Towards 2-derivators for formal $\infty$-category theory
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2023
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| _version_ | 1866915666836062208 |
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| author | Di Vittorio, Nicola |
| author_facet | Di Vittorio, Nicola |
| contents | Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or $(\infty, 1)$-category retains only limited information, the structured collection of homotopy 1-categories of diagram categories often suffices for many homotopical purposes. In this paper, we introduce a set of axioms for a 2-dimensional analog of derivators: a refinement of the homotopy 2-category of an enriched model category or $(\infty, 2)$-category into a coherent system of homotopy 2-categories of higher categories of diagrams. We show that these axioms are satisfied in a variety of models, including standard ones related to $(\infty, 1)$-category theory. Moreover, we prove that the axioms are preserved under a certain shift operation. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2309_05216 |
| institution | arXiv |
| publishDate | 2023 |
| record_format | arxiv |
| spellingShingle | Towards 2-derivators for formal $\infty$-category theory Di Vittorio, Nicola Category Theory Algebraic Topology 18N40 18N65 55U35 Derivators, introduced independently by Grothendieck and Heller in the 1980s, provide a categorical framework for studying homotopy theory. They are based on the idea that, while the homotopy 1-category of a single model category or $(\infty, 1)$-category retains only limited information, the structured collection of homotopy 1-categories of diagram categories often suffices for many homotopical purposes. In this paper, we introduce a set of axioms for a 2-dimensional analog of derivators: a refinement of the homotopy 2-category of an enriched model category or $(\infty, 2)$-category into a coherent system of homotopy 2-categories of higher categories of diagrams. We show that these axioms are satisfied in a variety of models, including standard ones related to $(\infty, 1)$-category theory. Moreover, we prove that the axioms are preserved under a certain shift operation. |
| title | Towards 2-derivators for formal $\infty$-category theory |
| topic | Category Theory Algebraic Topology 18N40 18N65 55U35 |
| url | https://arxiv.org/abs/2309.05216 |