Universal Properties of Variations of the Little Cubes Operads

Given a map $B\to B\mathrm{T}\mathrm{o}\mathrm{p}(n)$ of spaces, one can define a version ${\mathbb{E}}_B$ of the little cubes operad, whose construction is due to Lurie. We show that ${\mathbb{E}}_B$ enjoys the universal property that, for every $\mathrm{\infty }$-operad $\mathcal{O}$, an operad map ${\mathbb{E}}_B\to \mathcal{O}$ is equivalent to a $\mathrm{T}\mathrm{o}\mathrm{p}(n)$-equivariant map $B{\times }_{B\mathrm{T}\mathrm{o}\mathrm{p}(n)}E\mathrm{T}\mathrm{o}\mathrm{p}(n)\to \mathrm{Map}({\mathbb{E}}_n,\mathcal{O})$. This gives us an explicit diagram exhibiting ${\mathbb{E}}_B$ as a colimit of ${\mathbb{E}}_n$ parametrized by $B$. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka.