Universal Properties of Variations of the Little Cubes Operads

Given a map $B\to B\mathrm{Top}(n)$ 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 $\infty$-operad $\mathcal{O}$, an operad map $\mathbb{E}_{B}\to\mathcal{O}$ is equivalent to a $\mathrm{Top}(n)$-equivariant map $B\times_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to\operatorname{Map}(\mathbb{E}_{n},\mathcal{O})$. 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.