A straightening-unstraightening equivalence for $\infty$-operads
Francesca Pratali
TL;DR
This work extends the classical straightening/unstraightening correspondence from ∞-categories to ∞-operads. By combining the Hinich–Moerdijk comparison between dendroidal and Lurie operads with the symmetric monoidal envelope, the authors construct a monoidal straightening/unstraightening framework for ∞-operads, establishing an equivalence between the ∞-category of operadic left fibrations over a Lurie ∞-operad 𝒪^⊗ and the ∞-category of 𝒪^⊗-algebras in spaces. They further characterize strong symmetric monoidal left fibrations as the essential image of the unstraightening of strong monoidal functors, and provide an explicit formula for the straightening in the discrete case. The resulting equivalence, St^{𝒪}, is obtained by composing the symmetric monoidal envelope with the monoidal straightening, yielding a robust un/straightening toolkit for ∞-operads with potential applications to how operadic actions live in ∞-spaces. Overall, the paper unifies operadic left fibrations and algebra-valued functors under a single equivalence, advancing the operadic Grothendieck construction in the ∞-categorical setting.
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 $\mathcal{C}^\otimes\to \mathcal{S}^\times$.