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

This article is cited in 1 scientific paper (total in 1 paper)

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.

Keywords: pairing of functors, adjoint functors, weak (co)monads, Frobenius monads, firm modules, cofirm comodules, separability.

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Bibliographic databases:
MSC: 18A40, 18C20, 16T1
Received: 28.12.2015
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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}


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    Citing articles on Google Scholar: Russian citations, English citations
    Related articles on Google Scholar: Russian articles, English articles

    This publication is cited in the following articles:
    1. R. Wisbauer, “A categorical approach to algebras and coalgebras”, Int. Electron. J. Algebr., 24 (2018), 153–173  crossref  mathscinet  zmath  isi  scopus
  • Algebra and Discrete Mathematics
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