\(E_ n\)-monoidal categories and their group completions
type: Item
instance of: scholarly article
title: \(E_ n\)-monoidal categories and their group completions
zbMATH Open document ID: 0809.18007
DOI: 10.1016/0022-4049(94)90116-3
author: Gerald Dunn
published in: Journal of Pure and Applied Algebra
publication date: 1994-09-22T00:00:00Z
review text: Let \(\mathbb{R}_ +\) be the additive topological monoid of nonnegative reals and \(\mathbb{R}^*_ +\) the multiplicative topological group of positive reals, considered as permutative topological categories with all morphisms identities. An \(E_ n\)-monoidal category is a based topological category \({\mathcal A}\) with an \(\mathbb{R}^*_ +\)-action together with continuous \(\mathbb{R}^*_ +\)-equivariant based functors \(\mu_ i : {\mathcal A}^ 2 \to {\mathcal A}\), \(i = 1,2, \dots, n\), such that \(({\mathcal A}, \mu_ i,*)\) is strict monoidal for all \(i\), and \(({\mathcal A}, \mu_ i,*)\) contains \(\mathbb{R}_ +\) as a monoidal retract; moreover, the \(\mu_ i\) satisfy a certain interchange condition. Let \({\mathcal M}_ n {\mathcal C}at\) denote the category of \(E_ n\)-monoidal categories and structure preserving functors. In the first part of the paper the author relates \({\mathcal M}_ n {\mathcal C}at\) to categories of \(C\)-objects in \({\mathcal C}at\), the category of topological categories, where \(C\) is a suitable operad, and of \({\mathbf C}\)-algebras in \({\mathcal C}at\), where \({\mathbf C}\) is a suitable monad. He proves that \({\mathcal M}_ \infty {\mathcal C}at\) is equivalent to the category of topological permutative categories, the equivalences preserving the obvious notions of weak equivalence of these categories. An analogous statement is shown for \({\mathcal M}_ 2 {\mathcal C}at\) and the category of topological braided tensor categories and explained for \({\mathcal M}_ 1 {\mathcal C}at\) and the category of topological strict monoidal categories. In the second part the author defines an \(i\)-fold delooping functor \[ W_ i : {\mathcal M}_ n {\mathcal C}at \to {\mathcal M}_{n - i} {\mathcal C}at \] in the following sense: After inverting weak equivalences there is a natural map \(B {\mathcal A} \to \Omega^ i BW_ i {\mathcal A}\) which is a group completion \((B\) is the classifying space functor). This provides a unified proof of th…
Mathematics Subject Classification ID: 18D10
Mathematics Subject Classification ID: 55P47
zbMATH DE Number: 643790
zbMATH Keywords: permutative topological categories
zbMATH Keywords: braided tensor categories
zbMATH Keywords: monoidal categories
zbMATH Keywords: delooping
zbMATH Keywords: permutative category
zbMATH Keywords: infinite loop space
reviewed by: Rainer Vogt
MaRDI profile type: Publication
full work available at URL: 0022-4049(94)90116-3
OpenAlex ID: W1987353700
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description: scientific article; zbMATH DE number 643790