2-Segal spaces as invertible infinity-operads

TL;DR

This paper establishes deep connections between various $ abla$-categories and $ ext{2}$-Segal spaces, revealing equivalences and localizations that unify operadic and cyclic structures in higher category theory.

Contribution

It demonstrates that the simplex and Segal categories are localizations of dendroidal categories, and shows an equivalence between invertible $ ext{infinity}$-operads and $ ext{2}$-Segal spaces, also introducing a cyclic dendroidal category.

Findings

$ ext{Delta}$ and $ ext{Gamma}$ are $ ext{infinity}$-localizations of dendroidal categories.

Invertible $ ext{infinity}$-operads are equivalent to $ ext{2}$-Segal spaces.

A cyclic dendroidal category localizes to Connes's cyclic category $ ext{Lambda}$.

Abstract

We exhibit the simplex category $\Delta$ and Segal's category $\Gamma$ as $\infty$-categorical localizations of the dendroidal categories $\Omega_\pi$ and $\Omega$ introduced by Moerdijk and Weiss. As an application we obtain an equivalence of $\infty$-categories between invertible $\infty$-operads and the $2$-Segal spaces of Dyckerhoff and Kapranov. Finally, we describe a cyclic version of the dendroidal category and explain how it $\infty$-localizes to Connes's cyclic category $\Lambda$.