A straightening-unstraightening equivalence for $\infty$-operads

Abstract

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 ${\mathcal{C}}^{\otimes }\to {\mathcal{S}}^{\times }$.

Version two, comments are welcome!