## Homology, Homotopy and Applications

- Homology Homotopy Appl.
- Volume 12, Number 2 (2010), 211-230.

### Homotopy theory of posets

#### Abstract

This paper studies the category of posets Pos as a model for the homotopy theory of spaces. We prove that: (i) Pos admits a (cofibrantly generated and proper) model structure and the inclusion functor Pos → 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 Pos or Cat where the weak equivalences are defined by the nerve functor. We also discuss the homotopy theory of posets from the viewpoint of Alexandroff T0-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.

#### Article information

**Source**

Homology Homotopy Appl. Volume 12, Number 2 (2010), 211-230.

**Dates**

First available in Project Euclid: 28 January 2011

**Permanent link to this document**

http://projecteuclid.org/euclid.hha/1296223882

**Mathematical Reviews number (MathSciNet)**

MR2721035

**Zentralblatt MATH identifier**

05817844

**Subjects**

Primary: 18G55: Homotopical algebra 18B35: Preorders, orders and lattices (viewed as categories) [See also 06-XX] 55U35: Abstract and axiomatic homotopy theory 54G99: None of the above, but in this section

**Keywords**

Model category locally presentable category poset small category Alexandroff space classifying space

#### Citation

Raptis, George. Homotopy theory of posets. Homology Homotopy Appl. 12 (2010), no. 2, 211--230. http://projecteuclid.org/euclid.hha/1296223882.