On tensor products with equivariant commutative operads

Abstract

We affirm and generalize a conjecture of Blumberg and Hill: unital weak $\mathcal{N}_\infty$-operads are closed under $\infty$-categorical Boardman-Vogt tensor products and the resulting tensor products correspond with joins of weak indexing systems; in particular, we acquire a natural $G$-symmetric monoidal equivalence \[ \underline{\mathrm{CAlg}}^{\otimes}_{I} \underline{\mathrm{CAlg}}^{\otimes}_{J} \mathcal{C} \simeq \underline{\mathrm{CAlg}}^{\otimes}_{I \vee J} \mathcal{C}. \] We accomplish this by showing that $\mathcal{N}_{I\infty}^{\otimes}$ is $\otimes$-idempotent and $\mathcal{O}^{\otimes}$ is local for the corresponding smashing localization if and only if $\mathcal{O}$-monoid $G$-spaces satisfy $I$-indexed Wirthm\"uller isomorphisms. Ultimately, we accomplish this by advancing the equivariant higher algebra of cartesian and cocartesian $I$-symmetric monoidal $\infty$-categories. Additionally, we acquire a number of structural results concerning $G$-operads, including a canonical lift of $\otimes$ to a presentably symmetric monoidal structure and a general disintegration and assembly procedure for computing tensor products of non-reduced unital $G$-operads. All such results are proved in the generality of atomic orbital $\infty$-categories. We also achieve the expected corollaries for (iterated) Real topological Hochschild and cyclic homology and construct a natural $I$-symmetric monoidal structure on right modules over an $\mathcal{N}_{I\infty}$-algebra.