Cofibrant generation of pure monomorphisms in presheaf categories
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arXiv
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| Format: | Preprint |
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2025
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| author | Cox, Sean Feigert, Jonathan Kamsma, Mark Mazari-Armida, Marcos Rosický, Jiří |
| author_facet | Cox, Sean Feigert, Jonathan Kamsma, Mark Mazari-Armida, Marcos Rosický, Jiří |
| contents | We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant generation of pure monomorphisms between sets with an $S$-action in terms of $S$: this happens if and only if for all $a, b \in S$ there is $c \in S$ such that $a = cb$ or $ca = b$. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_20278 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Cofibrant generation of pure monomorphisms in presheaf categories Cox, Sean Feigert, Jonathan Kamsma, Mark Mazari-Armida, Marcos Rosický, Jiří Category Theory Logic 18C05, 20M50 (Primary), 03C48, 03C60, 18C35, 20M30 (Secondary) We characterise when the pure monomorphisms in a presheaf category $\mathbf{Set}^\mathcal{C}$ are cofibrantly generated in terms of the category $\mathcal{C}$. In particular, when $\mathcal{C}$ is a monoid $S$ this characterises cofibrant generation of pure monomorphisms between sets with an $S$-action in terms of $S$: this happens if and only if for all $a, b \in S$ there is $c \in S$ such that $a = cb$ or $ca = b$. We give a model-theoretic proof: we prove that our characterisation is equivalent to having a stable independence relation, which in turn is equivalent to cofibrant generation. As a corollary, we show that pure monomorphisms in acts over the multiplicative monoid of natural numbers are not cofibrantly generated. |
| title | Cofibrant generation of pure monomorphisms in presheaf categories |
| topic | Category Theory Logic 18C05, 20M50 (Primary), 03C48, 03C60, 18C35, 20M30 (Secondary) |
| url | https://arxiv.org/abs/2506.20278 |