Internal higher topos theory
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2023
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| _version_ | 1866912280705236992 |
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| author | Martini, Louis Wolf, Sebastian |
| author_facet | Martini, Louis Wolf, Sebastian |
| contents | We develop the theory of topoi internal to an arbitrary $\infty$-topos $\mathcal B$. We provide several characterisations of these, including an internal analogue of Lurie's characterisation of $\infty$-topoi, but also a description in terms of the underlying sheaves of $\infty$-categories, and we prove a number of structural results about these objects. Furthermore, we show that the $\infty$-category of topoi internal to $\mathcal B$ is equivalent to the $\infty$-category of $\infty$-topoi over $\mathcal B$, and use this result to derive a formula for the pullback of $\infty$-topoi. Lastly, we use our theory to relate smooth geometric morphisms of $\infty$-topoi to internal locally contractible topoi. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2303_06437 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Internal higher topos theory Martini, Louis Wolf, Sebastian Category Theory Algebraic Topology We develop the theory of topoi internal to an arbitrary $\infty$-topos $\mathcal B$. We provide several characterisations of these, including an internal analogue of Lurie's characterisation of $\infty$-topoi, but also a description in terms of the underlying sheaves of $\infty$-categories, and we prove a number of structural results about these objects. Furthermore, we show that the $\infty$-category of topoi internal to $\mathcal B$ is equivalent to the $\infty$-category of $\infty$-topoi over $\mathcal B$, and use this result to derive a formula for the pullback of $\infty$-topoi. Lastly, we use our theory to relate smooth geometric morphisms of $\infty$-topoi to internal locally contractible topoi. |
| title | Internal higher topos theory |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2303.06437 |