$\infty$-operadic foundations for embedding calculus

Abstract

Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of $\infty$-categories of truncated right-modules over a unital $\infty$-operad $\mathcal{O}$. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as $\mathcal{O}$ varies, and generalise these results to the level of Morita $(\infty,2)$-categories. Applied to the ${\rm BO}(d)$-framed $E_d$-operad, this extends Goodwillie-Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the $E_d$-operad, it yields new versions of embedding calculus, such as one for topological embeddings, based on ${\rm BTop}(d)$, or one similar to Boavida de Brito-Weiss' configuration categories, based on ${\rm BAut}(E_d)$. In addition, we prove a delooping result in the context of embedding calculus, establish a convergence result for topological embedding calculus, improve upon the smooth convergence result of Goodwillie, Klein, and Weiss, and deduce an Alexander trick for homology 4-spheres.