Equivariant dendroidal Segal spaces and $G$-$\infty$-operads

TL;DR

This paper develops equivariant analogues of Segal spaces and categories for $G$-$\\infty$-operads, establishing model categories and Quillen equivalences, advancing the understanding of equivariant operadic structures.

Contribution

It introduces equivariant complete Segal spaces and Segal categories for $G$-$\infty$-operads, and proves their model categories are Quillen equivalent to existing models.

Findings

Model categories for equivariant Segal spaces and categories are constructed.

Quillen equivalences between these models and existing $G$-$\infty$-operad models are established.

Variants for Blumberg-Hill indexing systems are also developed.

Abstract

We introduce the analogues of the notions of complete Segal space and of Segal category in the context of equivariant operads with norm maps, and build model categories with these as the fibrant objects. We then show that these model categories are Quillen equivalent to each other and to the model category for $G$-$\infty$-operads built in a previous paper. Moreover, we establish variants of these results for the Blumberg-Hill indexing systems. In an appendix, we discuss Reedy categories in the equivariant context.