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

TL;DR

This paper establishes a fundamental equivalence in the theory of $$-operads, connecting operadic fibrations with algebraic structures in spaces through a new straightening-unstraightening adjunction.

Contribution

It introduces a straightening-unstraightening equivalence for $$-operads, linking operadic fibrations with $$-algebras, and characterizes the unstraightening functor's essential image.

Findings

Proves an equivalence between operadic left fibrations and $$-algebras in spaces.

Establishes an equivalence between operadic and dendroidal left fibrations.

Characterizes the essential image of the monoidal unstraightening functor.

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