Cyclic homology arising from adjunctions

TL;DR

This paper explores the construction of cyclic homology from adjunctions involving monads and comonads, providing a unified framework that generalizes known results for associative and Hopf algebras.

Contribution

It introduces a 2-categorical perspective on cyclic homology arising from adjunctions and clarifies the role of bimonads in this context.

Findings

Reproduces cyclic homology for associative and Hopf algebras within a monadic framework.

Establishes conditions for the invertibility of morphisms between duplicial objects.

Provides a self-dual setting linking distributive laws, bimonads, and cyclic homology.

Abstract

Given a monad and a comonad, one obtains a distributive law between them from lifts of one through an adjunction for the other. In particular, this yields for any bialgebroid the Yetter-Drinfel'd distributive law between the comonad given by a module coalgebra and the monad given by a comodule algebra. It is this self-dual setting that reproduces the cyclic homology of associative and of Hopf algebras in the monadic framework of Boehm and Stefan. In fact, their approach generates two duplicial objects and morphisms between them which are mutual inverses if and only if the duplicial objects are cyclic. A 2-categorical perspective on the process of twisting coefficients is provided and the role of the two notions of bimonad studied in the literature is clarified.