$2$–Segal spaces as invertible infinity-operads

Abstract

We exhibit the simplex category $\mathrm{\Delta }$ and Segal’s category $\mathrm{\Gamma }$ as $\infty$–categorical localizations of the dendroidal categories ${\mathrm{\Omega }}_{\pi }$ and $\mathrm{\Omega }$ 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’ cyclic category $\mathrm{\Lambda }$.