Universal Properties of Variations of the Little Cubes Operads

Abstract

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 $B{\times }_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to \mathrm{Map}({\mathbb{E}}_n,\mathcal{O})$. 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.

21 pages. Some arguments simplified and exposition improved, thanks to comments from the editor and referee. Identical to the version accepted in the Münster Journal of Mathematics, except for formatting