Little cubes operads were introduced by Boardman-Vogt and May to study algebraic structures of iterated loop spaces. Over time, many variations of these operads have emerged, such as the little disks operads and their framed counterparts. Motivated by these examples, Lurie introduced an even broader class of variations of the little cubes operads, which play a central role in the definition of topological chiral homology, also known as factorization homology.

While it has long been known that these variants are colimits of some diagrams of the little cubes operads, the exact diagrams remained unclear. This is unsatisfactory from a conceptual viewpoint, as the power of colimits lies in colimiting diagrams and not in the resulting object. In this talk, I will explain my recent work [arXiv:2406.01084], in which I constructed explicit diagrams whose colimits yield Lurie's variations. I will also discuss applications to gluing locally constant factorization algebras.