Algebraic patterns and additivity of equivariant little disk operads

In the equivariant algebraic topology surrounding manifolds with smooth $G$-action, Pontryagin-Thom techniques demand we study "$V$-loop spaces," where $V$ is a real $G$-representation. Moreover, in finite-group-equivariant higher algebra, these are corpresentable as suitably grouplike algebras over a "$G$-operad" called $\mathbb{E}_V$; in this talk, I will discuss some aspects of how to construct an equivalence to the category of objects with coherently interchanging $\mathbb{E}_V$-algebra and $\mathbb{E}_W$-algebra structures from the category of $\mathbb{E}_{V \oplus W}$-algebras.

This aformentioned $(\infty,2)$-category of $G$-operads turns out to be presented as the "fibrous patterns" associated with an "algebraic pattern," generalizing the passage from $\mathrm{Fin}_* = \Gamma^{\mathrm{op}}$ to the $(\infty,2)$-category of operads. In this talk, I will more precisely describe a rich Morita theory of suitably nice fibrous patterns, and indicate how to use it to reduce the above additivity theorem to elementary manifold topology.