Equivariant operads, symmetric sequences, and Boardman-Vogt tensor products

TL;DR

This paper develops the theory of $G$-operads within $$-categories, introducing new constructions and extending existing frameworks to encompass a broader class of operads and their tensor products.

Contribution

It constructs the underlying $G$-symmetric sequence of a $G$-operad, extends the operadic nerve to a conservative functor, and defines a homotopy-commutative Boardman-Vogt tensor product for $G$-operads.

Findings

Established a monadic functor for $G$-symmetric sequences.

Extended Blumberg-Hill's $ abla_ ext{infty}$-operad program.

Defined a homotopy-commutative Boardman-Vogt tensor product.

Abstract

We advance the foundational study of be Nardin-Shah's $\infty$-category of $G$-operads and their associated $\infty$-categories of algebras. In particular, we construct the underlying $G$-symmetric sequence of a (one color) $G$-operad, yielding a monadic functor; we use this to lift Bonventre's genuine operadic nerve to a conservative functor of $\infty$-categories, restricting to an equivalence between categories of discrete $G$-operads. Using this, we extend Blumberg-Hill's program concerning $\mathcal{N}_\infty$-operads to arbitrary sub-operads of the terminal $G$-operad, which we show are equivalent to weak indexing systems. We then go on to define and characterize a homotopy-commutative and closed Boardman-Vogt tensor product on $\mathrm{Op}_G$; in particular, this specializes to a $G$-symmetric monoidal $\infty$-category of $\mathcal{O}$-algebras in a $G$-symmetric monoidal $\infty$-category whose $\mathcal{P}$-algebras are objects with interchanging $\mathcal{O}$-algebra and $\mathcal{P}$-algebra structures.