A straightening-unstraightening equivalence for $\infty$-operads
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866909511497809920 |
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| author | Pratali, Francesca |
| author_facet | Pratali, Francesca |
| contents | We provide a straightening-unstraightening adjunction for $\infty$-operads in Lurie's formalism, and show it establishes an equivalence between the $\infty$-category of operadic left fibrations over an $\infty$-operad $\mathcal{O}^\otimes$ and the $\infty$-category of $\mathcal{O}^\otimes$-algebras in spaces. In order to do so, we prove that the Hinich-Moerdijk comparison functors induce an equivalence between the $\infty$-categories of operadic left fibrations and dendroidal left fibrations over an $\infty$-operad, and we characterize, for any symmetric monoidal $\infty$-category $\mathcal{C}^\otimes$, the essential image of the monoidal unstraightening functor restricted to strong monoidal functors $\mathcal{C}^\otimes\to \mathcal{S}^\times$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2501_05263 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A straightening-unstraightening equivalence for $\infty$-operads Pratali, Francesca Algebraic Topology Category Theory 18N70, 18N45, 18N55, 55P48 We provide a straightening-unstraightening adjunction for $\infty$-operads in Lurie's formalism, and show it establishes an equivalence between the $\infty$-category of operadic left fibrations over an $\infty$-operad $\mathcal{O}^\otimes$ and the $\infty$-category of $\mathcal{O}^\otimes$-algebras in spaces. In order to do so, we prove that the Hinich-Moerdijk comparison functors induce an equivalence between the $\infty$-categories of operadic left fibrations and dendroidal left fibrations over an $\infty$-operad, and we characterize, for any symmetric monoidal $\infty$-category $\mathcal{C}^\otimes$, the essential image of the monoidal unstraightening functor restricted to strong monoidal functors $\mathcal{C}^\otimes\to \mathcal{S}^\times$. |
| title | A straightening-unstraightening equivalence for $\infty$-operads |
| topic | Algebraic Topology Category Theory 18N70, 18N45, 18N55, 55P48 |
| url | https://arxiv.org/abs/2501.05263 |