Infinity‐operadic foundations for embedding calculus

Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of ∞-categories of truncated right modules over a uni-tal ∞-operad 𝒪. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as 𝒪 varies and gener-alise these results to the level of Morita (∞, 2)-categories. Applied to the 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 BTop(𝑑) — or one similarto Boavida de Brito–Weiss’ configuration categories —based on 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.