Special Open Sets in Manifold Calculus

TL;DR

This paper explores generalizations of the polynomial functor concept in manifold calculus by replacing the usual subposets with more general ones, maintaining the core properties of polynomial cofunctors.

Contribution

It introduces a broader class of subposets in manifold calculus and demonstrates that the polynomial functor notion remains consistent under these generalizations.

Findings

Polynomial cofunctors are preserved under more general subposet choices.

The traditional degree k polynomial functors can be characterized using these generalized subposets.

The approach broadens the applicability of manifold calculus techniques.

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 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.