A model structure on the category of small categories for coverings
type: Item
instance of: scholarly article
zbMATH Open document ID: 1284.18033
author: Kohei Tanaka
publication date: 2013-07-12T00:00:00Z
full work available at URL: https://arxiv.org/abs/0907.5339
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 18A22
zbMATH DE Number: 6187539
zbMATH Keywords: model category
zbMATH Keywords: small category
zbMATH Keywords: covering
zbMATH Keywords: 1-type
zbMATH Keywords: groupoid
MaRDI profile type: Publication
arXiv ID: 0907.5339
arXiv classification: math.CT
arXiv classification: math.AT
title: A model structure on the category of small categories for coverings
published in: Mathematical Journal of Okayama University
review text: This article provides two points of view on a model structure on the category of small categories. The homotopical one goes as follows. There is a model structure on \textbf{Cat}, due to \textit{A. Joyal} and \textit{M. Tierney} in a more general context [C. R. Math. Acad. Sci., Soc. R. Can. 13, No. 1, 2--6 (1991; Zbl 0733.18011)], where the weak equivalences are the categorical equivalences. This category is actually combinatorial, simplicial, and left proper so that one can apply the formalism of Bousfield localization with respect to the ``groupoidification'' \(\{ 0 \rightarrow 1\} \longrightarrow \{ 0 \leftrightarrows 1\}\). Weak equivalences are then \(1\)-equivalences, i.e. functors inducing isomorphisms on connected components and fundamental groups. This model category structure on small categories is in fact Quillen equivalent to the analogous one on simplicial sets via the nerve.NEWLINENEWLINEThe second point of view is categorical as all constructions can be made very explicitly. The groupoidification of a category is constructed and coverings are defined as having the unique right lifting property with respect to both inclusions of the trivial category \(*\) into \(\{ 0 \rightarrow 1\}\) so that a covering of categories corresponds exactly to coverings of simplicial sets via the nerve. In fact, coverings are fibrations with discrete fibers, the fibrant replacement of a category \(C\) is its groupoidification \(\pi C\), and the cofibrant replacement of a \textit{pointed} and connected category \(C\) is its universal cover \(\widetilde C\).
reviewed by: Jérôme Scherer
description: scientific article; zbMATH DE number 6187539