Homotopy theory of posets
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.
scientific article; zbMATH DE number 5817844