Homotopy theory of posets
type: Item
instance of: scholarly article
title: Homotopy theory of posets
zbMATH Open document ID: 1215.18017
author: George Raptis
published in: Homology, Homotopy and Applications
publication date: 2010-11-19T00:00:00Z
full work available at URL: a7
review text: It is a well known result that posets model all homotopy types via the nerve functor to simplicial complexes. Likewise posets can be considered as small categories and \textit{R. W. Thomason} [Cah. Topol. Géom. Différ. 21, 305--324 (1980; Zbl 0473.18012)] showed that \({\mathcal C}at\) admits a model category structure in which a functor is a weak equivalence if applying the nerve functor yields a weak equivalence of simplicial sets. In this paper, it is shown that the category, \({\mathcal P}os\), of posets also admits a model category structure and that the result is Quillen equivalent to the model category of simplicial sets. Further aspects of various different model category structures on \({\mathcal P}os\) are studied, especially with regard to the cofibrations, and then the link, via McCord's theorem giving an equivalence between \({\mathcal P}os\) and the category of Alexandroff \(T_0\)-spaces, is used to give a new proof of a classification theorem of Moerdijk and Weiss in the case of posets.
reviewed by: Timothy Porter
Mathematics Subject Classification ID: 18G55
zbMATH DE Number: 5817844
zbMATH Keywords: model category
zbMATH Keywords: locally presentable category
zbMATH Keywords: poset
zbMATH Keywords: small category
zbMATH Keywords: Alexandroff space
zbMATH Keywords: classifying space
MaRDI profile type: Publication
DOI: 10.4310/HHA.2010.V12.N2.A7
description: scientific article; zbMATH DE number 5817844