On bi-enriched $\infty$-categories
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arXiv
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| Format: | Preprint |
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2024
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| _version_ | 1866916911247261696 |
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| author | Heine, Hadrian |
| author_facet | Heine, Hadrian |
| contents | We extend Lurie's definition of enriched $\infty$-categories to notions of left enriched, right enriched and bienriched $\infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $\infty$-categories and share many desirable features with them. We use bienriched $\infty$-categories to endow the $\infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $\infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we construct enriched Kan-extensions from operadic Kan-extensions, compute the monad for enriched functors, prove an end formula for morphism objects of enriched $\infty$-categories of enriched functors and a coend formula for the relative tensor product of enriched profunctors and construct transfer of enrichment from scalar extension of presentably bitensored $\infty$-categories. In particular, we develop an independent theory of enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2406_09832 |
| institution | arXiv |
| publishDate | 2024 |
| record_format | arxiv |
| spellingShingle | On bi-enriched $\infty$-categories Heine, Hadrian Category Theory Algebraic Topology We extend Lurie's definition of enriched $\infty$-categories to notions of left enriched, right enriched and bienriched $\infty$-categories, which generalize the concepts of closed left tensored, right tensored and bitensored $\infty$-categories and share many desirable features with them. We use bienriched $\infty$-categories to endow the $\infty$-category of enriched functors with enrichment that generalizes both the internal hom of the tensor product of enriched $\infty$-categories when the latter exists, and the free cocompletion under colimits and tensors. As an application we construct enriched Kan-extensions from operadic Kan-extensions, compute the monad for enriched functors, prove an end formula for morphism objects of enriched $\infty$-categories of enriched functors and a coend formula for the relative tensor product of enriched profunctors and construct transfer of enrichment from scalar extension of presentably bitensored $\infty$-categories. In particular, we develop an independent theory of enriched $\infty$-categories for Lurie's model of enriched $\infty$-categories. |
| title | On bi-enriched $\infty$-categories |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2406.09832 |