The Rezk Completion for Elementary Topoi
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2026
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| Acceso en línea: | |
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| _version_ | 1866908753534648320 |
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| author | Wullaert, Kobe van der Weide, Niels |
| author_facet | Wullaert, Kobe van der Weide, Niels |
| contents | The development of category theory in univalent foundations and the formalization thereof is an active field of research. Categories in that setting are often assumed to be univalent which means that identities and isomorphisms of objects coincide. One consequence hereof is that equivalences and identities coincide for univalent categories and that structure on univalent categories transfers along equivalences. However, constructions such as the Kleisli category, the Karoubi envelope, and the tripos-to-topos construction, do not necessarily give univalent categories. To deal with that problem, one uses the Rezk completion, which completes a category into a univalent one. However, to use the Rezk completion when considering categories with structure, one also needs to show that the Rezk completion inherits the structure from the original category. In this work, we present a modular framework for lifting the Rezk completion from categories to categories with structure. We demonstrate the modularity of our framework by lifting the Rezk completion from categories to elementary topoi in manageable steps. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2601_04814 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | The Rezk Completion for Elementary Topoi Wullaert, Kobe van der Weide, Niels Logic in Computer Science Category Theory The development of category theory in univalent foundations and the formalization thereof is an active field of research. Categories in that setting are often assumed to be univalent which means that identities and isomorphisms of objects coincide. One consequence hereof is that equivalences and identities coincide for univalent categories and that structure on univalent categories transfers along equivalences. However, constructions such as the Kleisli category, the Karoubi envelope, and the tripos-to-topos construction, do not necessarily give univalent categories. To deal with that problem, one uses the Rezk completion, which completes a category into a univalent one. However, to use the Rezk completion when considering categories with structure, one also needs to show that the Rezk completion inherits the structure from the original category. In this work, we present a modular framework for lifting the Rezk completion from categories to categories with structure. We demonstrate the modularity of our framework by lifting the Rezk completion from categories to elementary topoi in manageable steps. |
| title | The Rezk Completion for Elementary Topoi |
| topic | Logic in Computer Science Category Theory |
| url | https://arxiv.org/abs/2601.04814 |