Comparison of Models for Equivariant Operads

We introduce and explore multiple categorical frameworks which allow for a general investigation of the homotopy theory of equivariant operads. We extend the Cisinski-Moerdijk-Weiss theory of dendroidal sets by suitably generalizing the category of trees in order to record equivariant composition information. This formalism of $G$-trees is then applied to a rebuilt free operad monad in order to endow the category of $G$-operads in $\V$ with an $\F$-semi-model structure for any weak indexing system $\F$ and fairly general model categories $\V$; as a consequence, we prove that all indexing systems can be realized as $N_\infty$-operads, confirming a conjecture of Blumberg-Hill.

Also using $G$-trees, we define appropriate notions of inner $G$-horns and $G$-$\infty$-operads. Inspired by the internal algebra of $G$-trees and $G$-$\infty$-operads, we extend $G$-operads to the new algebraic notion of genuine equivariant operads, which allow us to record the equivariance of our operadic compositions, while removing rigidity conditions on fixed points without relaxing the strictness of the composition laws. Lastly show that there is a natural homotopy strictification functor, sending $G$-$\infty$-operads to an associated genuine $G$-operad.