The nine model category structures on the category of sets

2025/08/21 by Omar Antolín-Camarena, Tobias Barthel, Antolín-Camarena, Omar +1 · 1 voice
#math.CT #math.AT #math.CO

paper · pdf · doi:10.48550/arxiv.2508.15731

Abstract

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

Citations

• Model structures on finite total orders
• Self-duality of the lattice of transfer systems via weak factorization\n systems

Discussions

• There are nine model structures on the category of sets, seven on pointed sets, and five on vector spaces. 🧮 arxiv.org/abs/2508.15731 [bsky, 5 points, 0 comments]

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