The nine model category structures on the category of sets
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866916911460122624 |
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| author | Antolín-Camarena, Omar Barthel, Tobias |
| author_facet | Antolín-Camarena, Omar Barthel, Tobias |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_15731 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | The nine model category structures on the category of sets Antolín-Camarena, Omar Barthel, Tobias Category Theory Algebraic Topology Combinatorics 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. |
| title | The nine model category structures on the category of sets |
| topic | Category Theory Algebraic Topology Combinatorics |
| url | https://arxiv.org/abs/2508.15731 |