Quillen closed model structures for sheaves
type: Item
instance of: scholarly article
title: Quillen closed model structures for sheaves
zbMATH Open document ID: 0828.18005
DOI: 10.1016/0022-4049(94)00033-F
author: Sjoerd E. Crans
published in: Journal of Pure and Applied Algebra
publication date: 1995-12-18T00:00:00Z
review text: Suppose that \({\mathcal C}\) is a category with a Quillen closed model structure and that there is a pair of functors \(L :{\mathcal C}\to{\mathcal D}\) and \(R :{\mathcal D}\to{\mathcal C}\) such that \(L\) is left adjoint to \(R\). Under fairly weak conditions, it is shown that \(D\) has an induced Quillen closed model structure. The result is applied with \({\mathcal C}\) the category of simplicial sheaves on a fixed site and \({\mathcal D}\) the category of sheaves of 2-groupoids, or of bisimplicial sheaves, or of simplicial sheaves of groupoids. In the first case, the homotopy category of \({\mathcal D}\) is equivalent to the full subcategory of the homotopy category of \({\mathcal C}\) generated by objects \(X\) such that \(\pi_nX = 0\) for \(n > 2\). In the remaining two cases, the homotopy category of \({\mathcal D}\) is equivalent to that of \({\mathcal C}\).
reviewed by: Richard John Steiner
Mathematics Subject Classification ID: 18G55
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 55U35
zbMATH DE Number: 797257
zbMATH Keywords: Quillen closed model category
zbMATH Keywords: simplicial sheaves
zbMATH Keywords: sheaves of 2- groupoids
zbMATH Keywords: bisimplicial sheaves
zbMATH Keywords: simplicial sheaves of groupoids
zbMATH Keywords: homotopy category
MaRDI profile type: Publication
cites work: Homotopy limits, completions and localizations
cites work: Abstract homotopy theory and generalized sheaf cohomology
cites work: Q4041756
cites work: On cat\(^ n\)-groups and homotopy types
cites work: Q5185712
cites work: Higher-dimensional crossed modules and the homotopy groups of \((n+1)\)- ads
cites work: Lokal präsentierbare Kategorien. (Locally presentable categories)
cites work: Q5580926
cites work: The loop space of the Q-construction
cites work: Homotopy theories
cites work: Q3744345
cites work: Simplicial presheaves
cites work: Q4145861
cites work: Classifying spaces for sheaves of simplicial groupoids
cites work: Strong stacks and classifying spaces
cites work: A combinatorial definition of homotopy groups
cites work: Spaces with finitely many non-trivial homotopy groups
cites work: Sheaves in geometry and logic: a first introduction to topos theory
cites work: On the 3-Type of a Complex
cites work: Accessible Categories: The Foundations of Categorical Model Theory
cites work: Q3490175
cites work: Algebraic classification of equivariant homotopy 2-types. I
cites work: Homotopical algebra
cites work: Rational homotopy theory
cites work: Higher algebraic K-theory: I
cites work: The algebra of oriented simplexes
cites work: Q3927379
cites work: Combinatorial homotopy. II
description: scientific article; zbMATH DE number 797257