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Algebra and Discrete Mathematics, 2013, Volume 15, Issue 1, Pages 127–154 (Mi adm415)  

This article is cited in 5 scientific papers (total in 5 papers)

RESEARCH ARTICLE

Regular pairings of functors and weak (co)monads

R. Wisbauer

Mathematisches Institut, Heinrich Heine University, 40225 Düsseldorf, Germany
Full-text PDF (305 kB) Citations (5)
References:
Abstract: For functors $L:\mathbb{A}\to \mathbb{B}$ and $R:\mathbb{B}\to \mathbb{A}$ between any categories $\mathbb{A}$ and $\mathbb{B}$, a pairing is defined by maps, natural in $A\in \mathbb{A}$ and $B\in \mathbb{B}$,
$$ \xymatrix{{\rm Mor}_\mathbb{B} (L(A),B) \ar@<0.5ex>[r]^{\alpha} & {\rm Mor}_\mathbb{A} (A,R(B))\ar@<0.5ex>[l]^{\beta}}. $$

$(L,R)$ is an adjoint pair provided $\alpha$ (or $\beta$) is a bijection. In this case the composition $RL$ defines a monad on the category $\mathbb{A}$, $LR$ defines a comonad on the category $\mathbb{B}$, and there is a well-known correspondence between monads (or comonads) and adjoint pairs of functors.
For various applications it was observed that the conditions for a unit of a monad was too restrictive and weakening it still allowed for a useful generalised notion of a monad. This led to the introduction of weak monads and weak comonads and the definitions needed were made without referring to this kind of adjunction. The motivation for the present paper is to show that these notions can be naturally derived from pairings of functors $(L,R,\alpha,\beta)$ with $\alpha = \alpha\cdot \beta\cdot \alpha$ and $\beta = \beta \cdot\alpha\cdot\beta$. Following closely the constructions known for monads (and unital modules) and comonads (and counital comodules), we show that any weak (co)monad on $\mathbb{A}$ gives rise to a regular pairing between $\mathbb{A}$ and the category of compatible (co)modules.
Keywords: pairing of functors; adjoint functors; weak monads and comonads; $r$-unital monads; $r$-counital comonads; lifting of functors; distributive laws.
Received: 24.08.2012
Revised: 12.09.2012
Bibliographic databases:
Document Type: Article
MSC: 18A40, 18C20, 16T15
Language: English
Citation: R. Wisbauer, “Regular pairings of functors and weak (co)monads”, Algebra Discrete Math., 15:1 (2013), 127–154
Citation in format AMSBIB
\Bibitem{Wis13}
\by R.~Wisbauer
\paper Regular pairings of functors and weak (co)monads
\jour Algebra Discrete Math.
\yr 2013
\vol 15
\issue 1
\pages 127--154
\mathnet{http://mi.mathnet.ru/adm415}
\mathscinet{http://mathscinet.ams.org/mathscinet-getitem?mr=3100133}
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  • https://www.mathnet.ru/eng/adm/v15/i1/p127
  • This publication is cited in the following 5 articles:
    Citing articles in Google Scholar: Russian citations, English citations
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