The nine model category structures on the category of sets

TL;DR

This paper classifies all nine model category structures on the category of sets, providing a detailed analysis of their relationships and equivalences, and clarifying the landscape of model structures in this fundamental category.

Contribution

It offers a complete classification of the nine model structures on Set and explores their Quillen equivalences, including examples of non-adjunctively equivalent models.

Findings

Precisely nine model structures on Set identified

Complete classification of weak factorization systems

Explicit examples of non-Quillen equivalent models

Abstract

We give a proof of the folklore theorem, attributed to Goodwillie, that there are precisely nine model structures on the category $\mathsf{Set}$ of sets. This result is deduced from a complete study of lifting problems and the ensuing classification of all weak factorization systems on $\mathsf{Set}$. Moreover, we determine the Quillen equivalences between these model structures and exhibit an explicit example of equivalent model structures that cannot be realized by a single Quillen adjunction.