| Embedding Calculus, as described by Weiss, is a calculus of functors, suitable for studying contravariant functors from the poset of open subsets of a smooth manifold M, denoted O (M), to a category of topological spaces (of which the functor Emb(--, N) for some fixed manifold N is a prime example). Polynomial functors of degree k can be characterized by their restriction to the subposet of O (M) consisting of open sets which are a disjoint union of at most k components, each diffeomorphic to the open unit ball. In this work, we consider the situation in which M is given as a codimension zero submanifold of a fixed Euclidean space. Then we can characterize polynomial functors by their behavior on the more restrictive subposet consisting of elements which are a disjoint union of actual (translations and scalings of) open balls. Furthermore, these subposets carry a natural topology which can be kept track of while forming polynomial approximations to functors. We show that the Taylor towers generated in this richer setting agree with the previous ones. |
