About Quillen's homotopy theorie of derivatives
type: Item
instance of: scholarly article
title: About Quillen's homotopy theorie of derivatives
zbMATH Open document ID: 1175.18008
author: Olivier Renaudin
published in: Journal of Pure and Applied Algebra
publication date: 2009-06-18T00:00:00Z
review text: This article is concerned with the 2-category of combinatorial left proper Quillen model categories, where the morphisms are Quillen pairs and the \(2\)-morphisms are natural transformations between left adjoints. A Quillen homotopy is such a natural transformation which is a weak equivalence on any cofibrant object. The author constructs path and cylinder model categories and shows that, by first inverting Quillen homotopies, one obtains a 2-category where Quillen equivalences admit a calculus of fractions as studied by \textit{D. A. Pronk} [Compos. Math. 102, No. 3, 243--303 (1996; Zbl 0871.18003)]. The corresponding bilocalization is the 2-category of combinatorial left proper Quillen homotopy theories. Next one can construct from a model category a prederivator. It associates to any small category \({\mathcal A}\) the homotopy category of diagrams of shape~\({\mathcal A}\). The good behavior between pull-back morphisms along functors \({\mathcal A}\to{\mathcal B}\) and their Kan adjoints make it actually into a derivator. The main result of this article is that the pseudo-functor sending combinatorial left proper Quillen homotopy theories to their associated derivators is a local equivalence, i.e., the categories of morphisms between model categories and between their associated derivators are equivalent. The proof is based on a representation result of \textit{D.-C. Cisinski} [Theory Appl. Categ. 20, 605--649 (2008; Zbl 1188.18009)] and the fact that combinatorial model categories have small presentations in the sense of \textit{D. Dugger} [Adv. Math. 164, No. 1, 177--201 (2001; Zbl 1001.18001)].
reviewed by: Jérôme Scherer
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 5566281
zbMATH Keywords: model category
zbMATH Keywords: Quillen equivalence
zbMATH Keywords: Grothendieck derivator
zbMATH Keywords: bilocalization
zbMATH Keywords: 2-category
MaRDI profile type: Publication
full work available at URL: j.jpaa.2009.02.014
OpenAlex ID: W2099229720
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DOI: 10.1016/J.JPAA.2009.02.014
description: scientific article; zbMATH DE number 5566281