Toposes with enough points as categories of étale spaces
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arXiv
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| Hauptverfasser: | , , |
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| Format: | Preprint |
| Veröffentlicht: |
2025
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| _version_ | 1866915811622387712 |
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| author | van Gool, Sam Marquès, Jérémie Tarantino, Umberto |
| author_facet | van Gool, Sam Marquès, Jérémie Tarantino, Umberto |
| contents | We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sense of Makkai, for geometric theories with enough Set-models. The same result has recently been obtained independently by Saadia (arXiv:2506.23935) and by Hamad (arXiv:2507.07922). Both of their proofs rely on groupoid representations of toposes, which our proof here does not assume. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2508_09604 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Toposes with enough points as categories of étale spaces van Gool, Sam Marquès, Jérémie Tarantino, Umberto Category Theory General Topology Logic 03G30, 18F10, 03G20, 54D80, 18F70, 03C99 We extend Makkai duality between coherent toposes and ultracategories to a duality between toposes with enough points and ultraconvergence spaces. Our proof generalizes and simplifies Makkai's original proof. Our main result can also be seen as an extension to ionads of Barr's equivalence between topological spaces and relational modules for the ultrafilter monad. In view of the correspondence between toposes and geometric theories, we obtain a strong conceptual completeness theorem, in the sense of Makkai, for geometric theories with enough Set-models. The same result has recently been obtained independently by Saadia (arXiv:2506.23935) and by Hamad (arXiv:2507.07922). Both of their proofs rely on groupoid representations of toposes, which our proof here does not assume. |
| title | Toposes with enough points as categories of étale spaces |
| topic | Category Theory General Topology Logic 03G30, 18F10, 03G20, 54D80, 18F70, 03C99 |
| url | https://arxiv.org/abs/2508.09604 |