Computations in topological cohochschild homology

In recent work, Hess and Shipley defined an invariant of coalgebra spectra called topological coHochschild homology (coTHH). In 2018, Bohmann-Gerhardt-Hogenhaven-Shipley-Ziegenhagen developed a coBokstedt spectral sequence to compute the homology of coTHH for coalgebras over the sphere spectrum. However, examples of coalgebras over the sphere spectrum are limited, and one would like to have computational tools to study coalgebras over other ring spectra. In this thesis, we construct a relative coBokstedt spectral sequence to study the topological coHochschild homology of more general coalgebra spectra. We consider H\uD835\uDD3D\uD835\uDC5D 2227HZ H\uD835\uDD3D\uD835\uDC5D and H\uD835\uDD3D\uD835\uDC5D 2227BP H\uD835\uDD3D\uD835\uDC5D for certain values of \uD835\uDC5B as H\uD835\uDD3D\uD835\uDC5D-coalgebras and compute the E2-term of the spectral sequence in these cases. Further, we show that this spectral sequence has additional algebraic structure, and exploit this structure to complete coTHH calculations. Finally, we show that coHochschild homology is a bicategorical shadow, in the sense of Ponto.