Pseudo-commutativity of KZ 2-monads
type: Item
instance of: scholarly article
title: Pseudo-commutativity of KZ 2-monads
zbMATH Open document ID: 1231.18006
published in: Advances in Mathematics
publication date: 2011-11-08T00:00:00Z
review text: The manuscript is advertised by its author as the first step in his attempt to get a rigid algebraic description of the situation, when certain colimits in a category commute with certain limits, the term ``first'' motivated by the fact that the main attention of the current paper is devoted to commutation of colimits with colimits. More precisely, given a complete and cocomplete symmetric monoidal closed \textbf{Cat}-enriched category \(\mathcal{W}\), the paper shows that a specific type of \(\mathcal{W}\)-monads on \(\mathcal{W}\)-categories, the so-called \textit{Kock-Zöberlein} (abbreviated \textit{KZ}) or \textit{lax-idempotent} \(\mathcal{W}\)-monads, are pseudo-commutative, the pseudo-commutativity in question being unique (Theorem~7.3). The main examples of such monads the author is interested in (considered in Section~8) are monads on \(\mathcal{V}\)-\textbf{Cat}, the algebras of which are \(\mathcal{V}\)-categories with a given small class \(\Phi\) of colimits, which essentially is a small class of weights \(\phi:D\rightarrow\mathcal{V}\). On the other hand, there exist pseudo-commutative 2-monads, which are not KZ (Example~7.5), e.g., the 2-monad on \textbf{Cat}, the algebras of which are symmetric strict monoidal categories. The paper starts with a thorough introduction into the main aspects of enriched category theory underlying its developments (Sections~2~-~6), provides its main achievements and their illustrating examples (Sections~7~-~8), and ends with rather routine \(\mathcal{W}\)-enriched versions of certain results concerning categories of algebras w.r.t. a monad (Appendix~A), e.g., the fact that a 2-monad on \(\mathcal{V}\)-\textbf{Cat}, whose algebras are \(\mathcal{V}\)-categories with chosen finite colimits, is finitary. The manuscript is quite technical in its nature, involving a significant number of enriched category-theoretic concepts and trying to provide a brief insight into each of them. The ultimate outcome of such an exposition, how…
Mathematics Subject Classification ID: 18D20
Mathematics Subject Classification ID: 18D05
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 18D15
Mathematics Subject Classification ID: 18C20
zbMATH DE Number: 5969495
zbMATH Keywords: 2-category
zbMATH Keywords: 2-monad
zbMATH Keywords: colimit
zbMATH Keywords: enriched category
zbMATH Keywords: fully faithful functor
zbMATH Keywords: KZ doctrine
zbMATH Keywords: monoidal category
zbMATH Keywords: multilinear map
zbMATH Keywords: pseudo-closed category
zbMATH Keywords: pseudo-commutative monad
zbMATH Keywords: \(T\)-algebra
zbMATH Keywords: tensor product
author: Ignacio López Franco
reviewed by: Sergejs Solovjovs
MaRDI profile type: Publication
full work available at URL: j.aim.2011.06.039
OpenAlex ID: W2148034910
cites work: Q4488088
cites work: Introduction to bicategories
cites work: Two-dimensional monad theory
cites work: Q3961629
cites work: Monoidal bicategories and Hopf algebroids
cites work: Q3349023
cites work: Adjoint triangles
cites work: Kan extensions in enriched category theory
cites work: Q5586461
cites work: Pseudo-commutative monads and pseudo-closed 2-categories
cites work: Q4828514
cites work: Braided tensor categories
cites work: Q4100669
cites work: Q4100674
cites work: Q3934579
cites work: Q3324132
cites work: Q4679165
cites work: Q4380422
cites work: Q4487535
cites work: \(\mathfrak V\)-Cat is locally presentable or locally bounded if \(\mathfrak V\) is so
cites work: Q5711517
cites work: Q4100671
cites work: Monads on symmetric monoidal closed categories
cites work: Closed categories generated by commutative monads
cites work: Strong functors and monoidal monads
cites work: Monads for which structures are adjoint to units
cites work: Codescent objects and coherence
cites work: A 2-Categories Companion
cites work: Q3159161
cites work: Q5655527
cites work: Q4342436
cites work: Q4487536
cites work: Q3876103
cites work: Doctrines on 2-categories
DOI: 10.1016/J.AIM.2011.06.039
description: scientific article; zbMATH DE number 5969495