Enriched $\infty$-categories as marked module categories
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arXiv
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| Auteurs principaux: | , |
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| Format: | Preprint |
| Publié: |
2025
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| _version_ | 1866929674959978496 |
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| author | Reutter, David Zetto, Markus |
| author_facet | Reutter, David Zetto, Markus |
| contents | We prove that an enriched $\infty$-category is completely determined by its enriched presheaf category together with a `marking' by the representable presheaves. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of $\mathcal{V}$-enriched $\infty$-categories and a certain full sub-category of the category of presentable $\mathcal{V}$-module categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-category theory to the theory of presentable $\infty$-categories.
As applications, we use Lurie's tensor product of presentable $\infty$-categories to construct a tensor product of enriched $\infty$-categories with many desirable properties -- including compatibility with colimits and appropriate monoidality of presheaf functors -- and compare it to existing tensor products in the literature. We also re-examine and provide a model-independent reformulation of the notion of univalence (or Rezk-completeness) for enriched $\infty$-categories.
Our comparison result relies on a monadicity theorem for presentable module categories which may be of independent interest. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2501_07697 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Enriched $\infty$-categories as marked module categories Reutter, David Zetto, Markus Algebraic Topology Category Theory We prove that an enriched $\infty$-category is completely determined by its enriched presheaf category together with a `marking' by the representable presheaves. More precisely, for any presentably monoidal $\infty$-category $\mathcal{V}$ we construct an equivalence between the category of $\mathcal{V}$-enriched $\infty$-categories and a certain full sub-category of the category of presentable $\mathcal{V}$-module categories equipped with a functor from an $\infty$-groupoid. This effectively allows us to reduce many aspects of enriched $\infty$-category theory to the theory of presentable $\infty$-categories. As applications, we use Lurie's tensor product of presentable $\infty$-categories to construct a tensor product of enriched $\infty$-categories with many desirable properties -- including compatibility with colimits and appropriate monoidality of presheaf functors -- and compare it to existing tensor products in the literature. We also re-examine and provide a model-independent reformulation of the notion of univalence (or Rezk-completeness) for enriched $\infty$-categories. Our comparison result relies on a monadicity theorem for presentable module categories which may be of independent interest. |
| title | Enriched $\infty$-categories as marked module categories |
| topic | Algebraic Topology Category Theory |
| url | https://arxiv.org/abs/2501.07697 |