Higher cyclic operads

TL;DR

This paper develops a new framework for weak cyclic operads using unrooted trees and Segal conditions, establishing a nerve theorem and a homotopy-theoretic model category for these structures.

Contribution

It introduces a category of trees related to rooted trees, proves a nerve theorem for cyclic operads, and constructs a model category for weak cyclic operads as homotopy generalizations.

Findings

Established a nerve theorem linking cyclic operads to presheaves on a new tree category.

Constructed a Quillen model category for weak cyclic operads.

Provided a homotopy-theoretic perspective on 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.