Adjoint monads and an isomorphism of the Kleisli categories
type: Item
instance of: scholarly article
title: Adjoint monads and an isomorphism of the Kleisli categories
zbMATH Open document ID: 0703.18004
DOI: 10.1016/0021-8693(90)90069-Z
published in: Journal of Algebra
publication date: 1990-01-01T00:00:00Z
review text: Let \({\mathbb{U}}\) be a category, \({\mathbb{F}}\) a monad on \({\mathbb{U}}\) and \({\mathbb{G}}\) a comonad on \({\mathbb{U}}\). If \({\mathbb{F}}\) is right adjoint to \({\mathbb{G}}\), then the Kleisli category \({\mathbb{U}}_{{\mathbb{F}}}\) of \({\mathbb{F}}\) is isomorphic to the Kleisli category \({\mathbb{U}}_{{\mathbb{G}}}\) of \({\mathbb{G}}\), or, alternatively, the category \({\mathbb{U}}_ 0^{{\mathbb{F}}}\) of free F-algebras is equivalent to the category \({\mathbb{U}}_ 0^{{\mathbb{G}}}\) of cofree \({\mathbb{G}}\)-coalgebras. This result is illustrated by an example where \({\mathbb{U}}\) is a category of modules.
Mathematics Subject Classification ID: 18C20
Mathematics Subject Classification ID: 18A40
zbMATH DE Number: 4152662
zbMATH Keywords: monad
zbMATH Keywords: comonad
zbMATH Keywords: Kleisli category
zbMATH Keywords: category of modules
author: Mark Kleiner
reviewed by: Yves Diers
MaRDI profile type: Publication
cites work: Almost split sequences in subcategories
cites work: Almost split sequences for relatively projective modules
cites work: On Tame Algebras and Bocses
cites work: Adjoint functors and triples
cites work: Every Standard Construction is Induced by a Pair of Adjoint Functors
cites work: Q5639839
cites work: The Predual Theorem to the Jacobson-Bourbaki Theorem
full work available at URL: 0021-8693(90)90069-z
OpenAlex ID: W1992155573
description: scientific article; zbMATH DE number 4152662