Duplicial functors, descent categories and generalized Hopf modules

Abstract

Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups ${\mathrm{HC}}^{\ddot{I}}{‡}_i(\mathrm{N},\mathrm{M})$ to right and left coalgebras $\mathrm{N}$ respectively $\mathrm{M}$ over a distributive law $\ddot{I}‡$ between two comonads. For the key example associated to a bialgebra $H$, right $\ddot{I}‡$-coalgebras have a description in terms of modules and comodules over $H$. The present article formulates conditions under which such a description is simultaneously possible for the left $\ddot{I}‡$-coalgebras. In the above example, this is the case when the bialgebra $H$ is a Hopf algebra with bijective antipode. We also discuss how the generalized Hopf module theorem by Mesablishvili and Wisbauer features both in theory and examples.

32 pages, many figures