Eckmann-Hilton arguments in equivariant higher algebra

TL;DR

This paper extends Eckmann-Hilton arguments to equivariant higher algebra, showing how tensor products of certain operads relate to connectivity and lifting properties, with implications for equivariant spectra and localization.

Contribution

It establishes connectivity bounds for tensor products of equivariant operads and characterizes their monoids' lifting properties, connecting to unital $ ext{N}_ ext{infinity}$-operads and semi-Mackey functors.

Findings

Tensor product of connected operads increases connectivity by two.

Unital $G$-operads correspond to unital $ ext{N}_ ext{infinity}$-operads via localizations.

Monoids in certain categories lift to semi-Mackey functors, generalizing Eckmann-Hilton arguments.

Abstract

Let $\mathcal{O}^{\otimes}$ and $\mathcal{P}^{\otimes}$ be $k$- and $\ell$-connected unital $G$-operads subject to the condition for all $S$ that $\mathcal{O}(S) = \emptyset$ if and only if $\mathcal{P}(S) = \emptyset$. We show that the Boardman-Vogt tensor product $\mathcal{O}^{\otimes} \otimes \mathcal{P}^{\otimes}$ is $(k + \ell + 2)$-connected; equivalently, $\mathcal{O} \otimes \mathcal{P}$-monoids in any $(k + \ell + 3)$-category lift uniquely to incomplete semi-Mackey functors. As a consequence, we show that the smashing localizations on unital $G$-operads correspond precisely to unital $\mathcal{N}_\infty$-operads, and hence to the (finite) poset of unital weak indexing systems by previous work of the author. Along the way we characterize $\ell$-connectivity of a unital $G$-operad $\mathcal{O}^{\otimes}$ equivalently as $\ell$-connectivity of $\mathcal{O}$-admissible Wirthm\"uller maps of $\mathcal{O}$-monoid spaces. In the discrete case, under no connectivity assumptions, $\mathcal{O} \otimes \mathcal{P}$-monoids lift uniquely to incomplete semi-Mackey functors, recovering an Eckmann-Hilton argument for "$C_p$-unital magmas." In the limiting case of infinite tensor powers, we take the loops out of equivariant infinite loop space theory, constructing algebraic approximations to incompletely stable $G$-spectra over arbitrary transfer systems.