A monoidal Grothendieck construction for $\infty$-categories
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arXiv
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| Format: | Preprint |
| Veröffentlicht: |
2022
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| _version_ | 1866917258136125440 |
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| author | Ramzi, Maxime |
| author_facet | Ramzi, Maxime |
| contents | We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty$-category of functors from $\mathbf C$ to $\mathbf{Cat}_\infty$.
In fact, we do this over any $\infty$-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty$-category $\mathbf C$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf{Cat}_\infty$ on the other hand. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2209_12569 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | A monoidal Grothendieck construction for $\infty$-categories Ramzi, Maxime Category Theory Algebraic Topology We construct a monoidal version of Lurie's un/straightening equivalence. In more detail, for any symmetric monoidal $\infty$-category $\mathbf C$, we endow the $\infty$-category of coCartesian fibrations over $\mathbf C$ with a (naturally defined) symmetric monoidal structure, and prove that it is equivalent the Day convolution monoidal structure on the $\infty$-category of functors from $\mathbf C$ to $\mathbf{Cat}_\infty$. In fact, we do this over any $\infty$-operad by categorifying this statement and thereby proving a stronger statement about the functors that assign to an $\infty$-category $\mathbf C$ its category of coCartesian fibrations on the one hand, and its category of functors to $\mathbf{Cat}_\infty$ on the other hand. |
| title | A monoidal Grothendieck construction for $\infty$-categories |
| topic | Category Theory Algebraic Topology |
| url | https://arxiv.org/abs/2209.12569 |