The nine model category structures on the category of sets

We give a proof of the folklore theorem, attributed to Goodwillie, that there are precisely nine model structures on the category $\mathsf{S}\mathsf{e}\mathsf{t}$ of sets. This result is deduced from a complete study of lifting problems and the ensuing classification of all weak factorization systems on $\mathsf{S}\mathsf{e}\mathsf{t}$. 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.