Spanier-Whitehead duality for topological coHochschild homology

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.