A local-global principle for parametrized $\infty$-categories
Fuente:
arXiv
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| Natura: | Preprint |
| Pubblicazione: |
2024
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| _version_ | 1866915740950462464 |
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| author | Heine, Hadrian |
| author_facet | Heine, Hadrian |
| contents | We prove a local-global principle for $\infty$-categories over any base $\infty$-category $\mathcal{C}$: we show that any $\infty$-category $\mathcal{B} \to \mathcal{C}$ over $\mathcal{C}$ is determined by the following data: the collection of fibers $\mathcal{B}_X$ for $X$ running through the set of equivalence classes of objects of $\mathcal{C}$ endowed with the action of the space of automorphisms $\mathrm{Aut}_X(\mathcal{B})$ on the fiber, the local data, together with a locally cartesian fibration $\mathcal{D} \to \mathcal{C}$ and $\mathrm{Aut}_X(\mathcal{B})$-linear equivalences $\mathcal{D}_X \simeq \mathcal{P}(\mathcal{B}_X)$ to the $\infty$-category of presheaves on $\mathcal{B}_X$, the gluing data. As applications we describe the $\infty$-category of small $\infty$-categories over $[1]$ in terms of the $\infty$-category of left fibrations and prove an end formula for mapping spaces of the internal hom of the $\infty$-category of small $\infty$-categories over $[1]$ and the conditionally existing internal hom of the $\infty$-category of small $\infty$-categories over any small $\infty$-category $\mathcal{C}.$ Considering functoriality in $\mathcal{C}$ we obtain as a corollary that the double $\infty$-category $\mathrm{CORR}$ of correspondences is the pullback of the double $\infty$-category $\mathrm{PR}^L$ of presentable $\infty$-categories along the functor $\infty\mathrm{Cat} \to \mathrm{Pr}^L$ taking presheaves. We deduce that $\infty$-categories over any $\infty$-category $\mathcal{C}$ are classified by normal lax 2-functors. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2409_05568 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | A local-global principle for parametrized $\infty$-categories Heine, Hadrian Algebraic Topology Category Theory We prove a local-global principle for $\infty$-categories over any base $\infty$-category $\mathcal{C}$: we show that any $\infty$-category $\mathcal{B} \to \mathcal{C}$ over $\mathcal{C}$ is determined by the following data: the collection of fibers $\mathcal{B}_X$ for $X$ running through the set of equivalence classes of objects of $\mathcal{C}$ endowed with the action of the space of automorphisms $\mathrm{Aut}_X(\mathcal{B})$ on the fiber, the local data, together with a locally cartesian fibration $\mathcal{D} \to \mathcal{C}$ and $\mathrm{Aut}_X(\mathcal{B})$-linear equivalences $\mathcal{D}_X \simeq \mathcal{P}(\mathcal{B}_X)$ to the $\infty$-category of presheaves on $\mathcal{B}_X$, the gluing data. As applications we describe the $\infty$-category of small $\infty$-categories over $[1]$ in terms of the $\infty$-category of left fibrations and prove an end formula for mapping spaces of the internal hom of the $\infty$-category of small $\infty$-categories over $[1]$ and the conditionally existing internal hom of the $\infty$-category of small $\infty$-categories over any small $\infty$-category $\mathcal{C}.$ Considering functoriality in $\mathcal{C}$ we obtain as a corollary that the double $\infty$-category $\mathrm{CORR}$ of correspondences is the pullback of the double $\infty$-category $\mathrm{PR}^L$ of presentable $\infty$-categories along the functor $\infty\mathrm{Cat} \to \mathrm{Pr}^L$ taking presheaves. We deduce that $\infty$-categories over any $\infty$-category $\mathcal{C}$ are classified by normal lax 2-functors. |
| title | A local-global principle for parametrized $\infty$-categories |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2409.05568 |