Extending conceptual completeness via virtual ultracategories
Fuente:
arXiv
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| Formato: | Preprint |
| Publicado: |
2025
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| _version_ | 1866912457997418496 |
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| author | Saadia, Gabriel |
| author_facet | Saadia, Gabriel |
| contents | We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $β$-algebras. From a categorical point of view, virtual ultracategories generalize ultracategories in the same way that multicategories generalize monoidal categories. From a logical point of view, whereas the points of a coherent topos form an ultracategory, the points of an arbitrary topos form a virtual ultracategory. We then extend Makkai--Lurie's conceptual completeness: a topos with enough points can be reconstructed from its virtual ultracategory of points. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2506_23935 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Extending conceptual completeness via virtual ultracategories Saadia, Gabriel Category Theory Logic 03G30 (Primary) 03C20, 18M65, 54D80 (Secondary) We introduce the notion of virtual ultracategory. From a topological point of view, this notion can be seen as a categorification of relational $β$-algebras. From a categorical point of view, virtual ultracategories generalize ultracategories in the same way that multicategories generalize monoidal categories. From a logical point of view, whereas the points of a coherent topos form an ultracategory, the points of an arbitrary topos form a virtual ultracategory. We then extend Makkai--Lurie's conceptual completeness: a topos with enough points can be reconstructed from its virtual ultracategory of points. |
| title | Extending conceptual completeness via virtual ultracategories |
| topic | Category Theory Logic 03G30 (Primary) 03C20, 18M65, 54D80 (Secondary) |
| url | https://arxiv.org/abs/2506.23935 |