Title: Polynomial functors in manifold calculus,
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.