$(\infty,1)$-Categorical Comprehension Schemes
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2020
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| _version_ | 1866913435732672512 |
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| author | Stenzel, Raffael |
| author_facet | Stenzel, Raffael |
| contents | We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing Bénabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural generalizations of smallness and relative definability in this context, and show for instance that the universal cartesian fibration is small. Furthermore, by building on Johnstone's notion of comprehension schemes for ordinary fibered categories, we characterize and relate numerous higher categorical properties and structures such as left exactness, local cartesian closedness, univalent morphisms and internal $(\infty,1)$-categories in terms of comprehension schemes. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2010_09663 |
| institution | arXiv |
| publishDate | 2020 |
| record_format | arxiv |
| spellingShingle | $(\infty,1)$-Categorical Comprehension Schemes Stenzel, Raffael Category Theory Algebraic Topology Logic 03G30, 18D30, 18N60, 18C50 We define and study notions of comprehension in $(\infty,1)$-category theory. In essence, we do so by implementing Bénabou's foundations of naive category theory in a univalent meta-theory. In particular, we develop natural generalizations of smallness and relative definability in this context, and show for instance that the universal cartesian fibration is small. Furthermore, by building on Johnstone's notion of comprehension schemes for ordinary fibered categories, we characterize and relate numerous higher categorical properties and structures such as left exactness, local cartesian closedness, univalent morphisms and internal $(\infty,1)$-categories in terms of comprehension schemes. |
| title | $(\infty,1)$-Categorical Comprehension Schemes |
| topic | Category Theory Algebraic Topology Logic 03G30, 18D30, 18N60, 18C50 |
| url | https://arxiv.org/abs/2010.09663 |