\(A\)-cellular homotopy theories
type: Item
instance of: scholarly article
title: \(A\)-cellular homotopy theories
zbMATH Open document ID: 0934.18011
DOI: 10.1016/S0022-4049(98)00021-8
author: Alexander Nofech
published in: Journal of Pure and Applied Algebra
publication date: 2000-04-26T00:00:00Z
review text: Given a suitable pointed closed simplicial model category \({\mathcal C}\) and a cofibrant object \(A\) in \({\mathcal C}\), the author proves the existence of a new closed model category structure in \({\mathcal C}\) in which cofibrant objects are built by attaching ``\(A\)-cells''. Especially, if \({\mathcal C}\) is the category of pointed spaces the \(A\)-cells are half-smash products \(A\rtimes \Delta[n]\), and the \(A\)-cellular structure coincides with the usual CW-structure when \(A=S^0\). The author uses this existence theorem in order to prove that there exist \(A\)-cellular closed simplicial model category structures in the following categories: (1) in the category of pointed spaces with respect to a cofibrant space \(A\), (2) in the category of pointed simplicial sets with respect to a pointed simplicial set \(A\), (3) in the category of simplicial groups with respect to a cofibrant simplicial group \(A\), (4) in the category of diagrams of pointed spaces over a small category \({\mathcal I}\) with respect to \(A\wedge({\mathcal I}/-)_+\), where \(({\mathcal I}/-)_+\) is the overcategory with a disjoint base point, (5) in the category of pointed simplicial sets over a small category \({\mathcal I}\) with respect to the diagram \(A\wedge({\mathcal I}/-)_+\), (6) in the category of diagrams of simplicial groups over a small category \({\mathcal I}\) with respect to \(A\otimes ({\mathcal I}/-)\), where \(A\) is a cofibrant simplicial group. For diagrams over a contractible category the closed model category structures have the property that under suitable conditions the homotopy limit functor commutes with a factorization of a map into a cofibration followed by a trivial fibration up to an equivalence.
Mathematics Subject Classification ID: 18G30
Mathematics Subject Classification ID: 55U35
Mathematics Subject Classification ID: 55P99
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 1340438
zbMATH Keywords: diagrams over a small category
zbMATH Keywords: small object argument
zbMATH Keywords: closed simplicial model category
zbMATH Keywords: cofibrant objects
zbMATH Keywords: simplicial sets
zbMATH Keywords: simplicial groups
zbMATH Keywords: homotopy limit
reviewed by: Götz Brunner
MaRDI profile type: Publication
full work available at URL: s0022-4049(98)00021-8
OpenAlex ID: W1970265631
description: scientific article; zbMATH DE number 1340438