Model Structures on Ind Categories and the Accessibility Rank of Weak Equivalences

TL;DR

This paper establishes intrinsic conditions under which weak fibration categories can be extended to model categories on pro-categories, and applies these to show the finite accessibility of weak equivalences in simplicial sets.

Contribution

It provides new intrinsic criteria for the two out of three property in weak fibration categories, enabling the construction of model structures on pro-categories and proving accessibility results.

Findings

Weak fibration categories can be completed into model categories under certain conditions.

The class of weak equivalences in simplicial sets is finitely accessible.

The results generalize recent findings by Raptis and Rosický.

Abstract

In a recent paper we introduced a much weaker and easy to verify structure than a model category, which we called a "weak fibration category". We further showed that a small weak fibration category can be "completed" into a full model category structure on its pro-category, provided the pro-category satisfies a certain two out of three property. In the present paper we give sufficient intrinsic conditions on a weak fibration category for this two out of three property to hold. We apply these results to prove theorems giving sufficient conditions for the finite accessibility of the category of weak equivalences in combinatorial model categories. We apply these theorems to the standard model structure on the category of simplicial sets, and deduce that its class of weak equivalences is finitely accessible. The same result on simplicial sets was recently proved also by Raptis and Rosick\'y, using different methods.