Models for cyclic infinity operads

TL;DR

This paper develops model structures for cyclic infinity operads using cyclic dendroidal sets and spaces, establishing their equivalence to simplicial cyclic operads, and advancing the understanding of cyclic $ extinfty$-operads.

Contribution

It constructs and proves the equivalence of model structures for cyclic dendroidal models of cyclic $ extinfty$-operads, answering open questions in the field.

Findings

Model structures on cyclic dendroidal sets and spaces are established.

Quillen equivalences to simplicial cyclic operads are proven.

Supports the completion of Walde's program on cyclic 2-Segal spaces.

Abstract

We construct model structures on cyclic dendroidal sets and cyclic dendroidal spaces for cyclic quasi-operads and complete cyclic dendroidal Segal spaces, respectively. We show these models are Quillen equivalent to the model structure for simplicial cyclic operads. This answers in the affirmative a question of the second author and Drummond-Cole concerning model structures for cyclic $\infty$-operads. We infer similar statements for planar cyclic $\infty$-operads, providing the model-categorical foundation needed to complete Walde's program on the relationship between cyclic 2-Segal spaces and planar cyclic $\infty$-operads.