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 Algebra Discrete Math.: Year: Volume: Issue: Page: Find

 Algebra Discrete Math., 2016, Volume 21, Issue 2, Pages 287–308 (Mi adm569)

RESEARCH ARTICLE

Weak Frobenius monads and Frobenius bimodules

Robert Wisbauer

Department of Mathematics, HHU, 40225 Düsseldorf, Germany

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.

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MSC: 18A40, 18C20, 16T1
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Citation: Robert Wisbauer, “Weak Frobenius monads and Frobenius bimodules”, Algebra Discrete Math., 21:2 (2016), 287–308

Citation in format AMSBIB
\Bibitem{Wis16} \by Robert~Wisbauer \paper Weak Frobenius monads and Frobenius bimodules \jour Algebra Discrete Math. \yr 2016 \vol 21 \issue 2 \pages 287--308 \mathnet{http://mi.mathnet.ru/adm569} \mathscinet{http://www.ams.org/mathscinet-getitem?mr=3537452} \isi{http://gateway.isiknowledge.com/gateway/Gateway.cgi?GWVersion=2&SrcApp=PARTNER_APP&SrcAuth=LinksAMR&DestLinkType=FullRecord&DestApp=ALL_WOS&KeyUT=000382847700010}