Weak Frobenius monads and Frobenius bimodules

TL;DR

This paper explores the conditions under which categories of modules and comodules are equivalent for non-unital monads and non-counital comonads, extending classical Frobenius theory.

Contribution

It introduces the concept of weak Frobenius monads and bimodules, generalizing Frobenius properties to non-unital and non-counital contexts.

Findings

Characterization of weak Frobenius monads

Conditions for equivalence of module and comodule categories

Extension of Frobenius theory to non-unital cases

Abstract

As shown by S. Eilenberg and J.C. Moore (1965), for a monad $F$ with right adjoint comonad $G$ on any catgeory $\mathbb{A}$, the category of unital $F$-modules $\mathbb{A}_F$ is isomorphic to the category of counital $G$-comodules $\mathbb{A}^G$. The monad $F$ is Frobenius provided we have $F=G$ and then $\mathbb{A}_F\simeq \mathbb{A}^F$. Here we investigate which kind of equivalences can be obtained for non-unital monads (and non-counital comonads).