Duplicial functors, descent categories and generalized Hopf modules
Ivan Bartulović, John Boiquaye, Ulrich Krähmer
TL;DR
The paper develops a dual theory to classify left χ-coalgebras in a two-comonad setup linked by a distributive law, extending the established right-χ coalgebra framework. It identifies precise adjunction-extension data (including a lax comonad isomorphism and a mate that is invertible) under which left χ-coalgebras on a target functor correspond to opcoalgebras for an extended comonad; this yields a symmetrical Hopf-module/descent-data perspective. The results are illustrated via entwined coalgebras, generalized Hopf module theorems, and concrete Hopf-algebra examples, while a nonexample (the non-empty list bimonad) highlights the boundaries of applicability. Overall, the work unifies cyclic-homology-type constructions with descent categories and Hopf-module theory, offering a dual, categorical lens on quantum-group–like phenomena and their cohomological invariants.
Abstract
Böhm and Ştefan have expressed cyclic homology as an invariant that assigns homology groups $\mathrm{HC}^χ_i(\mathrm N, \mathrm M)$ to right and left coalgebras $\mathrm N$ respectively $\mathrm M$ over a distributive law $χ$ between two comonads. For the key example associated to a bialgebra $H$, right $χ$-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 $χ$-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.