Universal Properties of Variations of the Little Cubes Operads
Kensuke Arakawa
TL;DR
The paper constructs and analyzes the infinity-operad $\mathbb{E}_{B}^{\otimes}$ associated to a map $B\to B\mathrm{Top}(n)$, proving a universal property that identifies operad maps $\mathbb{E}_{B}^{\otimes}\to \mathcal{O}^{\otimes}$ with $\mathrm{Top}(n)$-equivariant maps into $\operatorname{Map}(\mathbb{E}_{n}^{\otimes},\mathcal{O}^{\otimes})$; it shows that $\mathbb{E}_{B}^{\otimes}$ is the colimit of the diagram $B\to B\mathrm{Top}(n)\to \mathcal{O}\mathsf{p}_{\infty}$ of $\mathbb{E}_{n}^{\otimes}$ parametrized by $B$, and establishes a global universal property via the functor $\mathbb{E}_{\bullet}^{\otimes}$ together with left Kan extensions along the Yoneda embedding. This framework yields a new explicit description of algebras over $\mathbb{E}_{B}^{\otimes}$ and provides an alternative proof of Matsuoka’s gluing theorem, showing that locally constant factorization algebras satisfy descent. The work leverages deep machinery from the theory of $\infty$-categories and $\infty$-operads, including categorical patterns, universal weak equivalences, and the unstraightening/straightening correspondence to obtain colimit and descent results that clarify the interaction between ambient Top groups, tangent microbundles, and ambient ambient spaces. Overall, it clarifies the structure of $\mathbb{E}_{B}^{\otimes}$ and its algebras, with implications for factorization homology and topological quantum field theories.
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 $B\times_{B\mathrm{Top}(n)}E\mathrm{Top}(n)\to\operatorname{Map}(\mathbb{E}_{n},\mathcal{O})$. 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.