Théories homotopiques dans les topos. (Homotopy theories in topoi)
type: Item
instance of: scholarly article
title: Théories homotopiques dans les topos. (Homotopy theories in topoi)
zbMATH Open document ID: 1015.18009
DOI: 10.1016/S0022-4049(01)00176-1
author: Denis-Charles Cisinski
published in: Journal of Pure and Applied Algebra
publication date: 2002-12-02T00:00:00Z
review text: The central theme of the paper is to give a construction of a closed model structure [in the sense of \textit{D. Quillen}, ``Homotopical algebra'', Lect. Notes Math. 43 (1967; Zbl 0168.20903)] on any (Grothendieck) topos \(\mathcal E\), given a fixed class \(\mathcal W\) of morphisms (satisfying certain stability conditions) to play the role of weak equivalences and where the cofibrations are the monomorphisms in \(\mathcal E\). Although inspired by the closed model structure given by \textit{A. Joyal} on the category of simplicial sets (unpublished), it is carried out in any topos \(\mathcal E\), in particular in the topos of simplicial sets of any given topos \(\mathcal S\). The paper is strongly influenced by the work and approach of \textit{A. Grothendieck} but also utilizes some ideas which are typical of elementary treatment of topos theory, after \textit{F. W. Lawvere} and \textit{M. Tierney} (such as ``the Lawvere cylinder'', obtained from ``the Lawvere object'', which is the subobjects classifier). It is not clear, however, that everything in this paper could be carried out entirely in an elementary (first-order) manner, as some set-theoretic constructions are used throughout. A first section is devoted to a review of the basic notions of accessible functors [following \textit{A. Grothendieck} in SGA4, Lect. Notes Math. 269 (1972; Zbl 0234.00007)] and to the ``small object'' type of argument and of the factorization of an arrow associated with it. The main section in the paper is the second one, where from the abstract ``homotopy data'' determined by a cylinder in a topos \(\mathcal E\), the author defines anodyne extensions and in turn a closed model structure on \(\mathcal E\) as desired. It is pointed out that this construction, though ad hoc, does not depend on the choice of a cellular model. The rest of the paper is devoted to some refinements of the axiomatics (for instance in order to determine when is the closed model structure a ``proper'' one), an…
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18E35
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 18G30
zbMATH DE Number: 1839048
zbMATH Keywords: closed model structure
zbMATH Keywords: homotopy theory
zbMATH Keywords: simplicial sets
reviewed by: Marta Bunge
MaRDI profile type: Publication
cites work: A cellular nerve for higher categories
cites work: The localization of spaces with respect to homology
cites work: Quillen closed model structures for sheaves
cites work: Q5580926
cites work: Q4220599
cites work: Motivic symmetric spectra
cites work: Boolean localization, in practice
cites work: Q4246616
cites work: \(\mathbb{A}^1\)-homotopy theory of schemes
cites work: Homotopical algebra
full work available at URL: s0022-4049(01)00176-1
OpenAlex ID: W2033555113
description: scientific article; zbMATH DE number 1839048