Weak Frobenius monads and Frobenius bimodules

Abstract

As observed by Eilenberg and Moore (1965), for a monad F with right adjoint comonad G on any category A, the category of unital F-modules AF is isomorphic to the category of counital G-comodules AG. The monad F is Frobenius provided we have F=G and then AF≃AF. Here we investigate which kind of isomorphisms can be obtained for non-unital monads and non-counital comonads. For this we observe that the mentioned isomorphism is in fact an isomorphisms between AF and the category of bimodules AFF subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad (F,m,η) and a weak comonad (F,δ,ε) satisfying Fm⋅δF=δ⋅m=mF⋅Fδ and m⋅Fη=Fε⋅δ, the category of compatible F-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible F-comodules.