Special Open Sets in Manifold Calculus

Abstract

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 $\mathcal{O}(M)$, 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 $\mathcal{O}_k(M)$, the full subposet of $\mathcal{O}(M)$ 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 $\mathcal{O}_k(M)$ by more general subposets and see that we still recover the same notion of polynomial cofunctor.