On tensor products with equivariant commutative operads

We affirm and generalize a conjecture of Blumberg and Hill: unital weak ${\mathcal{N}}_{\mathrm{\infty }}$-operads are closed under $\mathrm{\infty }$-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 $${\underset{‾}{\mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}}}_I^{\otimes }{\underset{‾}{\mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}}}_J^{\otimes }\mathcal{C}\simeq {\underset{‾}{\mathrm{C}\mathrm{A}\mathrm{l}\mathrm{g}}}_{I\vee J}^{\otimes }\mathcal{C}\mathrm{.}$$ We accomplish this by showing that ${\mathcal{N}}_{I\mathrm{\infty }}^{\otimes }$ 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 Wirthmüller isomorphisms. Ultimately, we accomplish this by advancing the equivariant higher algebra of cartesian and cocartesian $I$-symmetric monoidal $\mathrm{\infty }$-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 $\mathrm{\infty }$-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 ${\mathcal{N}}_{I\mathrm{\infty }}$-algebra.