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 $\Xi$ of trees, which carries a tight relationship to the Moerdijk-Weiss category of rooted trees $\Omega$. We prove a nerve theorem exhibiting colored cyclic operads as presheaves on $\Xi$ 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.

This version has been accepted to AGT. Substantial updates throughout, including an alternative description (suggested by the referee) of the morphisms of $\Xi$, a new appendix, and various other improvements

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• Dwyer--Kan homotopy theory for cyclic operads
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• Categories of graphs for operadic structures
• Segal conditions for generalized operads