Infinity‐operadic foundations for embedding calculus

Abstract Motivated by applications to spaces of embeddings and automorphisms of manifolds, we consider a tower of ‐categories of truncated right modules over a unital ‐operad . We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as varies and generalise these results to the level of Morita ‐categories. Applied to the ‐framed ‐operad, this extends Goodwillie–Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the ‐operad, it yields new versions of embedding calculus, such as one for topological embeddings — based on — or one similar to Boavida de Brito–Weiss' configuration categories — based on . 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.