Presentability and topoi in internal higher category theory
Fuente:
arXiv
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| Autores principales: | , |
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| Formato: | Preprint |
| Publicado: |
2022
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| _version_ | 1866915209662169088 |
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| author | Martini, Louis Wolf, Sebastian |
| author_facet | Martini, Louis Wolf, Sebastian |
| contents | The goal of this article is to develop the theory of presentable categories and topoi internal to an arbitrary $\infty$-topos $\mathcal{B}$. Our main results are internal analogues of Lurie's and Lurie-Simpson's characterisations of presentable $\infty$-categories and $\infty$-topoi. In the process, we introduce a theory of internal filteredness and accessible internal categories and establish a number of structural results about presentable $\mathcal{B}$-categories such as adjoint functor theorems and the existence of an internal analogue of the Lurie tensor product. We also compare these internal notions with external variants. We show that $\mathcal{B}$-modules embed fully faithfully into presentable $\mathcal{B}$-categories and prove that there is an equivalence between topoi internal to $\mathcal{B}$ and $\infty$-topoi over $\mathcal{B}$. We also include a number of applications of our results, such as a general version of Diaconescu's theorem for $\infty$-topoi and a characterisation of locally contractible geometric morphisms in terms of smoothness. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_05103 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Presentability and topoi in internal higher category theory Martini, Louis Wolf, Sebastian Category Theory Algebraic Topology 18N60, 18N70 The goal of this article is to develop the theory of presentable categories and topoi internal to an arbitrary $\infty$-topos $\mathcal{B}$. Our main results are internal analogues of Lurie's and Lurie-Simpson's characterisations of presentable $\infty$-categories and $\infty$-topoi. In the process, we introduce a theory of internal filteredness and accessible internal categories and establish a number of structural results about presentable $\mathcal{B}$-categories such as adjoint functor theorems and the existence of an internal analogue of the Lurie tensor product. We also compare these internal notions with external variants. We show that $\mathcal{B}$-modules embed fully faithfully into presentable $\mathcal{B}$-categories and prove that there is an equivalence between topoi internal to $\mathcal{B}$ and $\infty$-topoi over $\mathcal{B}$. We also include a number of applications of our results, such as a general version of Diaconescu's theorem for $\infty$-topoi and a characterisation of locally contractible geometric morphisms in terms of smoothness. |
| title | Presentability and topoi in internal higher category theory |
| topic | Category Theory Algebraic Topology 18N60, 18N70 |
| url | https://arxiv.org/abs/2209.05103 |