Combinatorial model categories have presentations

TL;DR

This paper proves that all combinatorial model categories can be constructed from diagram categories of simplicial sets through localization, establishing that they have a presentation akin to generators and relations.

Contribution

It demonstrates that every combinatorial model category admits a presentation via localization of diagram categories, unifying their structure.

Findings

Every combinatorial model category can be obtained by localizing a diagram category of simplicial sets.

This provides a systematic way to understand and construct combinatorial model categories.

The result links combinatorial model categories to presentations similar to algebraic generators and relations.

Abstract

We show that every combinatorial model category can be obtained, up to Quillen equivalence, by localizing a model category of diagrams of simplicial sets. This says that any combinatorial model category can be built up from a category of `generators' and a set of `relations'---that is, any combinatorial model category has a presentation.