Atomic Toposes with Co-Well-Founded Categories of Atoms
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2024
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| _version_ | 1866909621554249728 |
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| author | Marquès, Jérémie |
| author_facet | Marquès, Jérémie |
| contents | The atoms of the Schanuel topos can be described as the pairs $(n,G)$ where $n$ is a finite set and $G$ is a subgroup of $\operatorname{Aut}(n)$. We give a general criterion on an atomic site ensuring that the atoms of the topos of sheaves on that site can be described in a similar fashion. We deduce that these toposes are locally finitely presentable. By applying this to the Malitz-Gregory atomic topos, we obtain a counter-example to the conjecture that every locally finitely presentable topos has enough points. We also work out a combinatorial property satisfied exactly when the sheaves for the atomic topology are the pullback-preserving functors. In this case, the category of atoms is particularly simple to describe. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_14346 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | Atomic Toposes with Co-Well-Founded Categories of Atoms Marquès, Jérémie Category Theory Logic 03G30 (Primary) 18B25, 18C35 (Secondary) The atoms of the Schanuel topos can be described as the pairs $(n,G)$ where $n$ is a finite set and $G$ is a subgroup of $\operatorname{Aut}(n)$. We give a general criterion on an atomic site ensuring that the atoms of the topos of sheaves on that site can be described in a similar fashion. We deduce that these toposes are locally finitely presentable. By applying this to the Malitz-Gregory atomic topos, we obtain a counter-example to the conjecture that every locally finitely presentable topos has enough points. We also work out a combinatorial property satisfied exactly when the sheaves for the atomic topology are the pullback-preserving functors. In this case, the category of atoms is particularly simple to describe. |
| title | Atomic Toposes with Co-Well-Founded Categories of Atoms |
| topic | Category Theory Logic 03G30 (Primary) 18B25, 18C35 (Secondary) |
| url | https://arxiv.org/abs/2406.14346 |