Homotopy theory of posets

This paper studies the category of posets $\mathcal{Pos}$ as a model for the homotopy theory of spaces. We prove that: (i) $\mathcal{Pos}$ admits a (cofibrantly generated and proper) model structure and the inclusion functor $\mathcal{Pos \to Cat}$ into Thomason’s model category is a right Quillen equivalence, and (ii) there is a proper class of different choices of cofibrations for a model structure on $\mathcal{Pos}$ or $\mathcal{Cat}$ where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the viewpoint of Alexandroff $T_0$-spaces, and we apply a result of McCord to give a new proof of the classification theorems of Moerdijk and Weiss in the case of posets.

Keywords

model category, locally presentable category, poset, small category, Alexandroff space, classifying space

2010 Mathematics Subject Classification

18B35, 18G55, 54G99, 55U35

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Published 1 January 2010