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 \(\mathbb{A}\),  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 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 \(\mathbb{A}_F\) and the category of bimodules \(\mathbb{A}^F_F\) subject to certain compatibility conditions (Frobenius bimodules). Eventually we obtain that for a weak monad \((F,m,\eta)\) and a weak comonad  \((F,\delta,\varepsilon)\)  satisfying \(Fm\cdot \delta F = \delta \cdot m = mF\cdot F\delta\) and \(m\cdot F\eta = F\varepsilon\cdot \delta\), the category of compatible \(F\)-modules is isomorphic to the category of compatible Frobenius bimodules and the category of compatible \(F\)-comodules.