Homology, Homotopy and Applications

Volume 12 (2010)

Number 2

Homotopy theory of posets

Pages: 211 – 230

DOI: http://dx.doi.org/10.4310/HHA.2010.v12.n2.a7

Author

George Raptis (Institut für Mathematik, Universität Osnabrück, Germany)

Abstract

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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