Universal Properties of Variations of the Little Cubes Operads

2024/06/03 by Kensuke Arakawa, Arakawa, Kensuke
Mathematics · #Mathematics and Applications

paper · doi:10.48550/arxiv.2406.01084

Abstract

Given a map B→ BTop(n) of spaces, one can define a version 𝔼_B of the little cubes operad, whose construction is due to Lurie. We show that 𝔼_B enjoys the universal property that, for every ∞-operad O, an operad map 𝔼_B\toO is equivalent to a Top(n)-equivariant map B×_BTop(n)ETop(n)\toMap(𝔼_n,O). This gives us an explicit diagram exhibiting 𝔼_B as a colimit of 𝔼_n parametrized by B. It also shows that locally constant factorization algebras satisfy descent, reproving a recent theorem of Matsuoka.

Related

• Two remarks on spaces of maps between operads of little cubes
• Relative (non-)formality of the little cubes operads and the algebraic Cerf Lemma
• Dimension vs. Genus: A surface realization of the little k-cubes and an E_∞-operad
• Operads and cosimplicial objects: an introduction
• Constructions of E_n Operads