Higher cyclic operads

Abstract

We introduce a convenient definition for weak cyclic operads, which is based on unrooted trees and Segal conditions. More specifically, we introduce a category $\mathrm{\Xi }$ of trees, which carries a tight relationship to the Moerdijk–Weiss category of rooted trees $\mathrm{\Omega }$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\mathrm{\Xi }$ which satisfy a Segal condition. Finally, we produce a Quillen model category whose fibrant objects satisfy a weak Segal condition, and we consider these objects as an up-to-homotopy generalization of the concept of cyclic operad.