(Lack of) Model Structures on the Category of Graphs

Abstract

In this article, we study model structures on the category of finite graphs with $\times$-homotopy equivalences as the weak equivalences. We show that there does not exist an analogue of Strøm-Hurewicz model structure on this category of graphs. More interestingly, we show that this category of graphs with $\times$-homotopy equivalences does not have a model structure whenever the class of cofibrations is a subclass of graph inclusions.

minor typos are corrected and a few definitions are added. To appear in the Applied Categorical Structures

References in corpus (2)

• Two Models for the Homotopy Theory of Cocomplete Homotopy Theories
• Lovász' original lower bound: Getting tighter bounds and Reducing computational complexity