Spanier-Whitehead duality for topological coHochschild homology

TL;DR

This paper extends the definition of topological coHochschild homology (coTHH) to $$-categories and demonstrates a duality relationship with topological Hochschild homology (THH) for certain coalgebras, providing new computational tools.

Contribution

It generalizes the definition of coTHH to $$-categories and establishes a Spanier-Whitehead duality between coTHH and THH for quasi-proper coalgebras.

Findings

Computed coTHH of the Steenrod algebra spectrum.

Extended coTHH definition to $$-categories using Nikolaus-Scholze approach.

Proved coTHH of quasi-proper coalgebras can be derived from THH via duality.

Abstract

In this work, we compute the topological coHochschild homology (coTHH) of interesting coalgebras such as the Steenrod algebra spectrum. For this, we start by extending the Hess-Shipley definition of coTHH to $\infty$-categories, following the Nikolaus-Scholze approach to THH. Furthermore, we prove that coTHH of what we call quasi-proper coalgebras can be obtained from THH via Spanier-Whitehead duality which provides further insight into coTHH and its relationship to THH.