2024/09/17 by Krannich, Manuel, Kupers, Alexander · 1 citation
#18F50 #18N70 #57S05 #58D10 #Algebraic Topology (math.AT) #Category Theory (math.CT) #FOS: Mathematics #Geometric Topology (math.GT)
paper · doi:10.48550/arxiv.2409.10991
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 O. We study monoidality and naturality properties of this tower, identify its layers, describe the difference between the towers as O varies, and generalise these results to the level of Morita (∞,2)-categories. Applied to the \rm BO(d)-framed Ed-operad, this extends Goodwillie-Weiss' embedding calculus and its layer identification to the level of bordism categories. Applied to other variants of the Ed-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(Ed). 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.