Polynomial functors in manifold calculus

TL;DR

This paper extends manifold calculus to cofunctors valued in any simplicial model category, characterizing polynomial and homogeneous cofunctors via configuration spaces and Kan extensions, broadening the framework's applicability.

Contribution

It generalizes polynomial cofunctor classification to arbitrary simplicial model categories and relates homogeneous cofunctors to linear ones on configuration spaces.

Findings

Polynomial cofunctors are determined by their values on subposets of open sets.

Homogeneous cofunctors of degree k are equivalent to linear cofunctors on configuration spaces.

Homotopy right Kan extensions preserve isotopy cofunctors in this setting.

Abstract

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Manifold calculus, due to Goodwillie and Weiss, is a calculus of functors suitable for studying contravariant functors (cofunctors) F: O(M)--> Top from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree <= k are determined by their values on O_k(M), where O_k(M) is the full subposet of O(M) whose objects are open subsets diffeomorphic to the disjoint union of at most k balls. Afterwards Pryor showed that one can replace O_k(M) by more general subposets and still recover the same notion of polynomial cofunctor. In this paper, we generalize these results to cofunctors from O(M) to any simplicial model category C. If conf(k, M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M) --> C of degree k is weakly equivalent to the category of linear cofunctors O(conf(k, M)) --> C provided that C has a zero object. Using a completely different approach, we also show that if C is a general model category and F: O_k(M) --> C is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion O_k(M) --> O(M) is also an isotopy cofunctor.