Special Open Sets in Manifold Calculus

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 O_k(M), the full 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 replace O_k(M) by more general subposets and see that we still recover the same notion of polynomial cofunctor.