Thomason-Type Model Structures on Simplicial Complexes and Graphs

TL;DR

This paper establishes new Thomason-type model structures on simplicial complexes and graphs, demonstrating their properties and equivalences, and characterizing cofibrant objects within these structures.

Contribution

It introduces and analyzes Thomason-type model structures on simplicial complexes and reflexive graphs, showing their cofibrant objects and Quillen equivalences.

Findings

Model structures are cofibrantly generated and proper.

All cofibrant simplicial complexes are flag complexes.

Forests are cofibrant objects.

Abstract

In this paper we show that the Matsushita model structure on loop graphs, which is right-transferred from the Kan-Quillen model structure on simplicial sets, factors through two other right-transferred model structures on simplicial complexes and reflexive graphs. We show that each Quillen adjunction between these right-transferred model categories is a Quillen equivalence. These model structures are analogous to the Thomason model structure on small categories, and we prove that they are all cofibrantly generated and proper. Furthermore we show that all cofibrant simplicial complexes are flag complexes, and all forests are cofibrant.