𝐴 ∈ Kom(𝐻𝑛−1ModG 𝑅 ) and 𝐵 ∈ Kom(𝐻𝑛ModG 𝑅 ). Then we have the following isomorphisms of complexes. Ñ é (cid:32) (cid:33) Hom𝑛 𝐴 , 𝜑Å B , (0,1) ã 𝐵 (cid:27) Hom𝑛−1 𝐴 , 𝐵 {−1, 0} and Ñ Hom𝑛 𝜑Å B , (0,1) é (cid:32) (cid:33) ã 𝐵 , 𝐴 (cid:27) Hom𝑛−1 𝐵 , 𝐴 {0, −1} . 144 Proof. Unlike the analogues of this result for even Khovanov homology [Hog19] and even Kho- vanov spectra [SW24], the fact that certain maps occur in disjoint disks does not mean that they commute, but rather that swapping the two changes the overall composition by an isomorphism induced by a locally vertical change of chronology. We will see that our G -shifting 2-system ac- counts for this difference, so that the above result holds with little alterations to the aforementioned proofs. We will prove the first isomorphism, leaving the second to the reader—notice that the grading shift by {−1, 0} in the former is replaced by a grading shift by {0, −1} in the latter. Suppose that Ñ é 𝑓 ∈ Hom𝑛 𝐴 ⊔ 1, 𝜑Å B , (0,1) ã𝐵 has homogenous I -degree Δ𝑣, so it is realized as a G -graded map 𝑓 : 𝜑Δ𝑣 𝐴 ⊔ 1 → 𝜑Å ã𝐵 (we do not have to pay attention to the homological degree). B , (0,1) Define 𝜙( 𝑓 ) ∈ Hom𝑛−1(𝐴, Tr(𝐵){−1, 0}) to be the composition 𝜆𝜙( 𝑓 ) ◦ F Ä ä ◦ 𝜑𝐻𝐵 𝜑Δ𝑣 𝐴 𝜑Δ𝑣+(−1,0) 𝐴 Tr( 𝑓 ) 𝐵 {−1, 0} where 𝜑𝐻𝐵 : Id ⇒ 𝜑−1 ◦ 𝜑−1 (cid:27) {−1, 0} ◦ {1, 0}. and 𝜆𝜙( 𝑓 ) is shorthand for the isomorphism which pushes the {−1, 0} shift after Δ𝑣; that is, 𝜆𝜙( 𝑓 ) = 𝛾(−1,0),Δ𝑣 ◦ 𝜆(𝑣, (−1, 0)). Schematically, 𝑣 𝜆𝜙( 𝑓 ) : ⇒ Δ (−1, 0) (−1, 0) 𝑣 . Δ Lastly, by Tr( 𝑓 ) we just mean 𝑓 ⊗ 1𝐷Tr 𝑛 . Notice that 𝜙( 𝑓 ) has the desired form since B • 1𝐷Tr 𝑛 = B is a split, so the shifting functor associated to it is the Z × Z-grading shift {0, −1}, thus canceling with the original Z × Z-grading shift of {0, 1}. Said another way, Tr( 𝑓 ) ∈ Hom𝑛 (cid:0)𝐴 ⊔ ⃝, Tr(𝐵)(cid:1). 145 Next, let 𝑔 ∈ Hom𝑛−1(𝐴, Tr(𝐵){−1, 0}) and denote the I -degree of 𝑔 by E𝑤. We define Ñ é 𝜓(𝑔) ∈ Hom𝑛 𝐴 ⊔ 1, 𝜑Å ã𝐵 to be the composition B , (0,1) 𝜑E 𝑤 𝐴 𝑔 ⊔ 1 𝐵 {−1, 0} ◦ 𝜑𝐻𝑆 {−1, 0} ◦ 𝜑−1 𝐵 B where 𝜑𝐻𝑆 : Id ⇒ 𝜑−1 ◦ 𝜑 𝐵 . 𝐵 Then, 𝜓(𝑔) has the desired form since 𝜑−1 𝐵 = 𝜑Ñ é 𝐵 , (1,1) composed with {−1, 0} is 𝜑Å ã. B , (0,1) Now, we compute 𝜓(𝜙( 𝑓 )) as the composition 𝜆𝜙( 𝑓 ) ◦ F Ä ä ◦ 𝜑𝐻𝐵 𝜑Δ𝑣 𝐴 𝜑Δ𝑣+(−1,0) 𝐴 Tr( 𝑓 ) ⊔ 1 𝐵 {−1, 0} ◦ 𝜑𝐻𝑆 𝜑Å ã 𝐵 B , (0,1) If we slide 𝑓 past the saddle, then the above complex is equivalent to the following one, where we 146 have compensated for the slide by a change of chronology 𝜑𝐻1. 𝜑Δ𝑣 𝐴 𝜆𝜙( 𝑓 ) ◦ F Ä ä ◦ 𝜑𝐻𝐵 𝜑Δ𝑣+(−1,0) 𝐴 𝜑𝐻𝑆 {−1, 0} ◦ 𝜑−1 𝐵 ◦ 𝜑 𝐵 ◦ 𝜑Δ𝑣 𝐴 𝜑𝐻1 {−1, 0} ◦ 𝜑−1 𝐵 ◦ 𝜑Δ𝑣 ◦ 𝜑 𝐴 𝐴 𝜑Å B , (0,1) ã ◦ 𝜑Δ𝑣 ◦ {−1, 0} 𝐴 𝜑Å ã 𝐵 B , (0,1) 𝑓 𝜑Å B , (0,1) ã ◦ 𝜑Δ𝑣 𝐴 The key observation is that 𝜆𝜙( 𝑓 )—which corresponds to sliding a shift by {−1, 0} through Δ𝑣—and 𝜑𝐻1—which corresponds to a change of chronology which pushes a saddle through Δ𝑣, at which point it is realized as a merge (and the grading shift associated to merges is {−1, 0})—are inverse to one another. After this, the birth and merge cancel with one another, and we conclude that 𝜓(𝜙( 𝑓 )) = 𝑓 . We play a similar game for 𝜙(𝜓(𝑔)): it is computed as 𝜆𝜓(𝑔) ◦ F Ä ä ◦ 𝜑𝐻𝐵 𝜑E 𝑤 𝐴 𝜑E 𝑤+(−1,0) 𝐴 Tr(𝑔 ⊔ 1) 𝐵 {−2, 0} ◦ 𝜑𝐻𝑆 𝐵 {−1, 0} 𝜑−1 𝐵 𝐵 {−2, 0} 147 where the last equality follows because 𝜑−1 (cid:27) {1, 0}. Now, slide 𝑔 before the birth; as 𝐵 before, to do so, we have to compensate by 𝜑𝐻2 : 𝜑E 𝑤 ◦ {1, 0} ⇒ {1, 0} ◦ 𝜑E 𝑤 . Here is the resulting composition. 𝜑E 𝑤 𝐴 𝜑𝐻𝐵 𝜑E 𝑤 ◦ {−1, 0} ◦ {1, 0} 𝐴 𝜆𝜙( 𝑓 ) {−1, 0} ◦ 𝜑E 𝑤 ◦ {1, 0} 𝐴 𝜑𝐻2 {−1, 0} ◦ {1, 0} ◦ 𝜑E 𝑤 𝐴 𝑔 {−2, 0} ◦ {1, 0} 𝐵 Ä ä F 𝐵 {−1, 0} 𝐵 {−2, 0} ◦ 𝜑𝐻𝑆 Now, 𝜆𝜙( 𝑓 ) and 𝜑𝐻2 are inverse to one another, since 𝜆((𝑎, 𝑏), (−1, 0)) = 𝑋 −𝑎 𝑍 𝑏 and 𝜆((𝑎, 𝑏), (1, 0)) = 𝑋 𝑎 𝑍 −𝑏. Again, the birth cancels with the merge, and we have that 𝜙(𝜓(𝑔)) = 𝑔, concluding the proof. Remark 8.1.6. Since Ñ deg𝑞 𝜑Å é ã = 0 B , (0,1) □ this result descends to Theorem 2.31 of [Hog19] if we collapse the G -grading to the 𝑞-grading. 8.2 Duals and mirrors Suppose 𝑅, 𝑆, and 𝑇 are C -graded algebras. Per usual, we expect that if 𝑀 is a C -graded (𝑅; 𝑆)- multimodule, and 𝑁 is a C -graded (𝑅; 𝑇)-multimodule, then Hom𝑅(𝑀, 𝑁) is an (𝑆; 𝑇)-multimodule 148 by 𝜌Hom 𝐿 (𝑠, 𝑓 )(𝑚) := 𝑓 (𝜌𝑀 𝑅 (𝑚, 𝑠)) and 𝜌Hom 𝑅 ( 𝑓 , 𝑡)(𝑚) := 𝜌𝑁 𝑅 ( 𝑓 (𝑚), 𝑡) for each 𝑓 ∈ Hom𝑅(𝑀, 𝑁), 𝑚 ∈ 𝑀, 𝑠 ∈ 𝑆 and 𝑡 ∈ 𝑇. However, Hom𝑅(𝑀, 𝑁) does not satisfy the axioms of a C -graded multimodule: by definition, Hom𝑅(𝑀, 𝑁) is graded by (cid:102)I ×Z = (I ⊔{Id})×Z, and the reader is invited to verify that • 𝜌Hom 𝐿 (𝑠1 · 𝑠2, 𝑓 )(𝑚) = 𝛼 (cid:0)|𝑚| ,|𝑠1| ,|𝑠2|(cid:1)−1 𝜌Hom 𝐿 (𝑠1, 𝜌Hom 𝐿 (𝑠2, 𝑓 ))(𝑚), • 𝜌Hom 𝑅 (𝜌Hom 𝑅 ( 𝑓 , 𝑡1), 𝑡2)(𝑚) = 𝛼 Ä(cid:12) (cid:12) 𝑓 (𝑚)(cid:12) (cid:12) ,|𝑡1| ,|𝑡2| ä 𝜌Hom 𝑅 ( 𝑓 , 𝑡1 · 𝑡2)(𝑚), and • 𝜌Hom 𝑅 (𝜌Hom 𝐿 (𝑠, 𝑓 ), 𝑡)(𝑚) = 𝜌Hom 𝐿 (𝑠, 𝜌Hom 𝑅 ( 𝑓 , 𝑡))(𝑚). Despite this ambiguity, we are able to give a type of duality statement which turns out to be a generalization of Theorem 4.12 in [Hog20]. This implies a unified analogue to Lemma 4.14 of [SW24], which is all we will need to prove the uniqueness of unified Cooper-Krushkal projectors. We dualize a flat diskular (𝑚; 𝑛)-tangle 𝑇 by the following operation, flipping radially, 𝑇 = × · · · 2𝑚· · · T · · · 2𝑛· · · × dualize −−−−−→ = 𝑇 ∨ × · · · 2𝑛· · · T · · · 2𝑚· · · × to obtain a diskular (𝑛; 𝑚)-tangle. Notice that if 𝑇 is a flat diskular 𝑛-tangle, then 𝑇 ∨ is a flat diskular (𝑛; 0)-tangle; this is the case we are most interested in. On cobordisms of 𝑇 embedded in [0, 1]3, (−)∨ acts by the transformation (𝑥, 𝑦, 𝑧) ↦→ (𝑥, 1 − 𝑦, 1 − 𝑧). Now we describe how (−)∨ establishes a contravariant functor Chom(𝑛)G → Chom(𝑛)G . On objects (which are chain complexes of summands of G -graded 𝐻𝑛-modules associated to flat diskular 𝑛-tangles with a differential of matrices of cobordisms), (−)∨ applies (−)∨ on each entry, reverses homological degree (i.e., (𝐴∨)𝑘 := (𝐴−𝑘 )∨), applies (−)∨ on each cobordism and takes the transpose of each matrix of cobordisms, and reverses G -degree. By this last point, we mean that 149 each cobordism shift 𝑊 is dualized (note that if 𝑊 : 𝑎 → 𝑏, then 𝑊 ∨ : 𝑏∨ → 𝑎∨) and Z × Z-degree is reversed: {𝑣1, 𝑣2}∨ = {−𝑣2, −𝑣1}. In particular, if 𝑑 𝐴 is the differential for 𝐴 ∈ Chom(𝑛)G , then (abusing notation), fix 𝑑 𝐴∨ = −(𝑑 𝐴)∨ ◦ 𝜑𝐻 where 𝜑𝐻 means that we are applying the change of chronology 𝜑𝐻 : Id ⇒ 𝜑−1 (𝑑 𝐴)∨ ◦ 𝜑(𝑑 𝐴)∨ on each entry of each matrix comprising 𝑑 𝐴∨. For example, the dual of the complex 𝜑 F Ç å {−1, 0} Å ã F −−−−−−−→ F Ç å {−1, 0} is the complex Ç F å {0, 1} Å ã ◦𝜑𝐻 F −−−−−−−−−−−→ 𝜑−1 F Ç å {0, 1} for 𝜑𝐻 : Id ⇒ 𝜑−1 ◦ 𝜑 . In particular, this is to say that Ç Kh å∨ × = Kh Ç å × as one might hope. Finally, on morphisms, to 𝑓 ∈ Hom𝑘 Chom(𝑛)(𝜑𝑊,(𝑣1,𝑣2) 𝐴, 𝐵) (where 𝑘 is the homological degree and (𝑊, (𝑣1, 𝑣2)) is the (cid:102)I -degree) we define 𝑓 ∨ ∈ Hom𝑘 Chom(𝑛)(𝜑𝑊 ∨,(𝑣2,𝑣1)𝐵∨, 𝐴∨) to be ( 𝑓 ∨)𝑖 = (−1)𝑖𝑘 ( 𝑓−𝑖−𝑘 )∨ following the commutativity of the square (𝐵∨)𝑖 ( 𝑓 ∨)𝑖 (𝐴∨)𝑖+𝑘 (−1)𝑖𝑘( 𝑓−𝑖−𝑘)∨ (𝐵−𝑖)∨ (𝐴−𝑖−𝑘 )∨ 150 As consequence of reversing G -degree, the (cid:102)I -degree of compositions of morphisms is also reversed; this is to say that (for, say, maps of homological degree zero) (𝑔 ◦G 𝑓 )∨ = 𝜑𝐻 𝑓 ,𝑔 𝑔∨ ◦G 𝑓 ∨ where 𝜑𝐻 𝑓 ,𝑔 denotes the change of chronology prioritizing the degree shift of 𝑔 before that of 𝑓 . Then, we have the following standard lemma. Lemma 8.2.1. For 𝐴, 𝐵 and 𝑓 , 𝑔 as above, 1. (−)∨ induces a degree-zero chain map HomChom(𝑛)(𝐴, 𝐵) → HomChom(𝑛)(𝐵∨, 𝐴∨); 2. (𝑔 ◦G 𝑓 )∨ = 𝜑𝐻 𝑓 ,𝑔(−1)| 𝑓 | ℎ|𝑔| ℎ 𝑓 ∨ ◦G 𝑔∨. The purpose of the rest of this section is to prove that Hom𝑛(𝐴 ⊗ 𝛿, 𝐵) (cid:27) Hom𝑛(𝐴, 𝐵 ⊗ 𝛿∨) for any 𝐴, 𝐵 ∈ Chom(𝑛) and 𝛿 any flat diskular 𝑛-tangle. In order to describe our logical process for proving this statement, we will introduce yet another tensor product which will not reappear anywhere else in this thesis. Definition 8.2.2. Suppose 𝐴, 𝐵 ∈ Chom(𝑛)G . Recall that we may represent, for example, 𝐴 and 𝐴∨ as 𝐴 = A · · · 2𝑛· · · × and 𝐴∨ = × · · · 2𝑛· · · . A We define two natural operations. By 𝐴 | 𝐵∨, we mean the tensor 𝐴 ⊗𝐻𝑛 𝐵∨; on the other hand, by 𝐴∨ | 𝐵, we mean the tensor 𝐴∨ ⊗𝐻0 𝐵. Diagramatically, A 𝐴 | 𝐵∨ (cid:27) · · · 2𝑛· · · and 𝐴∨ | 𝐵 (cid:27) B 151 × · · · 2𝑛· · · A B · · · 2𝑛· · · × by Theorem 6.2.4. Theorem 8.2.3 (cf. Theorem 4.12, [Hog20]). Suppose 𝐴, 𝐵 ∈ Chom(𝑛)G . Then there is an isomorphism of complexes Hom𝑛(𝐴, 𝐵) (cid:27) Hom0(∅, 𝐵 | 𝐴∨{−𝑛, 0}). This Theorem implies our goal for the section. Corollary 8.2.4. Suppose 𝛿 is a flat diskular 𝑛-tangle. Then Hom𝑛(𝐴 ⊗ F (𝛿), 𝐵) (cid:27) Hom𝑛(𝐴, 𝐵 ⊗ F (𝛿∨)). Proof. Writing 𝛿 for F (𝛿), we have Hom𝑛(𝐴 ⊗ 𝛿, 𝐵) (cid:27) Hom0(∅, 𝐵 | (𝐴 ⊗ 𝛿)∨{−𝑛, 0}) (cid:27) Hom0(∅, 𝐵 | (𝛿∨ ⊗ 𝐴∨){−𝑛, 0}) (cid:27) Hom0(∅, (𝐵 ⊗ 𝛿∨) | 𝐴∨{−𝑛, 0}) (cid:27) Hom𝑛(𝐴, 𝐵 ⊗ 𝛿∨). The first and last isomorphisms are provided by Theorem 8.2.3, while the second follows from the definition of (−)∨ and the third is an application of Theorem 6.2.4. □ We prove Theorem 8.2.3 in two steps. First, we prove an analogue of Theorem 8.2.3 for crossingless matchings. Then, we argue that this implies the general statement. Definition 8.2.5. Suppose 𝑎 is a crossingless matching on 2𝑛 points; i.e., a planar diskular 𝑛-tangle. In this definition, we will assume 𝑎 is indecomposable; that is, 𝑎 is void of circle components. 1. Define 𝜂𝑎 as the map 𝜂𝑎 : ∅ 𝜑𝐻𝜂𝑎 −−−−→ {−𝑛, 0} ◦ {𝑛, 0} ∅ −→ {−𝑛, 0} 𝑎 | 𝑎∨ consisting of 𝑛-many births (since 𝑎 | 𝑎∨ is exactly 𝑛-many circles). 152 2. Let 𝑠𝑎 denote the map 𝑠𝑎 : 𝜑Σ𝑎 𝑎∨ | 𝑎 → 1𝑛 defined by the minimal chronological cobordism Σ𝑎 given by contracting symmetric arcs, right-to-left, with framing pointed upwards. The following lemma is apparent. Lemma 8.2.6. Fix indecomposable crossingless parings on 2𝑛-points 𝑎, 𝑏. 1. (1𝑎 | 𝑠𝑎) ◦ (𝜂𝑎 | 1𝑎) = 1𝑎 and (𝑠𝑎 | 1𝑎∨) ◦ (1𝑎∨ ◦ 𝜂𝑎) = 1𝑎∨. 2. Suppose 𝑏 | 𝑎∨ consists of ℓ-many circles, 1 ≤ ℓ ≤ 𝑛 (note that ℓ = 𝑛 if and only if 𝑏 = 𝑎). Then 1𝑏 | 𝑠𝑎 : 𝑏 | 𝑎∨ | 𝑎 → 𝑏 consists of ℓ-many merges followed by a minimal cobordism 𝑊 : 𝑎 → 𝑏. Note that 𝑊 consists of (𝑛 − ℓ)-many saddles. We’ll write(cid:12) (cid:12)𝑏 | 𝑎∨(cid:12) (cid:12) to denote the number of loops in 𝑏 | 𝑎∨ (above,(cid:12) (cid:12)𝑏 | 𝑎∨(cid:12) (cid:12) = ℓ). We’ll denote crossingless matchings, pictorially, as 𝑎 𝑎 = and 𝑎∨ = 𝑎∨ . For example, part 2 of Lemma 8.2.6 describes a cobordism 𝑏 1𝑏 |𝑠𝑎 −−−−→ = 𝑏 . 𝑏 𝑎∨ 𝑎 While these pictures are a departure from the planar arc diagrams we are accustomed to, they are a little more natural for the proof of the following proposition. Proposition 8.2.7 (cf. Proposition 4.8, [Hog20]). Suppose 𝑎 and 𝑏 are crossingless matchings on 2𝑛 points (not necessarily indecomposable) and fix a minimal cobordism 𝑊 : 𝑎 → (cid:98)𝑏, where (cid:98) (cid:98) 𝑎, (cid:98)𝑏 are 𝑎 and 𝑏 with circle components removed. Then Ç Hom𝑛 å 𝜑(cid:16) (cid:12) (cid:12) 𝑊,(𝑛− 𝑎 (cid:12)(cid:98)𝑏| (cid:98) (cid:12) (cid:12) (cid:12) (cid:17) 𝑎, 𝑏 ,0) (cid:27) Hom0 (cid:0)∅, 𝑏 | 𝑎∨{−𝑛, 0}(cid:1) . 153 In pictures, Ç Hom𝑛 𝜑(cid:16) (cid:12) (cid:12) 𝑊,(𝑛− 𝑎 (cid:12)(cid:98)𝑏| (cid:98) (cid:12) (cid:12) (cid:12) (cid:17) ,0) å 𝑎 , 𝑏 Ö (cid:27) Hom0 ∅, è {−𝑛, 0} . 𝑏 𝑎∨ Proof. First, we can assume without loss of generality that 𝑎 and 𝑏 are both indecomposable—the general result follows immediately by delooping. Proceeding, we will frequently denote 𝜑(cid:16) (cid:17) by 𝜑𝑊 𝑁 . Notice that the (cid:102)I -degree of any (cid:12) (cid:12) 𝑊,(𝑛− 𝑎 (cid:12)(cid:98)𝑏| (cid:98) (cid:12) (cid:12) (cid:12) 𝑓 ∈ Hom𝑛 (cid:0)𝜑𝑊 𝑁 𝑎, 𝑏(cid:1) can be chosen to be described purely by a Z × Z-degree, since 𝑊 : 𝑎 → 𝑏 is minimal. Recall that this is also the case for any 𝑔 ∈ Hom0 (cid:0)∅, 𝑏 | 𝑎∨ {−𝑛, 0}(cid:1) since any ,0) grading shift associated to a cobordism between closed diagrams is canonically isomorphic to a pure Z × Z-shift. Thus, we will denote the homogeneous degree of 𝑓 and 𝑔 by 𝑣 𝑓 and 𝑤𝑔 ∈ Z × Z respectively. The rest of this proof proceeds like the proof of Theorem 8.1.5. To any 𝑓 ∈ Hom𝑛 (cid:0)𝜑𝑊 𝑛 𝑎, 𝑏(cid:1), define 𝜙( 𝑓 ) ∈ Hom0(∅, 𝑏 | 𝑎∨ {−𝑛, 0}) as the composition 𝜂𝑎 𝑣 𝑓 ∅ 𝑣 𝑓 ◦ {−𝑛, 0} 𝜆(𝑣 𝑓 , (−𝑛, 0)) {−𝑛, 0} ◦ 𝑣 𝑓 𝑎 𝑎∨ 𝑎 𝑎∨ 𝑓 | 1𝑎∨ {−𝑛, 0} 𝑏 𝑎∨ To clear up any confusion, notice that the minimal cobordism 𝑊 : 𝑎 → 𝑏, which has Ä𝑛 −(cid:12) ä- (cid:12)𝑏 | 𝑎∨(cid:12) (cid:12) many saddles, extends to a cobordism 𝑊 • 1𝑎∨ : 𝑎 | 𝑎∨ → 𝑏 | 𝑎∨ in which all saddles are realized as merges. Thus 𝜑(𝑊•1𝑎∨ )𝑁 (cid:27) Id. Next, to 𝑔 ∈ Hom0(∅, 𝑏 | 𝑎∨ {−𝑛, 0}), define 𝜓(𝑔) ∈ Hom𝑛(𝜑𝑊 𝑁 𝑎, 𝑏) by 𝑤𝑔 ◦ 𝜑𝑊 𝑁 𝜆𝜓(𝑔) 𝑎 𝜑𝑊 𝑁 ◦ 𝑤𝑔 𝑔 | 1𝑎 𝜑𝑊 𝑁 ◦ {−𝑛, 0} 𝑎 𝑏 𝑎∨ 𝑎 𝛾𝑊,{−𝑛,0} 𝜑Ä𝑊,(−|𝑏|𝑎∨|,0)ä 𝑏 𝑎∨ 𝑎 1𝑏 | 𝑠𝑎 𝑏 154 where we set 𝜆𝜓(𝑔) := 𝛾𝑊 𝑁 ,𝑤𝑔 ◦ 𝜆(𝑁, 𝑤𝑔). Note that the last map is G -graded by part 2 of Lemma 8.2.6. We compute 𝜓(𝜙( 𝑓 )) as the composition 𝑣 𝑓 ◦ 𝜑𝑊 𝑁 𝑎 𝜆𝜓(𝜙( 𝑓 )) 𝜑𝑊 𝑁 ◦ 𝑣 𝑓 𝑎 𝜂𝑎 𝜑𝑊 𝑁 ◦ 𝑣 𝑓 ◦ {−𝑛, 0} 𝑎 𝑎∨ 𝑎 𝜆(𝑣 𝑓 , (−𝑛, 0)) 𝜑𝑊 𝑁 ◦ {−𝑛, 0} ◦ 𝑣 𝑓 𝑎 𝑎∨ 𝑎 1𝑏 | 𝑠𝑎 𝑏 𝜑Ä𝑊,(−|𝑏|𝑎∨|,0)ä 𝑏 𝑎∨ 𝑎 𝛾𝑊,{−𝑛,0} 𝜑𝑊 𝑁 ◦ {−𝑛, 0} 𝑏 𝑎∨ 𝑎 𝑓 | 1𝑎|𝑎∨ or 𝑣 𝑓 ◦ 𝜑𝑊 𝑁 𝑎 𝜆𝜓(𝜙( 𝑓 )) 𝜑𝑊 𝑁 ◦ 𝑣 𝑓 𝑎 𝜂𝑎 𝜑𝑊 𝑁 ◦ 𝑣 𝑓 ◦ {−𝑛, 0} 𝑏 𝑓 𝑣 𝑓 ◦ 𝜑𝑊 𝑁 1𝑎 | 𝑠𝑎 𝑎 𝑣 𝑓 ◦ 𝜑𝑊 𝑁 ◦ {−𝑛, 0} 155 𝑎 𝑎∨ 𝑎 𝜆(𝑣 𝑓 , (−𝑛, 0)) 𝑎 𝑎∨ 𝑎 𝜑𝑊 𝑁 ◦ {−𝑛, 0} ◦ 𝑣 𝑓 𝜑𝐻1 𝑎 𝑎∨ 𝑎 obtained by sliding 𝑓 past the saddle, which introduces a change of chronology 𝜑𝐻1 which in turn cancels with 𝜆(𝑣 𝑓 , (−𝑛, 0)) and 𝜆𝜓(𝜙( 𝑓 )). A discerning eye notices that this change of chronology also kills the 𝛾𝑊,{−𝑛,0} term, since the roles of 𝜑𝑊 𝑁 and {−𝑛, 0} are interchanged during this change of chronology. Now, notice that 1𝑎 | 𝑠𝑎 consists of 𝑛 merges, so the penultimate arrow makes sense. Then, 1. of Lemma 8.2.6 gives us that 𝜓(𝜙( 𝑓 )) = 𝑓 . On the other hand, 𝜙(𝜓(𝑔)) is rather easy to compute; the reader is invited to verify that this composition simplifies to 𝜂𝑎 𝑤𝑔 ∅ 𝑤𝑔 ◦ {−𝑛, 0} 𝑎 𝑎∨ 𝜆(𝑤𝑔, (−𝑛, 0)) {−𝑛, 0} ◦ 𝑤𝑔 𝑎 𝑔 | 1𝑎|𝑎∨ {−2𝑛, 0} 𝑎∨ 𝑏 𝑎∨ 𝑎 𝑎∨ 1𝑏 | 𝑠𝑎 | 1𝑎∨ {−𝑛, 0} 𝑏 𝑎∨ Then, pushing 𝑔 before the birth introduces a change of chronology 𝜑𝐻2 equal to 𝜆((−𝑛, 0), 𝑤𝑔). This is inverse to 𝜆(𝑤𝑔, (−𝑛, 0)), so that the new composition is 𝑏 𝑔 𝑤𝑔 ∅ {−𝑛, 0} 𝑎∨ 1𝑏|𝑎∨ | 𝜂𝑎 {−2𝑛, 0} 𝑏 𝑎∨ 𝑎 𝑎∨ 1𝑏 | 𝑠𝑎 | 1𝑎∨ {−𝑛, 0} 𝑏 𝑎∨ which simplifies to 𝑔 by Lemma 8.2.6. This concludes the proof. □ Remark 8.2.8. Since a minimal cobordism 𝑎 → 𝑏 consists of (𝑛 −(cid:12) (cid:12)𝑏 | 𝑎∨(cid:12) (cid:12))-many saddles, Ç deg𝑞 𝜑(cid:16) (cid:12) (cid:12) 𝑊,(𝑛− 𝑎 (cid:12)(cid:98)𝑏| (cid:98) (cid:12) (cid:12) (cid:12) (cid:17) ,0) å = 0 156 and we obtain a generalization of Proposition 4.8 in [Hog20]. Proof of Theorem 8.2.3. Recall that Hom𝑛(𝐴, 𝐵), for 𝐴 and 𝐵 G -graded dg-𝐻𝑛-modules, is the chain complex of bihomogeneous (that is, homogeneous in homological degree and purely homo- geneous in (cid:102)I -degree) maps 𝑓 of arbitrary (Z × (cid:102)I )-degree. So, we can view Hom𝑛-complexes as bigraded abelian groups Hom𝑛(𝐴, 𝐵)𝑘 (𝑊,𝑣) (cid:27) (cid:214) Hom𝑛 Ä𝜑(𝑊,𝑣) 𝐴ℓ, 𝐵ℓ+𝑘 ä . ℓ∈Z However, notice that for each ℓ, 𝑘, and (𝑊, 𝑣), Proposition 7.1.2 says that 𝜑(𝑊,𝑣) (cid:27) 𝜑(𝑊 𝑘 minimal cobordism 𝐴ℓ → 𝐵ℓ+𝑘 and 𝑣′ = 𝑣 + 𝜏𝑊 (−1, −1). This means that Hom𝑛(𝜑(𝑊,𝑣) 𝐴ℓ, 𝐵ℓ+𝑘 ) is ,𝑣′) for 𝑊 𝑘 ℓ a ℓ canonically isomorphic to Hom𝑛 (cid:16) 𝜑(𝑊 𝑘 ℓ ,𝑣′) 𝐴ℓ, 𝐵ℓ+𝑘 (cid:17) . Set 𝑣 𝑘 ℓ := (𝑛 − (cid:12) (cid:12) (cid:12) 𝐴ℓ | (𝐵ℓ+𝑘 )∨(cid:12) (cid:12) (cid:12) , 0); we conclude that Hom𝑛 Ä𝜑(𝑊,𝑣) 𝐴ℓ, 𝐵ℓ+𝑘 ä (cid:27) Hom𝑛 (cid:16) 𝜑(𝑊 𝑘 ℓ ,𝑣 𝑘 ℓ ) 𝐴ℓ {𝑣′ − 𝑣 𝑘 ℓ }, 𝐵ℓ+𝑘 (cid:17) . Thus, in the G -graded case, we can absorb the first coordinate of the (cid:102)I -grading into the homological degree and view Hom𝑛(𝐴, 𝐵) as bigraded by Z × Z2. Then Hom𝑛 (𝐴, 𝐵)𝑘 (cid:27) (cid:214) ℓ∈Z (cid:27) (cid:214) ℓ∈Z (cid:16) Hom𝑛 𝜑(𝑊 𝑘 ℓ ,𝑣 𝑘 ℓ ) 𝐴ℓ, 𝐵ℓ+𝑘 (cid:17) Hom0 Ä∅, 𝐵ℓ+𝑘 | (𝐴ℓ)∨ {−𝑛, 0} ä (cid:27) Hom0 (cid:0)∅, 𝐵 | 𝐴∨ {−𝑛, 0}(cid:1) where the second isomorphism follows from Proposition 8.2.7. This proves the isomorphism on the level of bigraded abelian groups. The rest of the statement follows from the argument provided in the proof of Theorem 4.12 in [Hog20]. We will not review the proof here, but for the argument to apply we must show that (𝑔 | 1𝑎∨) ◦G 𝜙( 𝑓 ) = 𝜙(𝑔 ◦G 𝑓 ) = (1𝑐 | 𝑓 ∨) ◦G 𝜙(𝑔) where 𝑓 ∈ Hom𝑛(𝜑(𝑊1,𝑁1)𝑎, 𝑏), 𝑔 ∈ Hom𝑛(𝜑(𝑊2,𝑁2)𝑏, 𝑐), and 𝜙 : Hom𝑛(𝜑𝑊 𝑁 𝑎, 𝑐) → Hom0(∅, 𝑐 | 𝑎∨ {−𝑛, 0}) is the isomorphism from the proof of Proposition 8.2.7. Here, 𝑊1 : 𝑎 → 𝑏 and 157 𝑊2 : 𝑏 → 𝑐 are minimal cobordisms, and 𝑁1 = (𝑛 −(cid:12) (cid:12) , 0), thus 𝑔 ◦G 𝑓 ∈ Hom𝑛 (cid:0)𝜑(𝑊2◦𝑊1,𝑁1+𝑁2)𝑎, 𝑐(cid:1). The equality on the left-hand side is immediate. We will (cid:12) , 0) and 𝑁2 = (𝑛 −(cid:12) (cid:12)𝑏 | 𝑎∨(cid:12) (cid:12)𝑐 | 𝑏∨(cid:12) content ourselves by proving the right-hand side. To start, we claim that ( 𝑓 | 1𝑎∨) ◦ 𝜂𝑎 = (1𝑏 | 𝑓 ∨) ◦ 𝜂𝑎. Notice that the claim holds trivially when 𝑓 is a dot. When 𝑓 is a saddle, both 𝑓 and 𝑓 ∨ are necessarily merge, and their (cid:102)I -degree is Id. Thus, in this case, isotopy invariance implies the equality. Indeed, for any 𝑓 ∈ Hom𝑛(𝜑(𝑊1,𝑁1)𝑎, 𝑏), the (cid:102)I -degree of 𝑓 | 1𝑎∨ is supported entirely in the Z × Z-coordinate; the same is true for 1𝑏∨ | 𝑓 ∨. We denote this degree by 𝑣 𝑓 and, in this case, we have that 𝑣 𝑓 = 𝑣 𝑓 ∨. To conclude the proof of the claim, we have to show the equality holds for compositions 𝑔 ◦G 𝑓 , for 𝑓 and 𝑔 as above. First, notice that (𝑔 ◦G 𝑓 ) | 1𝑎∨ = (𝑔 | 1𝑎∨) ◦G ( 𝑓 | 1𝑎∨) by Proposition 5.3.6 (here, Ξ = 1 since 1𝑎∨ is two of the four inputted maps). On the other hand, (𝑔 | 1𝑎∨) ◦G ( 𝑓 | 1𝑎∨) = (𝑔 | 1𝑎∨) ◦ ( 𝑓 | 1𝑎∨) since each map in the composite has trivial (cid:102)I -degree. So, we have (𝑔 | 1𝑎∨) ◦ ( 𝑓 | 1𝑎∨) ◦ 𝜂𝑎 = (𝑔 | 1𝑎∨) ◦ (1𝑏 | 𝑓 ∨) ◦ 𝜂𝑏 = (1𝑐 | 𝑓 ∨) ◦ (𝑔 | 1𝑏∨) ◦ 𝜆(𝑤𝑔, 𝑣 𝑓 ) ◦ 𝜂𝑏 = (1𝑐 | 𝑓 ∨) ◦ (1𝑐 | 𝑔∨) ◦ 𝜆(𝑤𝑔, 𝑣 𝑓 ) ◦ 𝜂𝑐. The first and last equalities are by assumption. The second equality follows from applying a change of chronology. Notice that 𝜆(𝑤𝑔, 𝑣 𝑓 ) is, in this setting, equal to the value 𝜑𝐻 𝑓 ,𝑔. Then, again applying Proposition 5.3.6, we conclude that ((𝑔 ◦G 𝑓 ) | 1𝑎∨) ◦ 𝜂𝑎 = (1𝑐 | ( 𝑓 ∨ ◦G 𝑔∨)) ◦ 𝜑𝐻 𝑓 ,𝑔 ◦ 𝜂𝑐 = (1𝑐 | (𝑔 ◦G 𝑓 )∨) ◦ 𝜂𝑐. 158 We leave it to the reader to verify that one application of this claim implies that 𝜙(𝑔 ◦G 𝑓 ) = (1𝑐 | 𝑓 ∨) ◦G 𝜙(𝑔) concluding our proof. □ 8.3 Definition and properties of unified projectors Recall that the through-degree of a Temperley-Lieb diagram 𝛿, denoted 𝜏(𝛿), is the number of strands with endpoints on opposite ends of the disk. We say that 𝐴 ∈ Chom(𝑛)G has through-degree less than 𝑘 if 𝐴 is homotopy equivalent to a colimit of G -graded dg-modules F (𝛿) for Temperley- Lieb diagramas 𝛿 with 𝜏(𝛿) < 𝑘. In this case, we also write 𝜏(𝐴) < 𝑘. Since the tensor product commutes with colimits, we have that 𝜏(𝐴 ⊗ 𝐵) ≤ min{𝜏(𝐴), 𝜏(𝐵)}. Definition 8.3.1. We say that 𝐴 ∈ Chom(𝑛)G kills turnbacks from above if, for each 𝐵 ∈ Chom(𝑛)G with 𝜏(𝐵) < 𝑛, we have 𝐵 ⊗ 𝐴 ≃ ∗. Similarly, 𝐴 ∈ Chom(𝑛) kills turnbacks from below if, for each 𝐵 with 𝜏(𝐵) < 𝑛, 𝐴 ⊗ 𝐵 ≃ ∗. Since all Temperley-Lieb diagrams with through-degree less than 𝑘 can be built by stacking various generators 𝑒𝑖 of the Temperley-Lieb algebra, we have the following (stated without proof). Proposition 8.3.2. Let 𝑒𝑖 denote a standard generator of the Temperley-Lieb algebra. Then any object 𝐴 of Chom(𝑛)G kills turnbacks from above (resp. below) if and only if F (𝑒𝑖) ⊗ 𝐴 ≃ ∗ (resp. 𝐴 ⊗ F (𝑒𝑖) ≃ ∗. Definition 8.3.3. A unified Cooper-Krushkal projector (or simply unified projector) is a pair (𝑃𝑛, 𝜄) consisting of an object 𝑃𝑛 ∈ Chom(𝑛)G and a morphism 𝜄 : I𝑛 → 𝑃𝑛, called the unit of the projector, so that (CK1) Cone(𝜄) has through-degree less than 𝑛, and (CK2) the G -graded dg-module 𝑃𝑛 kills turnbacks (from above and below). 159 Lemma 8.3.4. If (𝑃𝑛, 𝜄) is a unified projector, there is a homotopy equivalence Hom𝑛(𝑃𝑛, 𝑃𝑛) → Hom𝑛(I𝑛, 𝑃𝑛) induced by 𝜄. Proof. Specifically, we will show that the pullback 𝜄∗ : Hom𝑛(𝑃𝑛, 𝑃𝑛) → Hom𝑛(I𝑛, 𝑃𝑛) is a homotopy equivalence. It suffices to show that Cone(𝜄∗) is contractible. We compute Cone(𝜄∗) ≃ Hom𝑛(Cone(𝜄), 𝑃𝑛) ≃ Hom𝑛(colim(F (𝛿)𝑖), 𝑃𝑛) (CK1), 𝜏(𝛿) < 𝑛 for all 𝑖 ≃ lim ←−− ≃ lim ←−− ≃ lim ←−− (Hom𝑛(F (𝛿)𝑖, 𝑃𝑛)) (Hom𝑛(I𝑛, 𝑃𝑛 ⊗ F (𝛿∨)𝑖)) (Hom𝑛(I𝑛, ∗)) ≃ ∗ as desired. Corollary 8.2.4 (CK2) □ Proposition 8.3.5 (Properties of unified projectors). Suppose (𝑃𝑛, 𝜄) and (𝑃′ 𝑛, 𝜄′) are two unified projectors of Chom(𝑛)G . 1. (Uniqueness) 𝑃𝑛 ≃ 𝑃′ 𝑛 ⊗ 𝑃𝑛 ≃ 𝑃′ 𝑛, and there is a homotopy equivalence ℎ : 𝑃𝑛 → 𝑃′ 𝑛 satisfying ℎ ◦ 𝜄 ≃ 𝜄′. 2. (Idempotence) (𝑃𝑛 ⊗ 𝑃𝑛, 𝜄 ⊗ 𝜄) is a projector; thus, by uniqueness, 𝑃𝑛 ⊗ 𝑃𝑛 ≃ 𝑃𝑛. 3. (Generalized absorbtion) More generally, for ℓ ≤ 𝑛 𝑃𝑛 ⊗ (cid:0)𝑃ℓ ⊔ I𝑛−ℓ(cid:1) ≃ 𝑃𝑛 ≃ (cid:0)𝑃ℓ ⊔ I𝑛−ℓ(cid:1) ⊗ 𝑃𝑛. 160 Proof. Consider the following G -graded commutative diagram. I𝑛 𝜄 𝜄′ 𝑃𝑛 𝑃′ 𝑛 I𝑛 ⊗ 𝑃𝑛 idI𝑛 ⊗𝜄 𝜄′⊗id𝑃𝑛 I𝑛 ⊗ I𝑛 𝜄′⊗𝜄 𝑃′ 𝑛 ⊗ 𝑃𝑛 𝜄′⊗idI𝑛 id𝑃′ 𝑛 ⊗𝜄 𝑃′ 𝑛 ⊗ I𝑛 The unmarked arrows are isomorphisms coming from multigluing (or, if one likes, the probable the monoidal structure of Chom(𝑛)G ). Since this diagram is G -graded commutative, it commutes up to homotopy, which is all we need going forward. For the proof of uniqueness, notice that 𝜄′ ⊗ id𝑃𝑛 is a homotopy equivalence, as Cone(𝜄′ ⊗ id𝑃𝑛) ≃ Cone(𝜄′) ⊗ 𝑃𝑛 ≃ ∗, using (CK1) and (CK2). By the same reasoning, id𝑃′ 𝑛 ⊗ 𝜄 is a homotopy equivalence, thus 𝑃𝑛 ≃ 𝑃′ 𝑛 ⊗ 𝑃𝑛 ≃ 𝑃′ 𝑛. Then, since both these maps are homotopy equivalences, choosing a homotopy inverse for, say, id𝑃′ 𝑛 ⊗ 𝜄 induces a (class of) homotopy equivalence(s) ℎ : 𝑃𝑛 → 𝑃′ 𝑛 satisfying ℎ ◦ 𝜄 ≃ 𝜄′. To see that ℎ is unique up to homotopy, suppose ℎ1, ℎ2 are two homotopy equivalences satisfying, for 𝑖 = 1, 2, ℎ𝑖 ◦ 𝜄 ≃ 𝜄′, and that ℎ2 is a homotopy inverse for ℎ2. Then (𝜄 − ℎ2 ◦ ℎ1 ◦ 𝜄) = (id𝑃𝑛 − ℎ2 ◦ ℎ1) ◦ 𝜄 ∈ Hom𝑛(I𝑛, 𝑃𝑛) is nullhomotopic, so Lemma 8.3.4 implies that id𝑃𝑛 − ℎ2 ◦ ℎ1 is as well; thus ℎ1 ≃ ℎ2. For idempotence, replace 𝑃′ 𝑛 in the diagram with 𝑃𝑛 everywhere. Then we have that 𝑃𝑛 ⊗ 𝑃𝑛 ≃ 𝑃𝑛. More generally, that 𝑃𝑛 ⊗ 𝑃𝑛 kills turnbacks is clear by the monoidal structure of Chom(𝑛). Then, since 𝜄 ⊗ id𝑃𝑛 is a homotopy equivalence, the homotopy commutativity of the diagram implies that Cone(𝜄 ⊗ 𝜄) ≃ Cone(idI𝑛 ⊗ 𝜄) ≃ ∗. More generally, for ℓ < 𝑛, 𝑃ℓ comes equipped with unit 𝜄ℓ : Iℓ → 𝑃ℓ. Then, it is clear that id𝑃𝑛 ⊗ (𝜄ℓ ⊔ idI𝑛−ℓ ) : 𝑃𝑛 ⊗ (𝑃ℓ ⊔ I𝑛−ℓ) −→ 𝑃𝑛 ⊗ I𝑛 ≃ 𝑃𝑛 is a homotopy equivalence (its cone is contractible by (CK2)). The other homotopy equivalence is analogous. □ 161 Remark 8.3.6. We can define projectors for the category Chom(𝑛)𝑞 similarly. Notice that projec- tors of Chom(𝑛)G descend to projectors of Chom(𝑛)𝑞; in addition, given any (𝑊, 𝑣) ∈ I with deg𝑞(𝜑𝑊 𝑣 ) = 0, 𝜑𝑊 𝑣 𝑃𝑛 defines a projector of Chom(𝑛)𝑞. In future work, we hope to find particular elements 𝑈𝑛 ∈ Hom𝑛(𝑃𝑛, 𝑃𝑛) coming from an action on 𝑃𝑛, as in [Hog19]. Fundamental to this study is the homotopy equivalence between the endomorphism complex of 𝑃𝑛 and (a shift of) the closure of 𝑃𝑛. We point out that Theorem 8.1.5 and Lemma 8.3.4 imply a generalization of this result in the unified setting; we state it for the 𝑞-graded category. For 𝐴, 𝐵 G -graded dg 𝐻𝑛-modules let Hom𝑞 𝑛(𝐴, 𝐵) denote the HOM-complex Hom𝑛(𝐴, 𝐵) obtained by collapsing G -grading. Corollary 8.3.7. If 𝑃𝑛 is a unified projector, it descends to a projector in Chom(𝑛)𝑞. We have that Hom𝑞 𝑛(𝑃𝑛, 𝑃𝑛) (cid:27) 𝑞−𝑛Tr𝑛(𝑃𝑛). Proof. Apply Lemma 8.3.4 and then apply Theorem 8.1.5 𝑛-times. □ 8.4 Explicit computations for the 2-stranded projector Finally, our previous work allows us to mimic [CK12] in the G -graded (that is, unified) setting. Consider the complex we will call 𝑃2, which has the form · · · 𝐶−4 𝜑Ä , (−2,−2)ä 𝐶−3 𝜑Ä , (−1,−1)ä 𝐶−2 𝜑 𝐶−1 where 𝐶𝑖 =    𝑖 = −1 • − • 𝑖 = −2𝑘 • + 𝑋𝑌 • 𝑖 = −2𝑘 − 1 for all 𝑖 < 0. Notice that taking 𝑋, 𝑌 , 𝑍 ↦→ 1 recovers a 2-strand projector of [CK12]; taking 𝑋, 𝑍 ↦→ 1 and 𝑌 ↦→ −1 recovers the one of [Sch22]. Proposition 8.4.1. 𝑃2 ∈ Chom(2)G . 162 Proof. For the first case, notice that 𝐶−1 ◦ 𝐶−2 = 0 just as in the even case: passing a dot below a saddle and then back up the opposing side introduces two changes of chronology whose evaluations are inverse to one another, since 𝜆(𝑣, 𝑢) = 𝜆(𝑢, 𝑣)−1. The other two cases are slightly different since dots may not move past each other freely, but rather by multiplication by 𝑋𝑌 : (cid:16) • + 𝑋𝑌 • (cid:17) ◦ (cid:16) • − • (cid:17) = top • • − bot = −𝑋𝑌 • • + 𝑋𝑌 • • • • + 𝑋𝑌 • • − 𝑋𝑌 • • as desired, recalling that two dots are evaluated as zero. The other composition is the same. □ Proposition 8.4.2. The chain complex 𝑃2 ∈ Chom(2)G is a unified Cooper-Krushkal projector. Proof. (CK1) is satisfied clearly. We must check (CK2), that 𝑃2 is killed by turnbacks. We will show that 𝑒1 ⊗ 𝑃2 ≃ 0; the other direction is totally similar. We have 𝑒1 ⊗ 𝜑Ä ,(−𝑛,−𝑛)ä (cid:27) 𝜑Ç å ,(−𝑛,−𝑛) . Thus, the previously ambiguous saddles appearing in the shifting functors of 𝑃2 are seen to be a merge upon tensoring with 𝑒1. Merges have the effect of shifting Z × Z degree by (−1, 0), so we conclude that 𝜑Ç ,(−𝑛,−𝑛) å (cid:27) {−(𝑛 + 1), −𝑛}. Consequently, the chain complex 𝑒1 ⊗ 𝑃2 has the form + 𝑋𝑌 • • − • • · · · {−3, −2} {−2, −1} {−1, 0} . 163 Delooping yields the complex · · · ⊕ 𝑌 𝑍 −1 • {−3, −3} 𝑋𝑌 𝑋 𝑍 −1 • {−2, −2} ⊕ 𝑌 𝑍 −1 • {−2, −2} −1 𝑋 𝑍 −1 • {−1, −1} 𝑋 𝑍 −1 • {−1, −1} ⊕ 1 where each of the maps down and to the right are zero and are therefore not pictured. Simplifying the maps after delooping is not difficult—one need only take caution when applying the S1 relation. Noting that each of the nonzero, diagonal maps are invertible, simultaneous Gaussian elimination (Proposition 2.2.5) implies that this complex is homotopy equivalent to the zero complex. □ 8.4.1 Homology of the trace As in the even case, the unified projector satisfies a categorification of the closure property ⟨Tr(𝑝𝑛)⟩ = [𝑛 + 1]. In the 𝑛 = 2 case, notice that 𝜑Ç å ,(−𝑛,−𝑛) = {−𝑛, −(𝑛 + 1)} because the typically ambiguous saddle is a split after taking closure. Then, we see that the complex Tr2(𝑃2) has the form · · · (1 + 𝑋𝑌 ) • {−2, −3} 0 {−1, −2} {0, −1} . (8.4.1) Then we compute 𝐻𝑛(Tr2(𝑃2)) =    𝑅{2, 0} ⊕ 𝑅{1, −1} 0 𝑛 = 0 𝑛 = −1 (8.4.2) 𝑅{−2𝑘 + 2, −2𝑘 } ⊕ 𝑅 (1+𝑋𝑌 )𝑅 {−2𝑘 + 1, −2𝑘 − 1} 𝑛 = −2𝑘 (1 − 𝑋𝑌 )𝑅{−2𝑘 + 1, −2𝑘 − 1} ⊕ 𝑅{−2𝑘, −2𝑘 − 2} 𝑛 = −2𝑘 − 1 164 whenever 𝑘 > 0. Note that we recover the solution in the even case (see Section 4.3.1 of [CK12]) when 𝑋, 𝑌 , 𝑍 ↦→ 1. In the odd case, we see that it is important to specialize coefficients before taking homology, since the dotted map is killed by setting 𝑌 = −1. In either case , the Euler characteristic reproduces [3] = 𝑞2 + 1 + 𝑞−2, despite infinite homology. 8.4.2 Unified Khovanov homology of the infinite 2-twist While we have succeeded in constructing a representative for the second projector by guessing based on the result in the even case, we will prove the existence of unified projectors in the following section based on the suspicion that it ought to correspond to the Khovanov complex of an infinite twist ([Roz14, Wil18, SW24]). We’ll illustrate this fact in the 𝑛 = 2 case, using multigluing to compute the Khovanov complex for 2-strand torus braids, yielding a unified Cooper-Krushkal projector. Perhaps it is interesting that the projector obtained in this way has a slightly different appearance compared to 𝑃2 in the previous sections, although the homotopy equivalence is obvious. To a single (negative) crossing we associate the complex 𝜑 Thus, to the torus 2-braid with two negative crossings we assoiciate the complex (cid:32) 𝜑 (cid:33) (cid:32) ⊗ 𝜑 (cid:33) , which is isomorphic to 𝜑 𝜑 𝜑 ⊕ 165 Focusing on the leftmost vertex, we’ve shown that 𝜑 (cid:27) 𝜑Ä ,(−1,0)ä. Moreover, delooping tells us that we have the following isomorphism for any 𝑛: 𝜑Ä ,(−𝑛,1−𝑛)ä • • 𝜑Ä ,(−𝑛,−𝑛)ä ⊕ 𝜑Ä ,(1−𝑛,1−𝑛)ä Thus the original complex is isomorphic to the following complex. 𝜑Ä ,(−1,−1)ä 1 ⊕ 𝜑 𝑋 𝑍 • 𝑋 𝑍 • 1 𝜑 𝜑 ⊕ The 𝑋 𝑍 = 𝜆((−1, 0), (−1, −1)) factor comes from sliding a dot past a merge: • = 𝜆((−1, 0), (−1, −1)) = 𝑋 𝑍 • • Then, applying Gaussian elimination, we obtain the following complex. 𝜑Ä ,(−1,−1)ä 𝑋 𝑍 • − 𝑋 𝑍 • 𝜑 To stack with another crossing means to tensor this complex with the original single crossing complex. After delooping, this complex has the following form (arrows which are not pictured are 166 zero; dotted arrows are ones which die during Gaussian elimination). 𝜑Ä ,(−2,−2)ä 𝑋𝑌 𝑍 2 • 𝑋 𝑍 • −𝑋 𝑍 𝜑Ä ,(−1,−1)ä 𝜑Ä ,(−1,−1)ä 𝜑 1 𝑋 𝑍 • 𝑋 𝑍 • 1 1 𝜑Ä ,(−1,−1)ä 𝜑 𝜑 The arrows in the left-most column are obtained by applying sphere relations. Note that we can apply S1 and dot-sliding relations to obtain the following equivalences. • = 𝑌 𝑍 • and • = 𝑋 𝑍 • • • Applying Gaussian elimination, the above complex is homotopy equivalent to the following. 𝜑Ä ,(−2,−2)ä 𝑋𝑌 𝑍 2 • + 𝑍 2 • 𝜑Ä ,(−1,−1)ä 𝑋 𝑍 • − 𝑋 𝑍 • 𝜑 At this point a pattern emerges which controls the complex for any two stranded braid (although this might be easier to see computing the next case; we leave it to the reader). The complex has the form · · · 𝜑Ä ,(−3,−3)ä 𝐶−4 𝜑Ä ,(−2,−2)ä 𝐶−3 𝜑Ä ,(−1,−1)ä 𝐶−2 𝜑 𝐶−1 167 where 𝐶𝑖 =    𝑋 𝑍 2𝑘−1 (cid:16) • − • 𝑍 2𝑘 (cid:16) 𝑋𝑌 • + • (cid:17) (cid:17) 𝑖 = −1 𝑖 = −2𝑘 𝑖 = −2𝑘 − 1 for all 𝑖 < 0. As promised, this complex is homotopy equivalent to the 𝑃2 we guessed earlier on. 8.5 Existence of unified projectors In [Roz14], Rozansky showed that the Khovanov complex associated to an infinte twist on 𝑛 strands is a Cooper-Krushkal projector. In [Wil18], Willis generalized this argument to the spectral setting. His argument was further generalized in [SW24] for the setting of spectral multimodules. We will adapt the arguments of [SW24] to prove that unified Cooper-Krushkal projectors exist. As in the work of Stoffregen-Willis, the left-handed fractional twist complex, denoted T𝑛, is the complex associated to the diskular 𝑛-tangle shown below. Superscripts will indicate stacking: · · · ... · · · T 𝑚 𝑛 = T𝑛 ⊗ · · · ⊗ T𝑛 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:124) (cid:123)(cid:122) 𝑚-times with T 0 𝑛 = I𝑛. Notice that T 𝑛 𝑛 can be viewed as a pure braid; we call this the left-handed full twist complex. Finally, for any 𝑛 ∈ N, the left-handed infinite twist complex, denoted T ∞ 𝑛 , is defined as the colimit of the sequence 𝑛 = colim Ä T ∞ 𝑛 → T 1 T 0 𝑛 → · · · T 𝑚 𝑛 → · · · ä where each arrow comes from compositions of maps arising from the cofibration sequence Å F ã −→ F Å ã −→ 𝜑 F Å ã [1] 168 of Proposition 6.2.2. By the same proposition, · · · T 𝑛𝑘+𝑟 𝑛 = Cone         𝜑 · · · · · · T 𝑛𝑘+𝑟 𝑛 · · · −−−−−−−−−→ T 𝑛𝑘+𝑟 · · · 𝑛         We start our argument by computing a simplification of the term T 𝑛𝑘+𝑟 𝑛 ⊗ 𝑒𝑖, for 0 ≤ 𝑟 < 𝑛 and 1 ≤ 𝑖 ≤ 𝑛 − 1. Note that Å F ã (cid:27) F Å ã {−1, −1} and Å F ã (cid:27) 𝜑 Å F ã by delooping and Gaussian elimination. We’ll write 𝑒𝑖 as 𝑒top 𝑖 ⊗ 𝑒bot 𝑖 , although this tensor product is not exactly the same as the one in Definition 8.1.1; we do not belabor the point. Assume that 𝑟 = 0. Then T 𝑛𝑘 𝑛 is 𝑘-full twists, and we have that 𝑛 ⊗ 𝑒𝑖 = 𝑒top T 𝑛𝑘 𝑖′ ⊗ 𝜑𝑊 𝑛𝑘 𝑛 (𝑛−2)𝑘 T 𝑛−2 {−2𝑘, −2𝑘 } ⊗ 𝑒bot 𝑖 where 𝑊 𝑛𝑘 𝑛 is a cobordism consisting of 2𝑘(𝑛 − 2) saddles (for the 2𝑘(𝑛 − 2)-many Reidemeister II moves performed) and 𝑖′ = 𝑖 + 𝑟 mod 𝑛. There are also 2𝑘 Reidemeister I moves, accounting for the Z × Z-shift. To aid in comprehending 𝜑𝑊 𝑛𝑘 𝑛 , consider Figure 8.1. We remark that the tensor on the left is vertical stacking as in definition 8.1.1, and the one on the right is as in the writing of 𝑒top 𝑖 ⊗ 𝑒bot 𝑖 . Notice that 𝑒top 0 is allowed; by this we mean the following picture. 𝑒top 0 := · · · Now, for 1 ≤ 𝑟 < 𝑛, there are three cases. 1. If 𝑖 < 𝑛 − 𝑟, the extra isotopy contains no Reidemeister I moves, but it does consist of 𝑟-many Reidemeister II moves. 2. If 𝑖 = 𝑛 − 𝑟, the isotopy contains (𝑟 − 1) more Reidemeister II moves and exactly 1 more Reidemeister I move. Note that 𝑖′ = 0 in this case. 169 RII RII RI {−1, −1} RII {−2, −2} RI {−1, −1} RII {−1, −1} Figure 8.1 Computing the grading shift on T 4 4 ⊗ 𝑒1. 3. If 𝑖 > 𝑛 − 𝑟, the isotopy contains the addition of a sequence of (𝑟 − 2) many Reidemeister II moves, then 1 Reidemeister I move, followed by (𝑛 − 2) more Reidemeister II moves, and another lone Reidemeister I move; that is, (𝑛+𝑟 −4) Reidemeister II moves and 2 Reidemeister I moves. So, we have proven the following. Lemma 8.5.1. For any 0 ≤ 𝑟 < 𝑛 and 0 < 𝑖 < 𝑛, T 𝑛𝑘+𝑟 𝑛 ⊗ 𝑒𝑖 ≃ 𝑒top 𝑖′ ⊗ 𝜑𝑊 𝑛𝑘+𝑟 𝑛 (𝑛−2)𝑘+𝑟𝑖 T 𝑛−2 {−(2𝑘 + 𝑘𝑖), −(2𝑘 + 𝑘𝑖)} ⊗ 𝑒bot 𝑖 170 where 𝑖′ = 𝑖 + 𝑟 mod 𝑛 and 1. if 𝑖 < 𝑛 − 𝑟, 𝑊 𝑛𝑘+𝑟 𝑛 consists of 2𝑘(𝑛 − 2) + 𝑟 saddles, 𝑟𝑖 = 𝑟, and 𝑘𝑖 = 0; 2. if 𝑖 = 𝑛 − 𝑟, 𝑊 𝑛𝑘+𝑟 𝑛 consists of 2𝑘(𝑛 − 2) + (𝑟 − 1) saddles, 𝑟𝑖 = 𝑟 − 1, and 𝑘𝑖 = 1; 3. if 𝑖 > 𝑛 − 𝑟, 𝑊 𝑛𝑘+𝑟 𝑛 consists of 2𝑘(𝑛 − 2) + (𝑛 + 𝑟 − 4) saddles, 𝑟𝑖 = 𝑟 − 2, and 𝑘𝑖 = 2. In each of these cases, 𝑊 𝑛𝑘+𝑟 𝑛 is a cobordism in the style of Figure 8.1. We’ll denote by 𝑠𝑖 the number of additional saddles depending on 𝑟. That is, 𝑊 𝑛𝑘+𝑟 𝑛 consists of 2𝑘(𝑛 − 2) + 𝑠𝑖 saddles, where 1. 𝑠𝑖 = 𝑟 if 𝑖 < 𝑛 − 𝑟; 2. 𝑠𝑖 = 𝑟 − 1 if 𝑖 = 𝑛 − 𝑟; 3. 𝑠𝑖 = 𝑛 + 𝑟 − 4 if 𝑖 > 𝑛 − 𝑟. Note that our 𝑠𝑖 is not the same as the one appearing in [SW24]. We will use this Lemma to prove the existence of projectors. First, we would like to draw some connections between our work and computations found in Section 5 of [SW24]. Consider the complex C𝑚+1 defined as the cone C𝑚+1 := Cone(T 𝑚 𝑛 → T 𝑚+1 𝑛 ). Then, C𝑚+1 looks like (that is, is homotopy equivalent to) a cube of resolutions for T 1 𝑛 with T 𝑚 𝑛 stacked on top, modulo the identity term, which is taken to be zero. Any entry of the cube of resolutions for T𝑛 (apart from the identity entry, which we have avoided) is isomorphic to F (𝑒𝑖) ⊗ F (𝛿) for some flat diskular 𝑛-tangle 𝛿 and 1 ≤ 𝑖 ≤ 𝑛 − 1. Dropping the F notation, this is to say that C𝑚+1 is homotopy equivalent to a colimit in which all nontrivial terms are of the form 𝜑𝛼1 𝑛 T 𝑚 𝑛 ⊗ 𝑒𝑖 ⊗ 𝛿 171 where 𝜑𝛼1 𝑛 denotes the grading shift coming from the cube of resolutions for T 1 𝑛 . Writing 𝑚 = 𝑛𝑘 +𝑟, Lemma 8.5.1 says that this term is equivalent to 𝜑𝛼1 𝑛 Ä𝑒top 𝑖′ ⊗ 𝜑𝑊 𝑛𝑘+𝑟 𝑛 (𝑛−2)𝑘+𝑟𝑖 T 𝑛−2 {−(2𝑘 + 𝑘𝑖), −(2𝑘 + 𝑘𝑖)} ⊗ 𝑒bot 𝑖 ä ⊗ 𝛿. (8.5.1) As in [SW24], we want to provide a bound on grading shifts. On one hand, given a G -graded dg 𝐻𝑛-module 𝐴, by a global upper 𝑞-bound on G -grading shifts, we mean some 𝐵 ∈ Z so that, for each entry 𝐴𝑖 of 𝐴 with grading shift 𝜑𝑊 𝑣 , deg𝑞(𝜑𝑊 𝑣 ) ≤ 𝐵. For example, we can compute an upper bound of a complex with G -grading finding the minimum number of saddles appearing in each grading shift and maximizing the Z × Z-degree. We define global lower bounds similarly. This definition extends to a stricter notion on objects of Kom(𝐻𝑛ModG ) by taking the minimum (resp. maximum) among all global upper (resp. global lower) bounds for each complex 𝐴′ homotopy equivalent to 𝐴. Referring again to Proposition 6.2.2, to any diskular tangle 𝑇, F (𝑇) has an entry with trivial G -grading; this is to say that a global upper bound on F (𝑇) is 0. Similarly, a global lower bound is given by −𝑐(𝑇), for 𝑐(𝑇) the number of crossings in the diagram for 𝑇. Note that 𝜑𝛼1 𝑛 always consists of at least one saddle, by construction. Then, we can compute the 𝑞-grading shift on (8.5.1) on a case-by-case basis via Lemma 8.5.1 and conclude that C𝑚+1 is homotopy equivalent to a complex with global upper bound on G -grading 𝑏𝜖 ≤ 𝐵𝑚+1 := −2𝑛𝑘 − 𝑟 − 1. Observe that this bound is similar to the one provided in [SW24]. Remark 8.5.2. As in [SW24], we can present a model in which T ∞ 𝑛 is an iterated mapping cone. Start by setting A1 = T 1 𝑛 and, inductively, assume A2, . . . , A𝑚 have been constructed, each 𝑛 . We construct A𝑚+1 as follows. From the definition of C𝑚+1, there is an exact satisfying Aℓ ≃ T ℓ triangle T 𝑚 𝑛 T 𝑚+1 𝑛 C𝑚+1, thus there is a map 𝜓𝑚 so that T 𝑚+1 𝑛 ≃ Cone(C𝑚+1 𝜓𝑚 −−→ T 𝑚 𝑛 ). 172 Now, using Lemma 8.5.1, we have argued that C𝑚+1 is homotopy equivalent to a complex we’ll call C′ 𝑚+1 with glabal upper bound 𝐵𝑚+1. Let 𝜓′ 𝑚 denote the map defined by the commutative square C𝑚+1 ∼ C′ 𝑚+1 𝜓𝑚 𝜓′ 𝑚 T 𝑚 𝑛 ∼ A𝑚 where each vertical arrow is a homotopy equivalence. Then, set A𝑚+1 := Cone(C′ 𝑚+1 𝜓′ 𝑚−−→ A𝑚). Unfurling definitions and homotopy equivalences, it follows that A𝑚+1 ≃ T 𝑚+1 𝑛 . In particular, A𝑚+1 is obtained from A𝑚 by including finitely many new entries with G -grading shifts bounded from above by 𝐵𝑚+1. As 𝑚 → ∞, 𝐵𝑚+1 → −∞, and we obtain a model for 𝑛 ≃ A∞ as an iterated mapping cone. T ∞ On the other hand, define a global upper Z × Z-bound on G -grading shifts to be some (𝐵1, 𝐵2) ∈ Z × Z so that, for each 𝐴𝑖 of 𝐴 with grading shift 𝜑𝑊 (𝑣1,𝑣2), we can find a simplification of 𝜑𝑊 (𝑣1,𝑣2), written 𝜑 ˇ𝑊 (𝑣′ for ˇ𝑊 a minimal cobordism void of births, deaths, and unambiguous saddles. 2 ≤ 𝐵2. By a simplification, we that 𝜑𝑊 (𝑣1,𝑣2) (cid:27) 𝜑 ˇ𝑊 (𝑣′ 1 ≤ 𝐵1 and 𝑣′ ), in which 𝑣′ ,𝑣′ 2 ,𝑣′ 2 1 1 ) Notice that, since 𝜑𝑊 𝑛𝑘+𝑟 𝑛 consists only of saddles, we have that (−2𝑘, −2𝑘) provides a global upper Z × Z-bound on G -grading shifts for a complex homotopy equivalent to T 𝑛𝑘+𝑟 𝑛 ⊗ 𝑒𝑖. Theorem 8.5.3. For each 𝑛, T ∞ 𝑛 is a unified projector. Proof. Recall that T ∞ 𝑛 is defined as the colimit 𝑛 = colim Ä T ∞ 𝑛 → T 1 T 0 𝑛 → · · · T 𝑚 𝑛 → · · · ä which we’ll write colim(T 𝑛𝑘+𝑟 𝑛 ). Axiom (CK1) is apparent by definition, so we will content ourselves with a proof of (CK2). First, notice that colim(T 𝑛𝑘+𝑟 𝑛 ) ⊗ 𝑒𝑖 ≃ colim(T 𝑛𝑘+𝑟 𝑛 ⊗ 𝑒𝑖) 173 so if the homology of the colimit on the right-hand side is trivial, we can conclude that the colimit itself is contractible, thus T ∞ 𝑛 ≃ ∗. Recall that any homology class of the colimit arises as a homology class in a piece of the colimit. However, by Lemma 8.5.1, this colimit is built from complexes with a global upper Z × Z-bound of (−2𝑘, −2𝑘). As 𝑚 → ∞, 𝑘 → ∞, and the global upper bound goes to (−∞, −∞), so any nontrivial homology class must die in the colimit. □ 8.6 A unified colored link homology With very little work, the existence of unified projectors (together with multigluing) implies the existence of a unified colored link homology specializing to an even one ([CK12], see also [Kho05, BW08] by way of [BHPW23]), but also specializing to a new odd version. Recall the following definition, adapted from Definition 5.1 of [CK12]. Definition 8.6.1. For any 𝑛 ∈ N and m = (𝑚1, . . . , 𝑚𝑛) ∈ N𝑛, we denote by Chomm(𝑛)G the category where • ob(Chomm(𝑛)G ) = ob(Chom(𝑛)G ) and • HomChomm(𝑛)G (𝐴, 𝐵) = HomChom(𝑀𝑛)G (Πm(𝐴), Πm(𝐵)) where 𝑀 = (cid:205)𝑖 𝑚𝑖 and Πm replaces the 𝑖th strand in each diagram with its 𝑚𝑖th parallel composed with a copy of the 𝑚𝑖th projector. We define Chomm(𝑛)𝑞 by taking objects and morphisms of Chomm(𝑛)G and collapsing degree, as usual. We will represent projectors by small boxes, e.g., 𝑃𝑛 = 𝑛 . We will define the operation Π𝑚 on links, via operations on diskular tangles, as follows. As an example, if 𝐾 is a knot, let ˚𝐾 denote the diskular 1-tangle 𝐾 × and suppose ˚𝐾 𝑚 denotes its 𝑚th parallel. Then Π𝑚(𝐾) = Tr𝑚(Kh𝑞( ˚𝐾 𝑚) ⊗ 𝑃𝑚) More generally, if 𝐿 is an 𝑛-component link, we use multigluing. Let m = (𝑚1, . . . , 𝑚𝑛) ∈ N𝑛, and denote by 𝑇 m 𝐿 the result of taking 𝑚𝑖 parallel copies of the 𝑖th component of 𝐿 and then removing 174 a small diskular region from each of the original components (again, see Figure 1.1). Then, set Πm(𝐿) := (𝑃𝑚1, . . . , 𝑃𝑚𝑛) ⊗(𝐻𝑚1 ,...,𝐻𝑚𝑛 ) Kh𝑞(𝑇 m 𝐿 ) where each of the 𝑃𝑚𝑖 is viewed as an object of Chom(𝑚𝑖)𝑞 𝑅. Lemma 8.6.2. We have the following isomorphisms in Kom(𝐻𝑚+𝑛Mod)G : and 𝑚 (cid:27) 𝑚 (cid:27) 𝑚 𝑚 𝑛 𝑛 (cid:27) 𝑛 (cid:27) 𝑛 That is, free (parallel) strands can be moved over or under projectors in Kom(𝐻2𝑛Mod)G . Proof. We’ll explain the first homotopy equivalence; the others are proven with the same procedure. The trick is to start with the middle complex: using (CK1), 𝑚 𝑚 is homotopy equivalent to the complex of complexes 𝑚 −→ 𝑚 𝑐 where 𝑐 = Cone(𝜄). Again by (CK1), 𝑐 has through degree < 𝑚, so it contains some turnback. Pushing the turnback through the parallel overstrands induces nontrivial G -grading shifts (see Lemma 7.2.4), but after it passes through all 𝑛 overstrands, (CK2) tells us that that the entire complex on the right is contractible, and we’re done. □ Using this Lemma, together with multiguling (Theorem 6.2.4) and idempotence (Proposition 8.3.5), Πm can be described up to homotopy as sending ↦→ 𝑚𝑖 and ↦→ 𝑚𝑖 𝑚𝑗 𝑚𝑗𝑚𝑖 on the 𝑖th strand and each crossing of the 𝑖th strand under the 𝑗th. Theorem 8.6.3. The category Komm(𝑛)𝑞 contains invariants of framed tangles. 175 Proof. Applying Πm to the following typical diskular 2-tangle and applying idempotence (𝑃𝑛 ⊗𝑃𝑛 ≃ 𝑃𝑛) and Lemma 8.6.2, we obtain ↦→ Taking Kh (after picking any orientation), we know that (cid:32) Kh (cid:33) Ç ≃ 𝜑𝑊 𝑣 Kh ≃ å . Ç å ≃ 𝜑𝑊 𝑣 Kh where 𝜑𝑊 𝑣 is the grading shift obtained by 𝑚𝑖𝑚 𝑗 Reidemeister II moves (appeal to Lemma 7.2.4 for an exact value, if desired). We have that deg𝑞(𝜑𝑊 𝑣 ) = 0 by Theorem 7.2.9 which concludes the argument for the first framed tangle move. The argument for Reidemeister III moves is similar and left to the reader. □ If 𝐿 is a link, we denote by H (𝐿; m) the homology of Πm(𝐿). Moreover, denote by Πm 𝑒 (𝐿) and Πm 𝑜 (𝐿) the complexes obtained from Πm(𝐿) by taking 𝑋, 𝑌 , 𝑍 ↦→ 1 and 𝑋, 𝑍 ↦→ 1 and 𝑌 ↦→ −1 respectively. These complexes are also invariants of the framed link (𝐿; m); denote their respective homology by H𝑒(𝐿; m) and H𝑜(𝐿; m). We write 𝜒𝑞 to denote the graded Euler characteristic which records only the 𝑞-grading associated to a particular G -grading or G -grading shift. By definition, 𝜒𝑞(H𝑒(𝐿; m)) = 𝐽(𝐿; m)(𝑞) = 𝜒𝑞(H𝑜(𝐿; m)) where 𝐽(𝐿; m)(𝑞) denotes the colored Jones polynomial with indeterminate 𝑞. While H𝑒(𝐿; m) is the colored link homology of [CK12], H𝑜(𝐿; m) provides a new categorification of the colored Jones polynomial of 𝐿. To verify that the two homologies are distinct, recall that the computation in §8.4.1 implies that H𝑒(𝑈; 2) ≇ H𝑜(𝑈; 2) for 𝑈 the unknot. 176 CHAPTER 9 TOWARD A HOCHSHILD (CO)HOMOLOGY FOR C-GRADED ALGEBRAS We conclude this thesis with a chapter initiating future investigations concerning C-graded struc- tures. Namely, in this chapter, we provide a generalization of Hochschild homology which extends to C-graded algebras 𝐴 with coefficients in a C-graded (𝐴, 𝐴)-bimodule. This work is presented in more detail in [Spy25], where the constructions are applied to the unified Khovanov theory for tangles, C = G. In this chapter, we assume that (C, 𝛼) is a grading category. As a lead-in, we will eventually need to assume that our unitors are picked in a canonical manner. Recall that the category ModC is monoidal: define a monoidal product 𝑀 ⊗ 𝑁 by 𝑀 ⊗ 𝑁 := (cid:202) 𝑔∈Mor(C) (𝑀 ⊗ 𝑁)𝑔 where (𝑀 ⊗ 𝑁)𝑔 := (cid:202) 𝑔=𝑔2◦𝑔1 𝑀𝑔1 ⊗ 𝑁𝑔2. The coherence isomorphism is induced by the associator: fix 𝛼 : (𝑀1 ⊗ 𝑀2)⊗ 𝑀3 → 𝑀1 ⊗(𝑀2 ⊗ 𝑀3) by (cid:12)𝑦(cid:12) (𝑥 ⊗ 𝑦) ⊗ 𝑧 ↦→ 𝛼(|𝑥| ,(cid:12) (cid:12) ,|𝑧|)𝑥 ⊗ (𝑦 ⊗ 𝑧) for homogeneous elements 𝑥, 𝑦, and 𝑧. The fact that 𝛼 satisfies the pentagon relation follows directly from the cocycle condition of the grading category. The unit object is given by where Id𝑋 denotes the identity morphisms in C on 𝑋. In general, left- and right-unitors L : 𝐼C := (cid:202) KId𝑋 𝑋∈Ob(C) 𝐼C ⊗ 𝑀 → 𝑀 and R : 𝑀 ⊗ 𝐼C → 𝑀 are given by any isomorphisms satisfying the triangle relation: 𝛼 (𝑀 ⊗ 𝐼C) ⊗ 𝑁 𝑀 ⊗ (𝐼C ⊗ 𝑁) R⊗1𝑁 1𝑀 ⊗L 𝑀 ⊗ 𝑁 When needed, we will denote the chosen unitors for ModC by LC and RC. Indeed, the unitors can be chosen to be induced by the associator. For example, one can take • L : 𝐼C ⊗ 𝑀 → 𝑀 by (𝑘 ⊗ 𝑚) ↦→ L(|𝑘 | ,|𝑚|)𝑘𝑚, fixing L(|𝑘 | ,|𝑚|) := 𝛼(Id𝑋, Id𝑋,|𝑚|)−1, (9.0.1) 177 and • R : 𝑀 ⊗ 𝐼C → 𝑀 by (𝑚 ⊗ 𝑘) ↦→ R(|𝑚| ,|𝑘 |)𝑘𝑚, fixing R(|𝑚| ,|𝑘 |) := 𝛼(|𝑚| , Id𝑌 , Id𝑌 ), (9.0.2) where |𝑚| : 𝑋 → 𝑌 . To see that the triangle relation is satisfied, notice that for 𝑋 𝑔 −→ 𝑌 ℎ −→ 𝑍, 1 = 𝑑𝛼(𝑔, Id𝑌 , Id𝑌 , ℎ) = 𝛼(𝑔, Id𝑌 , Id𝑌 )𝛼(𝑔, Id𝑌 , ℎ)−1𝛼(Id𝑌 , Id𝑌 , ℎ). Notice that, in general, the cocycle relation implies 𝛼(𝑔, 𝑔, 𝑔) = 1 for any loop morphism 𝑔 : 𝑋 → 𝑋. In the case of the above choice of unitors, this means that whenever |𝑚| = Id𝑋 for any 𝑋 ∈ ob(C), we have that L(𝑘 ⊗ 𝑚) = 𝑘𝑚 = R(𝑚 ⊗ 𝑘). Provided that the coherence isomorphism of ModC is chosen to be the one induced by 𝛼, we say that the choice of unitor is typical if it satisfies L ≡ 1 and R ≡ 1 on any elements 𝑚 ∈ 𝑀Id𝑋 ⊂ 𝑀 for any 𝑋 ∈ ob(C). In general, the requirement that (1𝑀 ⊗ L) ◦ 𝛼 = R ⊗ 1𝑁 implies only that the values associated to L and R agree on 𝑚 ∈ 𝑀 with |𝑚| = Id𝑋. We call the unitors given by equations (9.0.1) and (9.0.2) above the typical unitors induced by 𝛼. In conclusion, we list a few quick computations regarding the associator which help to have in one’s back-pocket. Lemma 9.0.1. Let 𝑔, ℎ ∈ Mor(C) and 𝑔 : 𝑋 → 𝑌 and ℎ : 𝑌 → 𝑍. We have the following equivalences, with their paths pictured. (i) 𝛼(Id𝑋, 𝑔, Id𝑌 ) = 1 Id𝑋 𝑋 Id𝑌 𝑌 𝑔 (ii) 𝛼(Id𝑋, Id𝑋, ℎ ◦ 𝑔) = 𝛼(Id𝑋, 𝑔, ℎ)𝛼(Id𝑋, Id𝑋, 𝑔) Id𝑋 𝑋 𝑔 𝑌 ℎ 𝑍 178 (iii) 𝛼(ℎ ◦ 𝑔, Id𝑍 , Id𝑍 ) = 𝛼(𝑔, ℎ, Id𝑍 )𝛼(ℎ, Id𝑍 , Id𝑍 ) 𝑋 𝑔 ℎ 𝑌 Id𝑍 𝑍 (iv) 𝛼(𝑔, Id𝑌 , ℎ) = 𝛼(𝑔, Id𝑌 , Id𝑌 )𝛼(Id𝑌 , Id𝑌 , ℎ) Id𝑌 𝑌 𝑔 𝑋 ℎ 𝑍 Proof. Each of these are routine; we will prove (ii) as demonstration. We have 1 = 𝑑𝛼(Id𝑋, Id𝑋, 𝑔, ℎ) = 𝛼(Id𝑋, Id𝑋, 𝑔)𝛼(Id𝑋, Id𝑋, ℎ ◦ 𝑔)−1𝛼(Id𝑋, 𝑔, ℎ) as desired. □ The construction of the Hochschild complex is simple: given an algebra 𝐴, there is a special (𝐴, 𝐴)-bimodule B(𝐴), called the bar resolution of 𝐴. Since (𝐴, 𝐴)-bimodules are equivalent to 𝐴 ⊗ 𝐴op-modules, we can define 𝐻𝐶(𝐴, 𝑀) := B(𝐴) ⊗𝐴⊗ 𝐴op 𝑀 for any (𝐴, 𝐴)-bimodule 𝑀. So, in the C-graded scenario, there are three things to check: 1. There is some notion of C-graded 𝐴 ⊗ 𝐴op-modules equivalent to that of C-graded (𝐴, 𝐴)- bimodules; 2. There is a C-graded bar resolution B(𝐴) which has the structure of a C-graded DG (𝐴, 𝐴)- bimodule; 3. There is a notion of tensor product over 𝐴 ⊗ 𝐴op. These, respectively, are the subject of the next three sections. 179 9.1 More on C-graded algebras and bimodules For convenience, we relist the axioms of a C-graded algebra here. A C-graded algebra is a C-graded K-module 𝐴 = (cid:201) 𝑔∈Mor(C) 𝐴𝑔 endowed with a K-linear multiplication 𝜇 𝐴 : 𝐴 ⊗ 𝐴 → 𝐴 and unit element 1𝑋 ∈ 𝐴Id𝑋 for each 𝑋 ∈ ob(C) which satisfy each of the following. (A.I) 𝜇 𝐴 is a graded map; that is, for each homogeneous 𝑥, 𝑦 ∈ 𝐴,(cid:12) (cid:12)𝜇 𝐴(𝑥, 𝑦)(cid:12) (cid:12)𝑦(cid:12) (cid:12) = (cid:12) (cid:12) ◦|𝑥|. (A.II) 𝜇 𝐴 is graded associative; that is, for each homogeneous 𝑥, 𝑦, 𝑧 ∈ 𝐴, (cid:12)𝑦(cid:12) 𝜇 𝐴(𝜇 𝐴(𝑥, 𝑦), 𝑧) = 𝛼(|𝑥| ,(cid:12) (cid:12) ,|𝑧|)𝜇 𝐴(𝑥, 𝜇 𝐴(𝑥, 𝑦)). (A.III) For each homogeneous 𝑥 ∈ 𝐴, 𝜇 𝐴(1𝑋, 𝑥) = L(Id𝑋,|𝑥|)𝑥 and 𝜇 𝐴(𝑥, 1𝑌 ) = R(|𝑥| , Id𝑌 )𝑥 where |𝑥| : 𝑋 → 𝑌 . Notice that if our choice of unitors in ModC is typical, we have that 𝜇 𝐴(1𝑋, 1𝑋) = 1𝑋. Some of the usual operations performed on small categories can be extended to grading cat- egories. For motivation, suppose 𝐴 is a C-graded algebra, and consider 𝐴op. Recall that 𝐴op is simply 𝐴 but with multiplication defined by 𝜇 𝐴op(𝑥, 𝑦) := 𝜇 𝐴(𝑦, 𝑥). Then, notice that 𝐴op fails to be a C-graded algebra. However, 𝐴op has a natural description as a Cop-graded algebra. Recall that the category opposite C, denoted Cop, is the category with • ob(Cop) = ob(C), and • HomCop(𝑋, 𝑌 ) = HomC(𝑌 , 𝑋). Notice that, if 𝑋 𝑓 −→ 𝑌 𝑔 −→ 𝑍 is a sequence of morphisms in C, then (𝑋 𝑓 −→ 𝑌 𝑔 −→ 𝑍)op = 𝑍 𝑔op −−→ 𝑓 op −−→ 𝑋. That is, the functor op : C → Cop is contravariant, and (Cop)op = C. 𝑌 180 Definition 9.1.1. Let (C, 𝛼) be a grading category. Let (C, 𝛼)op := (Cop, 𝛼op) denote the opposite grading category, with 𝛼op : (Cop)[3] → K× defined by 𝛼op( 𝑓 op 3 , 𝑓 op 2 , 𝑓 op 1 ) := 𝛼( 𝑓1, 𝑓2, 𝑓3)−1. Remark 9.1.2. Notice that there is no real significance of change the underlying category—if 𝐴 is (C, 𝛼)-graded, we will see in the proof of the following proposition that 𝐴op is naturally (C, 𝛼−1)- graded. We make the choice to work with Cop so that there is no confusion when we say that something is a Cop-graded module/algebra. Proposition 9.1.3. Assume (C, 𝛼) is a grading category and 𝐴 is a C-graded algebra. Then (C, 𝛼)op is a grading category, and 𝐴op is a Cop-graded algebra. Proof. For the first claim, note that 𝑑(𝛼op)( 𝑓 op 4 , 𝑓 op 1 ) = 𝑑𝛼( 𝑓1, 𝑓2, 𝑓3, 𝑓4)−1, and the result follows by assumption that (C, 𝛼) is a grading category. For the second, given a decomposition 𝐴 = (cid:201) 𝑔∈Mor(C) 𝐴𝑔, choose the decomposition 𝐴op = (cid:201) 𝑔op∈Mor(Cop). Requirement (A.I) is satisfied 3 , 𝑓 op 2 , 𝑓 op since (cid:12)𝜇 𝐴op(𝑥, 𝑦)(cid:12) (cid:12) (cid:12)Cop = Ä(cid:12) (cid:12)𝜇 𝐴(𝑦, 𝑥)(cid:12) (cid:12)C äop = Ä (cid:12)𝑦(cid:12) |𝑥|C ◦(cid:12) (cid:12)C äop (cid:12)𝑦(cid:12) = (cid:12) (cid:12)Cop ◦|𝑥|Cop using the fact that (|𝑥|C)op = |𝑥|Cop. Requirement (A.II) is similar: 𝜇 𝐴op(𝜇 𝐴op(𝑥, 𝑦), 𝑧) = 𝜇 𝐴(𝑧, 𝜇 𝐴(𝑦, 𝑥)) (cid:12)𝑦(cid:12) = 𝛼(|𝑧|C ,(cid:12) ,|𝑥|C)−1𝜇 𝐴(𝜇 𝐴(𝑧, 𝑦), 𝑥) (cid:12)C (cid:12)𝑦(cid:12) = 𝛼op(|𝑥|Cop ,(cid:12) (cid:12)Cop ,|𝑧|Cop)𝜇 𝐴op(𝑥, 𝜇 𝐴op(𝑦, 𝑧)). Notice that this is why we must invert the associator to obtain a graded structure on 𝐴op. Finally, for (A.III), notice that the unit object 𝐼Cop is exactly 𝐼C. Then, sufficient unitors for ModCop are provided by fixing LCop = RC and RCop = LC. □ Indeed, as remarked earlier, notice that the categories ModC and ModCop differ cosmetically by reversing arrows in the grading structure, and substantively by inverting the coherence isomorphism. 181 Now, suppose 𝐴 and 𝐵 are C-graded and D-graded algebras respectively. Abusing notation, we will write 𝐴 ⊗ 𝐵 to denote the tensor product of 𝐴 and 𝐵 as K-modules. The graded structure on 𝐴 and 𝐵 induces one on 𝐴 ⊗ 𝐵 as follows. Recall that the product category C × D of two categories C and D is the one with • ob(C × D) = ob(C) × ob(D), • HomC×D((𝑋1, 𝑋2), (𝑌1, 𝑌2)) = HomC(𝑋1, 𝑌1) × HomD(𝑋2, 𝑌2), • composition defined by ( 𝑓2, 𝑔2) ◦ ( 𝑓1, 𝑔1) = ( 𝑓2 ◦ 𝑓1, 𝑔2 ◦ 𝑔1), and • identity morphisms Id(𝑋,𝑌 ) = (Id𝑋, Id𝑌 ). Definition 9.1.4. Given grading categories (C, 𝛼) and (D, 𝛽), define the product grading category (C, 𝛼) × (D, 𝛽) := (C × D, 𝛼 × 𝛽) where (𝛼 × 𝛽)(( 𝑓1, 𝑔1), ( 𝑓2, 𝑔2), ( 𝑓3, 𝑔3)) := 𝛼( 𝑓1, 𝑓2, 𝑓3)𝛽(𝑔1, 𝑔2, 𝑔3). Proposition 9.1.5. If (C, 𝛼) and (D, 𝛽) are grading categories, then so is (C × D, 𝛼 × 𝛽). Moreover, if 𝐴 is a (C, 𝛼)-graded algebra and 𝐵 is a (D, 𝛽)-graded algebra, then 𝐴 ⊗ 𝐵 is a (C × D, 𝛼 × 𝛽)- graded algebra. Proof. Again, the first claim is immediate. The second is routine: in general, we interpret 𝐴 ⊗ 𝐵 as a (C × D)-graded algebra by taking |𝑎 ⊗ 𝑏|C×D 𝜇 𝐴⊗𝐵 : (𝐴 ⊗ 𝐵) ⊗ (𝐴 ⊗ 𝐵) → 𝐴 ⊗ 𝐵 as := (|𝑎|C ,|𝑏|D) and defining the multiplication 𝜇 𝐴⊗𝐵(𝑎1 ⊗ 𝑏1, 𝑎2 ⊗ 𝑏2) := 𝜇 𝐴(𝑎1, 𝑎2) ⊗ 𝜇𝐵(𝑏1, 𝑏2). Then, for example, check (A.I) by computing (cid:12)𝜇 𝐴⊗𝐵(𝑎1 ⊗ 𝑏1, 𝑎2 ⊗ 𝑏2)(cid:12) (cid:12) (cid:12)𝜇 𝐴(𝑎1, 𝑎2) ⊗ 𝜇𝐵(𝑏1, 𝑏2)(cid:12) (cid:12)C×D = (cid:12) (cid:12)C×D ä Ä(cid:12) (cid:12)𝜇𝐵(𝑏1, 𝑏2)(cid:12) ,(cid:12) (cid:12)𝜇 𝐴(𝑎1, 𝑎2)(cid:12) (cid:12)D (cid:12)C (cid:1) = (cid:0)|𝑎2|C ◦|𝑎1|C ,|𝑏2|D ◦|𝑏1|D = = (|𝑎2|C ,|𝑏2|D) ◦ (|𝑎1|C ,|𝑏1|D) =: |𝑎2 ⊗ 𝑏2|C×D ◦|𝑎1 ⊗ 𝑏1|C×D . 182 Checking (A.II) is also routine. To check (A.III), we note that, as ModC×D inherits its coherence isomorphism from ModC and ModD, its unitors may also be chosen from these categories, defining LC×D := LC × LD, and similarly for the right unitor RC×D. Also fix unit elements 1(𝑋,𝑌 ) ∈ 𝐴Id( 𝑋,𝑌 ) to be 1𝑋 ⊗ 1𝑌 , recalling that, by definition, Id(𝑋,𝑌 ) = (Id𝑋, Id𝑌 ). Then the checks required for (A.III) are also routine: for example, 𝜇 𝐴⊗𝐵(1(𝑋,𝑌 ), 𝑎 ⊗ 𝑏) = 𝜇 𝐴(1𝑋, 𝑎) ⊗ 𝜇𝐵(1𝑌 , 𝑏) = LC(Id𝑋,|𝑎|C)LD(Id𝑌 ,|𝑏|D)𝑎 ⊗ 𝑏 = LC×D((Id𝑋, Id𝑌 ), (|𝑎|C ,|𝑏|D))𝑎 ⊗ 𝑏 = LC×D(Id(𝑋,𝑌 ),|𝑎 ⊗ 𝑏|C×D)𝑎 ⊗ 𝑏. The check for RC×D is totally analogous. □ Now, recall the definition of a C-graded bimodule. Suppose 𝐴 and 𝐵 are C-graded algebras. We define a C-graded (𝐴, 𝐵)-module as a C-graded K-module with graded, K-linear actions 𝜌𝐿 : 𝐴 ⊗ 𝑀 → 𝑀 and 𝜌𝑅 : 𝑀 ⊗ 𝐵 → 𝑀 which which satisfy the following axioms for each of 𝑎, 𝑎′ ∈ 𝐴, 𝑏, 𝑏′ ∈ 𝐵, and 𝑚 ∈ 𝑀. (B.I) 𝜌𝐿(𝜇 𝐴(𝑎, 𝑎′), 𝑚) = 𝛼(|𝑎| ,|𝑎′| ,|𝑚|)𝜌𝐿(𝑎, 𝜌𝐿(𝑎′, 𝑚)); (B.II) 𝜌𝑅(𝜌𝑅(𝑚, 𝑏), 𝑏′) = 𝛼(|𝑚| ,|𝑏| ,|𝑏′|)𝜌𝑅(𝑚, 𝜇 𝐴(𝑏, 𝑏′)); (B.III) 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏) = 𝛼(|𝑎| ,|𝑚| ,|𝑏|)𝜌𝐿(𝑎, 𝜌𝑅(𝑚, 𝑏)); (B.IV) 𝜌𝐿(1𝑋, 𝑚) = L(Id𝑋,|𝑚|)𝑚 and 𝜌𝑅(𝑚, 1𝑌 ) = R(|𝑚| , Id𝑌 )𝑚. We define a C-graded left 𝐴-module (resp. right 𝐵-module) as a C-graded (𝐴, 𝐼C)-bimodule (resp. (𝐼C, 𝐵)-bimodule)—in this case, the 𝜌𝑅 (resp. 𝜌𝐿) action is trivial. Equivalently, we can think of a left (resp. right) C-graded 𝐴-module as a C-graded K-module with a single graded, K-linear action 𝜌𝐿 (resp. 𝜌𝑅) satisfying (B.I) (resp. (B.II)) and the first (resp. second) half of (B.IV). 183 Proposition 9.1.6. 𝑀 is a C-graded left (resp. right) 𝐴-module if and only if it is a Cop-graded right (resp. left) 𝐴op-module. Proof. Assuming 𝑀 is a C-graded left 𝐴-module means that it has a left action 𝜌𝐿 : 𝐴 ⊗ 𝑀 → 𝑀 which satisfies and (cid:12)𝑦(cid:12) 𝜌𝐿(𝜇 𝐴(𝑥, 𝑦), 𝑚) = 𝛼(|𝑥|C ,(cid:12) (cid:12)C ,|𝑚|C)𝜌𝐿(𝑥, 𝜌𝐿(𝑦, 𝑚)) 𝜌𝐿(1𝑌 , 𝑚) = L(Id𝑌 ,|𝑚|C)𝑚. We want to show that 𝑀 has a natural definition as a Cop-graded right 𝐴op-module. First, if 𝑀 = (cid:201) 𝑔∈Mor(C) 𝑀𝑔, reverse arrows, as before, to get an induced grading by Cop; i.e., 𝑀 = (cid:201) 𝑔op∈Mor(Cop). Then, define 𝜌op 𝑅 : 𝑀 ⊗ 𝐴op → 𝑀 by 𝜌op 𝑅 (𝑚, 𝑎) := 𝜌𝐿(𝑎, 𝑚). We compute 𝑅 (𝜌op 𝜌op 𝑅 (𝑚, 𝑥), 𝑦) = 𝜌𝐿(𝑦, 𝜌𝐿(𝑥, 𝑚)) (cid:12)𝑦(cid:12) = 𝛼((cid:12) (cid:12)C ,|𝑥|C ,|𝑚|C)−1𝜌𝐿(𝜇 𝐴(𝑦, 𝑥), 𝑚) (cid:12)Cop)𝜌op (cid:12)𝑦(cid:12) = 𝛼op(|𝑚|Cop ,|𝑥|Cop ,(cid:12) 𝑅 (𝑚, 𝜇 𝐴op(𝑥, 𝑦)) and 𝜌op 𝑅 (𝑚, 1𝑋) = 𝜌𝐿(1𝑋, 𝑚) = LC(Id𝑋,|𝑚|C)𝑚 = RCop(|𝑚|Cop , Id𝑋)𝑚 as desired. The other checks are analogous. □ Assume 𝐴 and 𝐵 are both C-graded algebras. To conclude this section, we want there to be an equivalence between C-graded (𝐴, 𝐵)-bimodules and C-graded left 𝐴 ⊗ 𝐵op-modules. The problem is that our current definition of modules assumes that the algebra and the module share the same grading category—in the latter instance, 𝐴 ⊗ 𝐵op is a C × Cop-graded algebra. This prompts the following definition. Definition 9.1.7. Fix C-graded algebras 𝐴 and 𝐵. Define a C-graded left 𝐴 ⊗ 𝐵op-module to be a C-graded K-module 𝑀 with a left, K-linear action map 𝐿 : (𝐴 ⊗ 𝐵op) × 𝑀 → 𝑀 𝜌𝑒 184 which is graded in the sense that (cid:12) (cid:12) (cid:12) (cid:12) 𝜌𝑒 (cid:12) = |𝑏|C ◦|𝑚|C ◦|𝑎|C, and the following hold. 𝐿(𝑎 ⊗ 𝑏, 𝑚) (cid:12) (E.I) For 𝑎1, 𝑎2 ∈ 𝐴, 𝑏1, 𝑏2 ∈ 𝐵op, and 𝑚 ∈ 𝑀 homogeneous, 𝜌𝑒 𝐿(𝜇 𝐴⊗𝐵op(𝑎1 ⊗ 𝑏1, 𝑎2 ⊗ 𝑏2), 𝑚) = Δ(|𝑎1 ⊗ 𝑏1|C×Cop ,|𝑎2 ⊗ 𝑏2|C×Cop ,|𝑚|C) 𝐿(𝑎1 ⊗ 𝑏1, 𝜌𝑒 𝜌𝑒 𝐿(𝑎2 ⊗ 𝑏2, 𝑚)); (E.II) for (𝑋, 𝑌 ) ∈ ob(C × Cop), 𝜌𝑒 𝐿(1(𝑋,𝑌 ), 𝑚) = LC(Id𝑋,|𝑚|C)RC(|𝑚|C , Id𝑌 )𝑚 where Δ(|𝑎1 ⊗ 𝑏1|C×Cop ,|𝑎2 ⊗ 𝑏2|C×Cop ,|𝑚|C) is taken to be the value 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|)𝛼(|𝑚| ◦|𝑎2| ◦|𝑎1| ,|𝑏2| ,|𝑏1|)−1𝛼(|𝑎1| ,|𝑚| ◦|𝑎2| ,|𝑏2|) with all gradings taken in C, fixing |𝑏|C := (|𝑏|Cop)op. When 𝐵 = 𝐴, we write 𝐴𝑒 := 𝐴 ⊗ 𝐴op. Note that under the canonical identification |𝑚|C𝑜 𝑝 := (|𝑚|C)op, LC(Id𝑋,|𝑚|C)RC(|𝑚|C , Id𝑌 ) = LC×Cop(Id(𝑋,𝑌 ),|𝑚|C×Cop). Also note that the value for Δ can be obtained many different ways, and the cocycle relation implies that they all are equivalent. For example, the two paths ((𝑎1𝑎2)𝑚)(𝑏2𝑏1) 𝛼 (𝑎1(𝑎2𝑚))(𝑏2𝑏1) 𝛼 𝑎1((𝑎2𝑚)(𝑏2𝑏1)) 𝛼−1 ((𝑎1(𝑎2𝑚))𝑏2)𝑏1 𝛼−1 𝛼 𝑎1(((𝑎2𝑚)𝑏2)𝑏1) 𝛼−1 (𝑎1((𝑎2𝑚)𝑏2))𝑏1 yield equivalent values—the value provided in the definition is based on the lower path. Proposition 9.1.8. Suppose that 𝐴 and 𝐵 are C-graded algebras, and that the unitors of ModC are the typical unitors induced by 𝛼. Then, every C-graded left 𝐴 ⊗ 𝐵op-module can be given the structure of a C-graded (𝐴, 𝐵)-bimodule, and vice-versa. 185 Proof. The backwards direction is rigged to work. Given a C-graded (𝐴, 𝐵)-bimodule 𝑀, we give it the structure of a C-graded left 𝐴 ⊗ 𝐵op-module by defining 𝜌𝑒 𝐿 : (𝐴 ⊗ 𝐵op) ⊗ 𝑀 → 𝑀 by 𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚) := 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏). To verify (E.I), we compute 𝜌𝑒 𝐿(𝜇 𝐴⊗𝐵op(𝑎1 ⊗ 𝑏1, 𝑎2 ⊗ 𝑏2), 𝑚) = 𝜌𝑅(𝜌𝐿(𝜇 𝐴(𝑎1, 𝑎2), 𝑚), 𝜇 𝐴(𝑏2, 𝑏1)) = 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|)𝜌𝑅(𝜌𝐿(𝑎1, 𝜌𝐿(𝑎2, 𝑚)), 𝜇 𝐴(𝑏2, 𝑏1)) = 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|)𝛼(|𝑚| ◦|𝑎2| ◦|𝑎1| ,|𝑏1| ,|𝑏2|)−1 𝜌𝑅(𝜌𝑅(𝜌𝐿(𝑎1, 𝜌𝐿(𝑎2, 𝑚)), 𝑏2), 𝑏1) = 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|)𝛼(|𝑚| ◦|𝑎2| ◦|𝑎1| ,|𝑏1| ,|𝑏2|)−1 𝛼(|𝑎1| ,|𝑚| ◦|𝑎2| ,|𝑏2|)𝜌𝑅(𝜌𝐿(𝑎1, 𝜌𝑅(𝜌𝐿(𝑎2, 𝑚), 𝑏2)), 𝑏1) = Δ(|𝑎1 ⊗ 𝑏1| ,|𝑎2 ⊗ 𝑏2| ,|𝑚|)𝜌𝑒 𝐿(𝑎1 ⊗ 𝑏1, 𝜌𝑒 𝐿(𝑎2 ⊗ 𝑏2, 𝑚)) as desired. For (E.II), setting |𝑚| : 𝑋 → 𝑌 , 𝐿(1(𝑋,𝑌 ), 𝑚) = 𝜌𝑒 𝜌𝑒 𝐿(1𝑋 ⊗ 1𝑌 , 𝑚) = L(Id𝑋,|𝑚|)R(|𝑚| , Id𝑌 )𝑚 as well. For the other direction, assume 𝑀 is a C-graded 𝐴 ⊗ 𝐵op-module. If |𝑚| : 𝑋 → 𝑌 , define 𝜌𝐿(𝑎, 𝑚) := R(|𝑚| ◦|𝑎| , Id𝑌 )−1𝜌𝑒 𝐿(𝑎 ⊗ 1𝑌 , 𝑚) and 𝜌𝑅(𝑚, 𝑏) := L(Id𝑋,|𝑚|)−1𝜌𝑒 𝐿(1𝑋 ⊗ 𝑏, 𝑚) First we check that the axioms of a C-graded (𝐴, 𝐵)-bimodule are satisfied. We take the time to perform the checks arduously as to not take the result for granted, although the entire proof might be a bit pedantic. To check (B.I), assume that 𝑎1, 𝑎2 ∈ 𝐴 and 𝑚 ∈ 𝑀 are homogeneous so that 𝑊 |𝑎1| 𝑋 |𝑎2| 𝑌 |𝑚| 𝑍 186 We compute 𝜌𝐿(𝜇 𝐴(𝑎1, 𝑎2), 𝑚) = R(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑧)−1𝜌𝑒 𝐿(𝜇 𝐴(𝑎1, 𝑎2) ⊗ 1𝑍 , 𝑚) = R(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑧)−1𝜌𝑒 𝐿(𝜇 𝐴(𝑎1, 𝑎2) ⊗ 𝜇 𝐴op(1𝑍 , 1𝑍 ), 𝑚) = R(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑧)−1𝜌𝑒 𝐿(𝜇 𝐴⊗ 𝐴op(𝑎1 ⊗ 1𝑍 , 𝑎2 ⊗ 1𝑍 ), 𝑚) = R(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑧)−1Δ(|𝑎1 ⊗ 1𝑍 | ,|𝑎2 ⊗ 1𝑍 | ,|𝑚|) 𝐿(𝑎1 ⊗ 1𝑍 , 𝜌𝑒 𝜌𝑒 𝐿(𝑎2 ⊗ 1𝑍 , 𝑚)) = R(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑧)−1Δ(|𝑎1 ⊗ 1𝑍 | ,|𝑎2 ⊗ 1𝑍 | ,|𝑚|)R(|𝑚| ◦|𝑎2| , Id𝑍 ) R((cid:12) (cid:12)𝜌𝑒 𝐿(𝑎2 ⊗ 1𝑍 , 𝑚)(cid:12) (cid:12) ◦|𝑎1| , Id𝑍 )𝜌𝐿(𝑎1, 𝜌𝐿(𝑎2, 𝑚)). Notice that the second equivalence assumes that the unitors are typical. The first and the last term written as a function of R cancel each other since (cid:12) 𝜌𝑒 (cid:12) = |𝑚| ◦ |𝑎2|. Expanding the 𝐿(𝑎1 ⊗ 1𝑍 , 𝑚) (cid:12) remaining terms, Δ(|𝑎1 ⊗ 1𝑍 | ,|𝑎2 ⊗ 1𝑍 | ,|𝑚|) and R(|𝑚| ◦ |𝑎2| , Id𝑍 ), in terms of 𝛼 (using the fact (cid:12) (cid:12) (cid:12) that the right unitor is the typical one induced by 𝛼), we obtain 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|) 𝛼(|𝑚| ◦|𝑎2| ◦|𝑎1| , Id𝑍 , Id𝑍 )−1𝛼(|𝑎1| ,|𝑚| ◦|𝑎2| , Id𝑍 )𝛼(|𝑚| ◦|𝑎2| , Id𝑍 , Id𝑍 ) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:124) (cid:123)(cid:122) (∗) . Then, the terms labeled (∗) cancel by (iii) of Lemma 9.0.1, and we have that 𝜌𝐿(𝜇 𝐴(𝑎1, 𝑎2), 𝑚) = 𝛼(|𝑎1| ,|𝑎2| ,|𝑚|)𝜌𝐿(𝑎1, 𝜌𝐿(𝑎2, 𝑚)) as desired. Axiom (B.II) is very similar. Assume that 𝑏1, 𝑏2 ∈ 𝐵 and 𝑚 ∈ 𝑀 are homogeneous so that 𝑊 |𝑚| 𝑋 |𝑏2| 𝑌 |𝑏1| 𝑍 We leave it to the reader to verify that 𝜌𝑅(𝜌𝑅(𝑚, 𝑏2), 𝑏1) = L(Id𝑊 ,|𝑚|)−1L(Id𝑊 ,|𝑏2| ◦|𝑚|)−1Δ(|1𝑊 ⊗ 𝑏1| ,|1𝑊 ⊗ 𝑏2| ,|𝑚|)−1L(Id𝑊 ,|𝑚|) 𝜌𝑅(𝑚, 𝜇 𝐴(𝑏2, 𝑏1)). 187 The first and the last term which appear as a function of L cancel. Then, expanding the rest in terms of 𝛼 gives 𝛼(Id𝑊 , Id𝑊 ,|𝑏2| ◦|𝑚|)𝛼(Id𝑊 , Id𝑊 ,|𝑚|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(|𝑚| ,|𝑏2| ,|𝑏1|) 𝛼(Id𝑊 ,|𝑚| ,|𝑏2|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:124) (cid:123)(cid:122) (∗) The terms labeled (∗) cancel by (ii) of Lemma 9.0.1, so we are left with the desired result. Axiom (B.III) is exactly the same idea, but requires a little more computation. Now, pick 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵, and 𝑚 ∈ 𝑀 homogeneous so that 𝑊 |𝑎| 𝑋 |𝑚| 𝑌 |𝑏| 𝑍 We want to show that We compute 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏) = 𝛼(|𝑎| ,|𝑚| ,|𝑏|)𝜌𝐿(𝑎, 𝜌𝑅(𝑚, 𝑏)). 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1L(Id𝑊 ,|𝑚| ◦|𝑎|)−1𝜌𝑒 𝐿(1𝑊 ⊗ 𝑏, 𝜌𝑒 𝐿(𝑎 ⊗ 1𝑌 , 𝑚)) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1L(Id𝑊 ,|𝑚| ◦|𝑎|)−1Δ(|1𝑊 ⊗ 𝑏| ,|𝑎 ⊗ 1𝑌 | ,|𝑚|)−1 𝜌𝑒 𝐿(𝜇 𝐴⊗ 𝐴op(1𝑊 ⊗ 𝑏, 𝑎 ⊗ 1𝑌 ), 𝑚) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1L(Id𝑊 ,|𝑚| ◦|𝑎|)−1Δ(|1𝑊 ⊗ 𝑏| ,|𝑎 ⊗ 1𝑌 | ,|𝑚|)−1 𝜌𝑒 𝐿(𝜇 𝐴(1𝑊 , 𝑎) ⊗ 𝜇 𝐴(1𝑌 , 𝑏)), 𝑚) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1L(Id𝑊 ,|𝑚| ◦|𝑎|)−1Δ(|1𝑊 ⊗ 𝑏| ,|𝑎 ⊗ 1𝑌 | ,|𝑚|)−1L(Id𝑊 ,|𝑎|) L(Id𝑌 ,|𝑏|)𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚). Expanding the values on the last line in terms of 𝛼, we find 𝛼(|𝑚| ◦|𝑎| , Id𝑌 , Id𝑌 )−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(Id𝑊 , Id𝑊 ,|𝑚| ◦|𝑎|) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) 𝛼(Id𝑊 ,|𝑎| ,|𝑚|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) 𝛼(|𝑚| ◦|𝑎| , Id𝑌 ,|𝑏|) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(Id𝑊 ,|𝑚| ◦|𝑎| , Id𝑌 )−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗∗) 𝛼(Id𝑊 , Id𝑊 ,|𝑎|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) (∗∗) 𝛼(Id𝑌 , Id𝑌 ,|𝑏|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) . 188 The terms marked by (∗) cancel by (iv) of Lemma 9.0.1, those marked by (∗∗) cancel by (ii), and the (∗ ∗ ∗) is trivial by (i). On the other hand, one can verify in the same way that 𝜌𝐿(𝑎, 𝜌𝑅(𝑚, 𝑏)) = L(Id𝑋,|𝑚|)−1R(|𝑏| ◦|𝑚| ◦|𝑎| , Id𝑍 )−1Δ(|𝑎 ⊗ 1𝑍 | ,|1𝑋 ⊗ 𝑏| ,|𝑚|)−1R(|𝑎| , Id𝑋) R(|𝑏| , Id𝑍 )𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚). Then, expanding in terms of 𝛼, we have 𝛼(Id𝑋, Id𝑋,|𝑚|) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(|𝑏| ◦|𝑚| ◦|𝑎| , Id𝑍 , Id𝑍 )−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) 𝛼(|𝑎| , Id𝑋,|𝑚|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) (∗) 𝛼(|𝑚| ◦|𝑎| ,|𝑏| , Id𝑍 ) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) (∗∗) 𝛼(|𝑎| ,|𝑚||𝑏|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:125) (cid:123)(cid:122) (cid:124) (∗∗∗) 𝛼(|𝑎| , Id𝑋, Id𝑋) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(|𝑏| , Id𝑍 , Id𝑍 ) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) . The terms marked (∗) cancel by (iv) and the terms marked by (∗∗) cancel by (iii) of Lemma 9.0.1. The term marked (∗ ∗ ∗) remains, and we are left with the desired equality. Checking axiom (B.IV) is quickly verified. If |𝑚| : 𝑋 → 𝑌 , recall that 1𝑋 ⊗ 1𝑌 = 1(𝑋,𝑌 ) and 𝜌𝑒 𝐿(1(𝑋, 𝑌 ), 𝑚) = L(Id𝑋,|𝑚|)R(|𝑚| , Id𝑌 )𝑚. 𝜌𝐿(1𝑋, 𝑚) = R(|𝑚| , Id𝑌 )−1𝜌𝐿(1𝑋 ⊗ 1𝑌 , 𝑚) = L(Id𝑋,|𝑚|)𝑚 𝜌𝑅(𝑚, 1𝑌 ) = L(Id𝑋,|𝑚|)−1𝜌𝑒 𝐿(1𝑋 ⊗ 1𝑌 , 𝑚) = R(|𝑚| , Id𝑌 )𝑚 Then and as desired. Finally, we check that this assignment is inverse to the one 𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚) := 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏). Per usual, one direction is rigged to work: we have 𝜌𝐿(𝑎, 𝑚) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1𝜌𝑒 𝐿(𝑎 ⊗ 1𝑌 , 𝑚) = R(|𝑚| ◦|𝑎| , Id𝑌 )−1𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 1𝑌 ) = 𝜌𝐿(𝑎, 𝑚), since(cid:12) (cid:12)𝜌𝐿(𝑎, 𝑚)(cid:12) (cid:12) = |𝑚| ◦|𝑎|, and 𝜌𝑅(𝑚, 𝑏) = L(Id𝑋,|𝑚|)−1𝜌𝑒 𝐿(1𝑋 ⊗ 𝑏, 𝑚) = L(Id𝑋,|𝑚|)−1𝜌𝑅(𝜌𝐿(1𝑋, 𝑚), 𝑏) = 𝜌𝑅(𝑚, 𝑏). 189 For the other direction, we have to assume that the unitors are the typical ones induced by 𝛼. We assume the relevant gradings fit into the diagram Id𝑊 𝑊 |𝑎| |𝑚| 𝑋 Id𝑌 𝑌 |𝑎′ | 𝑍 First, we compute 𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚) = 𝜌𝑅(𝜌𝐿(𝑎, 𝑚), 𝑏) = L(Id𝑊 ,|𝑚| ◦|𝑎|)−1𝜌𝑒 𝐿(1𝑊 ⊗ 𝑏, 𝜌𝐿(𝑎, 𝑚)) = L(Id𝑊 ,|𝑚| ◦|𝑎|)−1R(|𝑚| ◦|𝑎| , Id𝑌 )−1𝜌𝑒 𝐿(1𝑊 ⊗ 𝑏, 𝜌𝑒 𝐿(𝑎 ⊗ 1𝑌 , 𝑚)) = L(Id𝑊 ,|𝑚| ◦|𝑎|)−1R(|𝑚| ◦|𝑎| , Id𝑌 )−1Δ(|1𝑊 ⊗ 𝑏| ,|𝑎 ⊗ 1𝑌 | ,|𝑚|)−1 𝜌𝑒 𝐿(𝜇 𝐴⊗𝐵op(1𝑊 ⊗ 𝑏, 𝑎 ⊗ 1𝑌 ), 𝑚) = L(Id𝑊 ,|𝑚| ◦|𝑎|)−1R(|𝑚| ◦|𝑎| , Id𝑌 )−1Δ(|1𝑊 ⊗ 𝑏| ,|𝑎 ⊗ 1𝑌 | ,|𝑚|)−1L(Id𝑊 ,|𝑎|) L(Id𝑌 ,|𝑏|) 𝜌𝑒 𝐿(𝑎 ⊗ 𝑏, 𝑚) where all gradings are taken in C, apart from the first two entries of Δ as per usual. Now we rewrite all the terms of the last line in terms of the associator to get the product 𝛼(Id𝑊 , Id𝑊 ,|𝑚| ◦|𝑎|) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(|𝑚| ◦|𝑎| , Id𝑌 , Id𝑌 )−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) 𝛼(Id𝑊 ,|𝑎| ,|𝑚|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(|𝑚| ◦|𝑎| , Id𝑌 ,|𝑏|) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) 𝛼(Id𝑊 ,|𝑚| ◦|𝑎| , Id𝑌 )−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗∗) 𝛼(Id𝑊 , Id𝑊 ,|𝑎|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗) 𝛼(Id𝑌 , Id𝑌 ,|𝑏|)−1 (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32)(cid:32) (cid:123)(cid:122) (cid:125) (cid:124) (∗∗) Then, the terms labeled (∗) cancel by (ii) of Lemma 9.0.1, the terms labeled (∗∗) cancel by (iv) of Lemma 9.0.1, and the term labeled (∗ ∗ ∗) is trivial by (i) of Lemma 9.0.1. □ We note that one can define C-graded right 𝐴 ⊗ 𝐵op-modules similarly, and it follows from the arguments above that they are equivalent to the notion of C-graded (𝐴, 𝐴)-modules. 190 9.2 A C-graded bar resolution We will use the following trivial example of a grading category to define C-graded differentially graded objects. Example 9.2.1. Consider the category Z := 𝐵Z with a single object ★ and HomZ(★, ★) = Z. Extend Z to a grading category trivially: that is, take 𝛼 ≡ 1. Thus, for the grading category (Z, 1), a Z-graded object is the same thing as a Z-graded object. In general, if 𝐵𝐺 denotes the category with a single object ★ and Hom𝐵𝐺(★, ★) = 𝐺 for 𝐺 a group, then we recover grading by arbitrary groups, as defined by Albequerque and Majid [AM99]. In addition, we will see that specializing C to Z will recover the ordinary Hochschild homology. Definition 9.2.2. A C-graded DG-(𝐴, 𝐵)-bimodule is a pair (𝑀, 𝜕𝑀) of a Z × C-graded (𝐴, 𝐵)- bimodule 𝑀 = (cid:201) 𝑔 and a K-linear map 𝜕𝑀 : 𝑀 → 𝑀, called the differential, 𝑛∈Z,𝑔∈Mor(C) 𝑀 𝑛 satisfying the following: (DG.I) 𝜕𝑀(𝑀 𝑛 𝑔 ) ⊂ 𝑀 𝑛−1 𝑔 ; (DG.II) 𝜕𝑀(𝜌𝐿(𝑎, 𝑚)) = 𝜌𝐿(𝑎, 𝜕𝑀(𝑚)); (DG.III) 𝜕𝑀(𝜌𝑅(𝑚, 𝑏)) = 𝜌𝑅(𝜕𝑀(𝑚), 𝑏); (DG.IV) 𝜕𝑀 ◦ 𝜕𝑀 = 0, for each 𝑎 ∈ 𝐴, 𝑏 ∈ 𝐵, and 𝑚 ∈ 𝑀. If 𝑚 ∈ 𝑀 is homogeneous with |𝑚| = (|𝑚|Z ,|𝑚|C), we call |𝑚|Z ∈ Z the homological degree of 𝑚. We call (𝑀, 𝜕𝑀) a C-graded chain complex if 𝐴 = 𝐵 = 𝐼C, so that the left- and right-actions are just scalar multiplication. A C-graded left DG-𝐴 ⊗ 𝐵op-module is a pair (𝑀, 𝜕𝑀) of a Z × C-graded left 𝐴 ⊗ 𝐵op-module which is defined exactly the same way, except that axioims (DG.II) and (DG.III) are replaced by the single axiom (DG.II’) 𝜕𝑀(𝜌𝐿(𝑎 ⊗ 𝑏, 𝑚)) = 𝜌𝐿(𝑎 ⊗ 𝑏, 𝜕𝑀(𝑚)). 191 Axiom (DG.I) says that the differential decreases homological degree by 1 and doesn’t have an effect on C-degree. For clarity, we note that we could have just as easily defined C-graded DG-(𝐴, 𝐵)-bimodules where axiom (DG.I) is replaced with the requirement that 𝜕𝑀(𝑀 𝑛 𝑔 ) ⊂ 𝑀 𝑛+1 𝑔 (see, for example, Definition 4.24 of [NP20]). The offered definition simply agrees with usual conventions for the bar resolution, defined shortly. Finally, note that, given a C-graded DG-(𝐴, 𝐵)- bimodule (𝑀, 𝜕𝑀), its homology 𝐻(𝑀, 𝜕𝑀) = ker(𝜕𝑀)/im(𝜕𝑀) is a Z × C-graded bimodule. Proposition 9.2.3. Suppose 𝐴 and 𝐵 are C-graded algebras, and that the unitors of ModC are the typical unitors induced by 𝛼. Then every C-graded left DG-𝐴 ⊗ 𝐵op-module can be given the structure of a C-graded DG-(𝐴, 𝐵)-bimodule, and vice-versa. Proof. This is a direct consequence of Proposition 9.1.8 and the proof thereof. It is an easy exercise, left to the reader, to verify that the actions defined there satisfy the new conditions. □ Let 𝐴 be a C-graded algebra. We introduce the bar resolution B(𝐴) of 𝐴 as a primary example of a C-graded DG-(𝐴, 𝐴)-bimodule. As a complex, it takes the following form. B(𝐴) := · · · (𝐴 ⊗ 𝐴) ⊗ 𝐴 𝐴 ⊗ 𝐴 0 with differential 𝜕 : 𝐴⊗(𝑛+2) → 𝐴⊗(𝑛+1) given by 𝜕(𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1) = 𝑛 ∑︁ 𝑖=0 (−1)𝑖𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝑎0 ⊗ · · · ⊗ 𝜇 𝐴(𝑎𝑖, 𝑎𝑖+1) ⊗ · · · ⊗ 𝑎𝑛+1 where we fix 𝛼(∅,|𝑎0| ,|𝑎1|) = 1 in the 𝑖 = 0 summand. The tensor product in 𝐴⊗𝑛 is the monoidal product of ModC; in particular, 𝐴⊗𝑛 is C-graded. We will view B(𝐴) as Z × C-graded taking |𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1|Z×C = (𝑛 + 1,|𝑎𝑛+1|C ◦ · · · ◦|𝑎1|C ◦|𝑎0|C). Then, we have that (B(𝐴), 𝜕) satisfies (DG.I) clearly. Lemma 9.2.4. If 𝐴 is a C-graded algebra, B(𝐴) is a chain complex; that is, 𝜕 ◦ 𝜕 = 0. Proof. Consider 𝜕(𝜕(𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1). We will denote summands in the ensuing expansion by pairs (𝑖, 𝑗), for 𝑖 = 0, 1, . . . , 𝑛 coming from the first differential and 𝑗 = 0, 1, . . . 𝑛 − 1 coming from 192 the second. Then, fixing 𝑖 ≤ 𝑗, observe that in the proof that the original bar complex is a chain complex, the (𝑖, 𝑗) summand cancels with the ( 𝑗 + 1, 𝑖) summand. We claim that this is also how terms cancel in the C-graded setting. Thus, since the signs are as they appear in the original setting, we do not need to keep track of them. There are three cases to consider. The first is when (𝑖, 𝑗) = (0, 0). This term is always 𝜇(𝜇(𝑎0, 𝑎1), 𝑎2) ⊗ 𝑎3 ⊗ · · · ⊗ 𝑎𝑛+1 and it clearly cancels with the ( 𝑗 + 1, 𝑖) = (1, 0) term 𝛼(|𝑎0| ,|𝑎1| ,|𝑎2|)𝜇(𝑎0, 𝜇(𝑎1, 𝑎2)) ⊗ 𝑎3 ⊗ · · · ⊗ 𝑎𝑛+1. For the second case, assume that 𝑖 < 𝑗. Then the (𝑖, 𝑗) term is 𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝛼((cid:12) (cid:12)𝑎 𝑗 (cid:12) ◦ · · · ◦(cid:12) (cid:12) (cid:12)𝜇(𝑎𝑖, 𝑎𝑖+1)(cid:12) (cid:12) ◦ · · · ◦|𝑎0| ,(cid:12) (cid:12)𝑎 𝑗+1 (cid:12) ,(cid:12) (cid:12) (cid:12)𝑎 𝑗+2 (cid:12) (cid:12)) times 𝑎0 ⊗ · · · ⊗ 𝜇(𝑎𝑖, 𝑎𝑖+1) ⊗ · · · ⊗ 𝜇(𝑎 𝑗+1, 𝑎 𝑗+2) ⊗ · · · ⊗ 𝑎𝑛+1. The ( 𝑗 + 1, 𝑖) term is clearly alike, with coefficient 𝛼((cid:12) (cid:12)𝑎 𝑗 (cid:12) ◦ · · · ◦|𝑎0| ,(cid:12) (cid:12) (cid:12)𝑎 𝑗+1 (cid:12) ,(cid:12) (cid:12) (cid:12)𝑎 𝑗+2 (cid:12) (cid:12))𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|) Thus, these two terms cancel, as(cid:12) (cid:12) = |𝑎𝑖+1| ◦|𝑎𝑖 |. Finally, suppose that 𝑖 = 𝑗 > 0. Then, the (𝑖, 𝑖)-term is (cid:12)𝜇(𝑎𝑖, 𝑎𝑖+1)(cid:12) 𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,(cid:12) (cid:12)𝜇(𝑎𝑖, 𝑎𝑖+1)(cid:12) (cid:12) ,|𝑎𝑖+2|) times 𝑎0 ⊗ · · · ⊗ 𝜇(𝜇(𝑎𝑖, 𝑎𝑖+1), 𝑎𝑖+2) ⊗ · · · ⊗ 𝑎𝑛+1, and the (𝑖 + 1, 𝑖)-term is 𝛼(|𝑎𝑖 | ◦|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖+1| ,|𝑎𝑖+2|)𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,(cid:12) (cid:12)𝜇(𝑎𝑖+1, 𝑎𝑖+2)(cid:12) (cid:12)) times 𝑎0 ⊗ · · · ⊗ 𝜇(𝑎𝑖, 𝜇(𝑎𝑖+1, 𝑎𝑖+2)) ⊗ · · · ⊗ 𝑎𝑛+1. Write 𝑓 = |𝑎𝑖−1| ◦ · · · ◦|𝑎0|, 𝑔 = |𝑎𝑖 |, ℎ = |𝑎𝑖+1| and ℓ = |𝑎𝑖+2|. Then, the cocycle relation 𝑑𝛼( 𝑓 , 𝑔, ℎ, ℓ) = 1 implies that these two terms are equivalent, since 𝜇(𝜇(𝑎𝑖, 𝑎𝑖+1), 𝑎𝑖+2) = 𝛼(|𝑎𝑖 | ,|𝑎𝑖+1| ,|𝑎𝑖+2|)𝜇(𝑎𝑖, 𝜇(𝑎𝑖+1, 𝑎𝑖+2)). This concludes the proof. □ 193 Suppose 𝑎, 𝑎0, 𝑎1, . . . , 𝑎𝑛+1 ∈ 𝐴. We define the following values: let Φ(|𝑎| ,|𝑎0| ,|𝑎1| , . . . ,|𝑎𝑛+1|) := 𝑛+1 (cid:214) 𝑖=1 𝛼(|𝑎| ,|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 |)−1 and Ψ(|𝑎0| , . . . ,|𝑎𝑛| ,|𝑎𝑛+1| ,|𝑎|) := 𝛼(|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1| ,|𝑎|). Proposition 9.2.5. If 𝐴 is a C-graded algebra, (B(𝐴), 𝜕) is a C-graded DG-(𝐴, 𝐴)-bimodule, with left-action 𝜌𝐿(𝑎, 𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1) := Φ(|𝑎| ,|𝑎0| ,|𝑎1| , . . . ,|𝑎𝑛+1|)𝜇 𝐴(𝑎, 𝑎0) ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1 and right-action 𝜌𝑅(𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1, 𝑎) := Ψ(|𝑎0| , . . . ,|𝑎𝑛| ,|𝑎𝑛+1| ,|𝑎|)𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝜇 𝐴(𝑎𝑛+1, 𝑎). Proof. After Lemma 9.2.4, we need to verify axioms (B.I)–(B.IV) and axioms (DG.II) and (DG.III). Like many proofs to this point, the argument is straightforward, but tedious. We’ll verify the more difficult (B.I), (B.III), and (DG.II), leaving the rest to the reader. These three are more tedious because of the involvement of the right-action. For (B.I), we must show that 𝜌𝐿(𝜇(𝑎, 𝑎′), 𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1) = Φ((cid:12) (cid:12)𝜇(𝑎, 𝑎′)(cid:12) (cid:12) ,|𝑎0| , . . . ,|𝑎𝑛+1|)𝜇(𝜇(𝑎, 𝑎′), 𝑎0) ⊗ 𝑎1 ⊗ . . . ⊗ 𝑎𝑛+1 is equal to 𝛼(|𝑎| ,(cid:12) (cid:12)𝑎′(cid:12) = 𝛼(|𝑎| ,(cid:12) (cid:12) ,|𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1|)𝜌𝐿(𝑎, 𝜌𝐿(𝑎′, 𝑎0 ⊗ . . . ⊗ 𝑎𝑛+1)) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1|)Φ(|𝑎| ,(cid:12) (cid:12)𝜇(𝑎′, 𝑎0)(cid:12) (cid:12) ,|𝑎1| , . . . ,|𝑎𝑛+1|)Φ((cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎0| , . . . ,|𝑎𝑛+1|) 𝜇(𝑎, 𝜇(𝑎′, 𝑎0)) ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1 Thus, it suffices to prove that 𝛼(|𝑎| ,(cid:12) (cid:12)𝑎′(cid:12) Φ((cid:12) (cid:12) ,|𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1|)× (cid:12)𝜇(𝑎, 𝑎′)(cid:12) (cid:12) ,|𝑎0| , . . . ,|𝑎𝑛+1|)−1Φ(|𝑎| ,(cid:12) (cid:12)𝜇(𝑎′, 𝑎0)(cid:12) (cid:12) ,|𝑎1| , . . . ,|𝑎𝑛+1|)Φ((cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎0| , . . . ,|𝑎𝑛+1|) 194 is equal to 𝛼(|𝑎| ,|𝑎′| ,|𝑎0|). This can be seen via an iterative process. Start with the “𝑛 + 1” terms from the expansions of each of the Φ products. These look like (cid:12)𝑎′(cid:12) 𝛼((cid:12) (cid:12) ◦|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ◦(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛+1|)−1𝛼((cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1. Taking 𝑓 = |𝑎|, 𝑔 = |𝑎′|, ℎ = |𝑎𝑛| ◦ · · · ◦ |𝑎0|, and ℓ = |𝑎𝑛+1|, inspecting the cocycle relation for 𝑑𝛼( 𝑓 , 𝑔, ℎ, ℓ), we see that the above is equal to 𝛼(|𝑎| ,(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛+1| ◦|𝑎𝑛| ◦ · · · ◦|𝑎0|)−1𝛼(|𝑎| ,(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛| ◦|𝑎𝑛−1| ◦ · · · ◦|𝑎0|). On one hand, the first term cancels with the original 𝛼(|𝑎| ,|𝑎′| ,|𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1|) term. On the other, consider the product of the second term with the “𝑛” terms from the Φ-expansions: these look like (cid:12)𝑎′(cid:12) 𝛼((cid:12) (cid:12) ◦|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ◦(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛+1|)−1𝛼((cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1. Then, taking 𝑓 = |𝑎|, 𝑔 = |𝑎′|, ℎ = |𝑎𝑛−1|◦· · ·◦|𝑎0|, and ℓ = |𝑎𝑛|, the cocycle relation for 𝑑𝛼( 𝑓 , 𝑔, ℎ, ℓ) tells us that this product is equal to 𝛼(|𝑎| ,(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ,|𝑎𝑛−1| ◦|𝑎𝑛−2| ◦ · · · ◦|𝑎0|). To conclude the proof, iterate this process, which terminates with leftover term 𝛼(|𝑎| ,|𝑎′| ,|𝑎0|). The proof of (B.II) is far easier given that Ψ is expressed by only one 𝛼 term. It follows by only one application of the cocycle relation. Similarly, though there are more terms, the proof of (B.III) requires only one application of the cocycle relation. Both are left to the reader. The proof of the first part of (B.IV) requires an iteration. By definition, we have 𝜌𝐿(1𝑋, 𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1) = Φ(Id𝑋,|𝑎0| , . . . ,|𝑎𝑛+1|)𝜇(1𝑋, 𝑎0) ⊗ 𝑎1 ⊗ . . . ⊗ 𝑎𝑛+1 = Φ(Id𝑋,|𝑎0| , . . . ,|𝑎𝑛+1|)L(Id𝑋,|𝑎0|)𝑎0 ⊗ · · · ⊗ 𝑎𝑛+1 195 By assumption, L(Id𝑋,|𝑎0|) = 𝛼(Id𝑋, Id𝑋,|𝑎0|)−1. Expanding Φ, we have 𝛼(Id𝑋,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1 · · · 𝛼(Id𝑋,|𝑎1| ◦|𝑎0| ,|𝑎2|)−1𝛼(Id𝑋,|𝑎0| ,|𝑎1|)−1𝛼(Id𝑋, Id𝑋,|𝑎0|)−1 = 𝛼(Id𝑋,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1 · · · 𝛼(Id𝑋,|𝑎1| ◦|𝑎0| ,|𝑎2|)−1𝛼(Id𝑋, Id𝑋,|𝑎1| ◦|𝑎0|) ... = 𝛼(Id𝑋, Id𝑋,|𝑎𝑛+1| ◦ · · · ◦|𝑎0|)−1 = L(Id𝑋,|𝑎0| ⊗ · · · ⊗ |𝑎𝑛+1|) by iterative applications of (ii) from Lemma 9.0.1. Similarly, the proof of the second half of (B.IV) follows from a single application of (iii) from Lemma 9.0.1 We proceed to proving the DG-axioms. As noted earlier, (DG.I) is immediate, and (DG.IV) is Lemma 9.2.4. Checking (DG.II) directly, we compute that 𝜕(𝜌𝐿(𝑎, 𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1)) = 𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1𝛼(|𝑎| ,|𝑎𝑛−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑛|)−1 · · · 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)−1 𝜕(𝜇(𝑎, 𝑎1) ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1) = 𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1𝛼(|𝑎| ,|𝑎𝑛−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑛|)−1 · · · 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)−1 𝜇(𝜇(𝑎, 𝑎0), 𝑎1) ⊗ 𝑎2 ⊗ · · · ⊗ 𝑎𝑛+1 + 𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1𝛼(|𝑎| ,|𝑎𝑛−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑛|)−1 · · · 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)−1 𝑛 ∑︁ (−1)𝑖𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ◦|𝑎| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝜇(𝑎, 𝑎0) ⊗ · · · ⊗ 𝜇(𝑎𝑖, 𝑎𝑖+1); ⊗ · · · ⊗ 𝑎𝑛+1. 𝑖=1 On the other hand, 𝜌𝐿(𝑎, 𝜕(𝑎0 ⊗ 𝑎1 ⊗ · · · ⊗ 𝑎𝑛+1)) = 𝜌𝐿(𝑎, 𝜇(𝑎0, 𝑎1) ⊗ · · · ⊗ 𝑎𝑛+1) 𝑛 ∑︁ (−1)𝑖𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝜌𝐿(𝑎, 𝑎0 ⊗ · · · ⊗ 𝜇(𝑎𝑖, 𝑎𝑖+1) ⊗ · · · ⊗ 𝑎𝑛+1) + 𝑖=1 196 which is equal to 𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1𝛼(|𝑎| ,|𝑎𝑛−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑛|)−1 · · · 𝛼(|𝑎| ,|𝑎1| ◦|𝑎0| ,|𝑎2|)−1 𝜇(𝑎, 𝜇(𝑎0, 𝑎1)) ⊗ 𝑎2 ⊗ · · · ⊗ 𝑎𝑛+1 + 𝑛 ∑︁ 𝑖=1 (−1)𝑖𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝛼(|𝑎| ,|𝑎𝑛| ◦ · · · ◦|𝑎0| ,|𝑎𝑛+1|)−1 · · · 𝛼(|𝑎| ,|𝑎𝑖+1| ◦|𝑎𝑖 | ◦|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖+2|)−1𝛼(|𝑎| ,|𝑎𝑖−1| ◦ · · ·|𝑎0| ,|𝑎𝑖+1| ◦|𝑎𝑖 |)−1 𝛼(|𝑎| ,|𝑎𝑖−2| ◦ · · · ◦|𝑎0| ,|𝑎𝑖−1|)−1 · · · 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)−1𝜇(𝑎, 𝑎0) ⊗ · · · ⊗ 𝜇(𝑎𝑖, 𝑎𝑖+1) ⊗ · · · ⊗ 𝑎𝑛+1 There are two cases to consider. First, observe that the coefficients leading the 𝑖 = 0 summands in both expansions are exactly the same outside of the 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)−1 appearing in front of the first, but not the second. But this is as we hoped, as 𝜇(𝜇(𝑎, 𝑎0), 𝑎1) ⊗ 𝑎2 ⊗ · · · ⊗ 𝑎𝑛+1 = 𝛼(|𝑎| ,|𝑎0| ,|𝑎1|)𝜇(𝑎, 𝜇(𝑎0, 𝑎1)) ⊗ 𝑎2 ⊗ · · · ⊗ 𝑎𝑛+1. In the second case, we can consider any of the summands when 𝑖 ≥ 1. The coefficients of these summands are exactly the same outside of the appearance of the terms 𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝛼(|𝑎| ,|𝑎𝑖 | ◦|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖+1|)−1𝛼(|𝑎| ,|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 |)−1 appearing in the first expansion, and the terms 𝛼(|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖 | ,|𝑎𝑖+1|)𝛼(|𝑎| ,|𝑎𝑖−1| ◦ · · · ◦|𝑎0| ,|𝑎𝑖+1| ◦|𝑎𝑖 |)−1 appearing tin the second. However, these values are equivalent by the cocycle condition. The proof of (DG.III) is similar but much less tedious, and is left to the reader. □ Remark 9.2.6. The Φ and Ψ terms are decided naturally by the following processes. The Φ term is chosen by following the path. 𝑎(((𝑎0𝑎1)𝑎2) · · · 𝑎𝑛) 𝛼−1 (𝑎((𝑎0𝑎1)𝑎2) · · · )𝑎𝑛 𝛼−1 𝛼−1 · · · (((𝑎(𝑎0𝑎1))𝑎2) · · · 𝑎𝑛) Φ 𝛼−1 ((((𝑎𝑎0)𝑎1)𝑎2) · · · )𝑎𝑛 Accordingly, the Ψ term is much simpler, since the necessary path is of length one. Ψ:((((𝑎0𝑎1)𝑎2) · · · )𝑎𝑛)𝑎′ 𝛼 (((𝑎0𝑎1)𝑎2) · · · )(𝑎𝑛𝑎′) 197 9.3 The universal trace and C-graded Hochschild homology Recall that, in general, the Hochschild homology of an algebra 𝐴 with coefficients in and (𝐴, 𝐴)-bimodule 𝑀 can be taken as the homology of the complex B(𝐴) ⊗𝐴⊗ 𝐴op 𝑀. Naisse and Putyra [NP20] describe the tensor product of two C-graded modules over an intermediary algebra. Suppose 𝐴, 𝐵, and 𝐶 are C-graded algebras, and that 𝑀 is a C-graded (𝐴, 𝐵)-bimodule and 𝑁 is a C-graded (𝐵, 𝐶)-bimodule. We view 𝑀 ⊗ 𝑁 as a C-graded (𝐴, 𝐶)-bimodule by defining actions 𝐴 ⊗ (𝑀 ⊗ 𝑁) 𝜌 𝑀 ⊗ 𝑁 𝐿 𝑀 ⊗ 𝑁 (𝑀 ⊗ 𝑁) ⊗ 𝐶 𝜌 𝑀 ⊗ 𝑁 𝑅 𝛼−1 (𝐴 ⊗ 𝑀) ⊗ 𝑁 𝜌 𝑀 𝐿 ⊗1𝑁 and 𝛼 𝑀 ⊗ (𝑁 ⊗ 𝐶) 𝑀 ⊗ 𝑁 1𝑀 ⊗𝜌 𝑁 𝑅 We define the tensor product of 𝑀 and 𝑁 over the intermediary algebra 𝐵 as 𝑀 ⊗𝐵 𝑁 := 𝑀 ⊗ 𝑁/ Ä𝜌𝑀 𝑅 (𝑚, 𝑏) ⊗ 𝑛 − 𝛼(|𝑚| ,|𝑏| ,|𝑛|)𝑚 ⊗ 𝜌𝑁 𝐿 (𝑏, 𝑛)ä for any 𝑚 ∈ 𝑀, 𝑏 ∈ 𝐵, and 𝑛 ∈ 𝑁. The C-graded (𝐴, 𝐶)-bimodule structure on 𝑀 ⊗ 𝑁 induces one on 𝑀 ⊗𝐵 𝑁. Finally, if 𝑀 and 𝑁 are C-graded DG-bimodules, we define their tensor product over 𝐵 as where (𝑀, 𝜕𝑀) ⊗𝐵 (𝑁, 𝜕𝑁 ) := (𝑀 ⊗𝐵 𝑁, 𝜕⊗) 𝜕⊗(𝑚 ⊗ 𝑛) := 𝜕𝑀(𝑚) ⊗ 𝑛 + (−1)|𝑚| Z 𝑚 ⊗ 𝜕𝑁 (𝑛). The issue with ⊗𝐴⊗ 𝐴op is that 𝐴 ⊗ 𝐴op is not canonically C-graded, but C × Cop-graded. Explicitly, to define a tensor product over 𝐴 ⊗ 𝐴op, we would like to take the coequalizer of the diagram, where 𝑀 (resp. 𝑁) is a C-graded right (resp. left) 𝐴 ⊗ 𝐴op-module. (cid:0)𝑀 × (𝐴 ⊗ 𝐴op)(cid:1) ⊗ 𝑁 𝜌𝑒 𝑅 ⊗1𝑁 Θ 𝑀 ⊗ 𝑁 𝑀 ⊗ (cid:0)(𝐴 ⊗ 𝐴op) × 𝑁(cid:1) 1𝑀 ⊗𝜌𝑒 𝐿 198 However, the connecting map Θ cannot be as simple as 𝛼: in the tensor product over 𝐴 ⊗ 𝐴op, we hope to identify 𝑅(𝑚, 𝑎 ⊗ 𝑎′) ⊗ 𝑛 ∼ 𝑚 ⊗ 𝜌𝑒 𝜌𝑒 𝐿(𝑎 ⊗ 𝑎′, 𝑛) up to some witness Θ. However the former has grading while the latter has grading |𝑛| ◦|𝑎| ◦|𝑚| ◦(cid:12) (cid:12)𝑎′(cid:12) (cid:12) (cid:12)𝑎′(cid:12) (cid:12) (cid:12) ◦|𝑛| ◦|𝑎| ◦|𝑚| . We see this as having two consequences. First, this means that the gradings of the elements involved must form a loop of length four: |𝑚| • • |𝑎′| • |𝑎| • |𝑛| else they are killed in the tensor over 𝐴 ⊗ 𝐴op. More interestingly, this also means that 𝑀 ⊗𝐴⊗ 𝐴op 𝑁, if it is definable, is not C-graded, but rather graded by the universal trace of C: Tr(C) := (cid:222) EndC(𝑋)(cid:14)𝑔 ◦ 𝑓 ∼ 𝑓 ◦ 𝑔. 𝑋∈Ob(C) Remark 9.3.1. The first of the two consequences is interesting, as it means that the “size” of the tensor product over 𝐴 ⊗ 𝐴op (and, thus, the Hochschild homology) in the C-graded setting depends largely on the abundance of loops in C. Notice that this doesn’t have any impact on the Z- or 𝐺-graded settings, as all paths are loops in 𝐵𝐺, thus nothing “extra” dies in the tensor. Fix a grading category (C, 𝛼). There is a canonical quotient map 𝑞 : (cid:222) 𝑋∈ob(C) EndC(𝑋) → Tr(C). 199 We’ll write ˆ𝑋 := 𝑞|EndC(𝑋) to denote the components of 𝑞. By the definition of the universal trace, we have that the diagram HomC(𝑋, 𝑌 ) × HomC(𝑌 , 𝑋) HomC(𝑌 , 𝑋) × HomC(𝑋, 𝑌 ) ◦ EndC(𝑋) ◦ EndC(𝑌 ) (9.3.1) ˆ𝑋 Tr(C) ˆ𝑌 commutes; in other words, for 𝑓 ∈ HomC(𝑋, 𝑌 ) and 𝑔 ∈ HomC(𝑌 , 𝑋), 𝑞(𝑔 ◦ 𝑓 ) = 𝑞( 𝑓 ◦ 𝑔). To extend to grading categories, we need a witness to the above diagram, extending the role played by the associator 𝛼. Let Ω𝑛C denote paths of length 𝑛 in C which form loops. Definition 9.3.2. A looper for a grading category (C, 𝛼) is a function 𝜀 : Ω2C → K× for which (i) 𝜀( 𝑓 , 𝑔)−1 = 𝜀(𝑔, 𝑓 ), and (ii) 𝜀 is coherent with 𝛼; that is, if ℎ ◦ 𝑔 ◦ 𝑓 is a loop of length three in C, then 𝛼( 𝑓 , 𝑔, ℎ)𝜀( 𝑓 , ℎ ◦ 𝑔)𝛼(𝑔, ℎ, 𝑓 )𝜀(𝑔, 𝑓 ◦ ℎ)𝛼(ℎ, 𝑓 , 𝑔)𝜀(ℎ, 𝑔 ◦ 𝑓 ) = 1 (9.3.2) If such an 𝜀 exists, we say that (C, 𝛼) admits a looper. 𝑋 𝑋 ℎ 𝑓 𝑍𝑍 𝑌 𝑌 𝑔 ◦ ◦ 𝛼(𝑔, ℎ, 𝑓 ) ◦ 𝑔 ◦ 𝑓 ◦ ℎ 𝑍𝑍 𝜀(𝑔, 𝑓 ◦ ℎ) 𝑌 𝑌 ˆ𝑍 ˆ𝑌 𝜀( 𝑓 , ℎ ◦ 𝑔) ˆ𝑋 𝑋 𝑋 ℎ ◦ 𝑔 ◦ 𝑓 𝜀(ℎ, 𝑔 ◦ 𝑓 ) 𝑓 ◦ ℎ ◦ 𝑔 ℎ ◦ 𝑓 𝑍𝑍 𝑌 𝑌 𝑔 𝛼(ℎ, 𝑓 , 𝑔) ◦ ◦ ◦ 𝑔 ◦ 𝑓 𝑋 𝑋 ℎ 𝑍𝑍 𝑓 𝑋 𝑋 ◦ ◦ 𝛼( 𝑓 , 𝑔, ℎ) ◦ 𝑌 𝑌 ℎ ◦ 𝑔 200 The the formula (9.3.2) above is called 𝛼-𝜀 coherence. It comes from the observation that the choices in “smoothing” a loop should be witnessed, and that the choice should ultimately be coherent with other choices. We view 𝛼-𝜀 coherence as instructions on the preceding cube, starting at the dotted path. As a natural extension of 𝛼 for loops, 𝛼-𝜀 coherence states that the following hexagon commutes: 𝛼( 𝑓 , 𝑔, ℎ) ℎ ◦ (𝑔 ◦ 𝑓 ) 𝜀(ℎ, 𝑔◦ 𝑓 ) (ℎ ◦ 𝑔) ◦ 𝑓 𝜀( 𝑓 , ℎ◦𝑔) 𝑓 ◦ (ℎ ◦ 𝑔) 𝛼(𝑔, ℎ, 𝑓 ) ( 𝑓 ◦ ℎ) ◦ 𝑔 𝜀(𝑔, 𝑓 ◦ℎ) (𝑔 ◦ 𝑓 ) ◦ ℎ 𝛼(ℎ, 𝑓 , 𝑔) 𝑔 ◦ ( 𝑓 ◦ ℎ) It is very useful to encode the binary matchings above via pictures, as so: • 𝑓 𝑔 • 𝛼( 𝑓 , 𝑔, ℎ) • ℎ 𝜀(ℎ, 𝑔 ◦ 𝑓 ) • 𝑓 • 𝑓 𝑔 • 𝑔 • • ℎ • ℎ 𝜀( 𝑓 , ℎ ◦ 𝑔) 𝛼(ℎ, 𝑓 , 𝑔) • 𝑓 • 𝑓 𝑔 • 𝑔 • • ℎ • ℎ 𝛼(𝑔, ℎ, 𝑓 ) • 𝑓 𝑔 • • ℎ 𝜀(𝑔, 𝑓 ◦ ℎ) Definition 9.3.3. We will call an element of Ω𝑛C an 𝑛-partitioning of a loop it represents in Tr(C). A presentation of an 𝑛-partition with a choice of 𝑛 − 1 binary matchings (frequently depicted as above) is called a topography on the 𝑛-partitioned loop. Denote the set of topographies on an arbitrary 𝑛-partition by 𝑇(𝑛). Lemma 9.3.4. Counting from one, the number of topographies on an 𝑛-partitioned loop is the 𝑛th central binomial coefficient: |𝑇(𝑛)| = Ç2(𝑛 − 1) 𝑛 − 1 å . 201 Proof. Choose a basepoint of an 𝑛-partitioned loop (there are 𝑛 choices). Doing so represents that loop as an element of EndC(𝑋) for some 𝑋 ∈ ob(C). Then, a choice of binary matchings after this first choice is equivalent to the number of distinct full binary trees on 𝑛 leaves, which is equal to the 𝑛 − 1st Catalan number. Thus |𝑇(𝑛)| = 𝑛 · 𝐶𝑛−1 = (cid:0)2(𝑛−1) 𝑛−1 (cid:1), as desired. □ The witnesses 𝛼 and 𝜀 satisfy another relation, which is perhaps obvious. This is, for paths large enough (at least 5), there are squares appearing of the following form: ((𝑎(𝑏𝑐))𝑑)𝑒 𝛼(𝑎, 𝑏, 𝑐) 𝛼(𝑐◦𝑏◦𝑎, 𝑑, 𝑒) (((𝑎𝑏)𝑐)𝑑)𝑒 (𝑎(𝑏𝑐))(𝑑𝑒) 𝛼(𝑐◦𝑏◦𝑎, 𝑑, 𝑒) 𝛼(𝑎, 𝑏, 𝑐) ((𝑎𝑏)𝑐)(𝑑𝑒) Now, this diagram commutes by the well-definedness of 𝛼, and the fact that it takes values in a commutative ring. We call this property distant commutativity for 𝛼. Similarly, there is 𝛼-𝜀 distant commutativity; for a loop partitioned into enough morphisms (at least four), diagrams of the following form start to appear. 𝑔 • 𝑓 • • ℎ • ℓ 𝛼( 𝑓 , 𝑔, ℎ) 𝜀(ℎ ◦ 𝑔 ◦ 𝑓 , ℓ) 𝑔 • 𝑓 𝑔 • 𝑓 • • • • ℎ • ℓ ℎ • ℓ 𝜀(ℎ ◦ 𝑔 ◦ 𝑓 , ℓ) 𝛼( 𝑓 , 𝑔, ℎ) 𝑔 • 𝑓 • • ℎ • ℓ We refer to both properties 𝛼(𝑎, 𝑏, 𝑐)𝛼(𝑐 ◦ 𝑏 ◦ 𝑎, 𝑑, 𝑒) = 𝛼(𝑐 ◦ 𝑏 ◦ 𝑎, 𝑑, 𝑒)𝛼(𝑎, 𝑏, 𝑐) 𝛼( 𝑓 , 𝑔, ℎ)𝜀(ℎ ◦ 𝑔 ◦ 𝑓 , ℓ) = 𝜀(ℎ ◦ 𝑔 ◦ 𝑓 , ℓ)𝛼( 𝑓 , 𝑔, ℎ) 202 ambiguously as distant commutativity. Definition 9.3.5. We denote by T (𝑛) the space of topographies associated to an arbitrary 𝑛- partitioned loop, defined as the following 2-dimensional CW-complex: 1. T (𝑛)0 := 𝑇(𝑛); 2. T (𝑛)1 is an (𝑛 − 1)-valent graph with|𝑇(𝑛)|-many vertices corresponding to changing a single binary matching (𝑛 − 2 correspond to a single application of 𝛼, and one of which corresponds to a basepoint change, i.e., an application of 𝜀); 3. T (𝑛)2 = T (𝑛) is obtained by gluing 2-cells along all words corresponding to a) the cocycle condition on 𝛼, b) 𝛼-𝜀 coherence, or c) distant commutativity. Theorem 9.3.6. Assume that (C, 𝛼) is a grading category admitting a grading by its trace via a looper 𝜀. Suppose that 𝐴 is a C-graded algebra and 𝑀 and 𝑁 are C-graded (𝐴, 𝐴)-bimodules, interpreting 𝑀 as a right C-graded 𝐴 ⊗ 𝐴op-module and 𝑁 as a left C-graded 𝐴 ⊗ 𝐴op-module. Assume Θ(|𝑚| ,|𝑎 ⊗ 𝑎′|C×Cop ,|𝑛|) witnesses a path from |𝑛| ◦ Ä(cid:0)|𝑎| ◦|𝑚|(cid:1) ◦(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ä → Ä(cid:12) (cid:12)𝑎′(cid:12) (cid:12) ◦ (cid:0)|𝑛| ◦|𝑎|(cid:1)ä ◦|𝑚| or, in terms of topographies, Θ(|𝑚| ,|𝑎 ⊗ 𝑎′|C×Cop ,|𝑛|): |𝑚| • (cid:12)𝑎′(cid:12) (cid:12) (cid:12) • • | 𝑎| • |𝑛| |𝑚| • (cid:12)𝑎′(cid:12) (cid:12) (cid:12) • • | 𝑎| • |𝑛| Then, 𝑀 ⊗𝐴⊗ 𝐴op 𝑁 is a Tr(C)-graded module, where 𝑀 ⊗𝐴⊗ 𝐴op 𝑁 := 𝑀 ⊗ 𝑁(cid:14) Ä𝜌𝑒 𝑅(𝑚, 𝑎 ⊗ 𝑎′) ⊗ 𝑛 − Φ(|𝑚| ,(cid:12) (cid:12)𝑎 ⊗ 𝑎′(cid:12) (cid:12)C×Cop ,|𝑛|)𝑚 ⊗ 𝜌𝑒 𝐿(𝑎 ⊗ 𝑎′, 𝑛)ä . 203 Proof. The result holds as long as the value Θ is well-defined. This holds as long as T (4) is simply connected. We can compute that T (4) ≃ 𝑆2; see Figure 9.1. By Lemma 9.3.4, we know that T (4) is (cid:1) = 20 vertices. We count twelve faces: four square, four pentagonal, and four a polyhedron with (cid:0)6 3 hexagonal. Each square face is seen to commute by 𝛼-𝜀 distant commutativity, each pentagonal face commutes by the cocycle condition, and each hexagonal face commutes by 𝛼-𝜀 coherence. □ Corollary 9.3.7. If (C, 𝛼) admits a looper, and 𝐴 is a C-graded algebra, then the Hochschild complex 𝐻𝐶(𝐴, 𝑀) := B(𝐴) ⊗𝐴⊗ 𝐴op 𝐴 is a well-defined (Z × Tr(C))-graded chain complex. 204 𝑔 • 𝑓 𝑔 • 𝑓 𝑔 • 𝑓 𝑔 • 𝑓 ) ℎ , 𝑓 ◦ 𝑔 , ℓ ( 𝛼 𝜀 𝜀 • • • • • • • • ℎ • ℓ ℎ • ℓ ℎ • ℓ ℎ • ℓ 𝛼(𝑔◦𝑓,ℎ,ℓ) 𝛼( 𝑓 , 𝑔, ℎ) 𝑔 • 𝑓 𝛼( 𝑓 , 𝑔, ℎ) 𝑔 • 𝑓 𝜀 • • • • ℎ • ℓ ℎ • ℓ 𝑔 • 𝑓 • • ℎ • ℓ 𝛼( 𝑓 , ℎ ◦ 𝑔, ℓ) 𝜀 𝛼(𝑓,𝑔,ℓ◦ℎ) 𝛼(𝑔, ℎ, ℓ) ℎ • ℓ 𝛼(𝑔, ℎ, ℓ) ℎ • ℓ 𝑔 • 𝑓 𝑔 • 𝑓 𝜀 • • • • 𝛼(ℓ, 𝑓 , ℎ ◦ ℎ) 𝛼(ℎ ◦ 𝑔, ℓ, 𝑓 ) 𝑔 • 𝑓 • • ℎ • ℓ 𝑔 • 𝑓 • • ℎ • ℓ 𝜀 • • ℎ • ℓ 𝑔 • 𝑓 𝛼(ℎ, ℓ, 𝑔 ◦ 𝑓 ) 𝛼(ℓ ◦ ℎ, 𝑓 , 𝑔) 𝛼(ℎ, 𝑓 ◦ ℓ, 𝑔) 𝛼( 𝑓 ◦ ℓ, 𝑔, ℎ) 𝛼(ℓ, 𝑓 , 𝑔) 𝑔 • 𝑓 𝛼(ℓ, 𝑓 , 𝑔) 𝑔 • 𝑓 𝜀 • • • • ℎ • ℓ ℎ • ℓ 𝛼(𝑔, ℎ, 𝑓 ◦ ℓ) 𝛼(ℎ, ℓ, 𝑓 ) ℎ • ℓ 𝛼(ℎ, ℓ, 𝑓 ) ℎ • ℓ 𝑔 • 𝑓 𝑔 • 𝑓 𝜀 • • • • Figure 9.1 The space of topographies for a 4-partitioned loop, T (4). 205 𝑔 • 𝑓 𝑔 • 𝑓 𝑔 • 𝑓 𝑔 • 𝑓 𝜀 𝜀 • • • • • • • • 𝛼 ( 𝑔 , ℓ ◦ ℎ , 𝑓 ) ℎ • ℓ ℎ • ℓ ℎ • ℓ ℎ • ℓ BIBLIOGRAPHY [AM99] Helena Albuquerque and Shahn Majid. Quasialgebra structure of the octonions. J. Algebra, 220(1):188–224, 1999. 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