Polynomial functors in manifold calculus

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)⟶Spaces from O(M) to the category of spaces. Weiss showed that polynomial cofunctors of degree ≤k are determined by their values on Ok(M), where Ok(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 Ok(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 M. If Fk(M) stands for the unordered configuration space of k points in M, we also show that the category of homogeneous cofunctors O(M)⟶M of degree k is weakly equivalent to the category of linear cofunctors O(Fk(M))⟶M provided that M has a zero object. Using a new approach, we also show that if M is a general model category and F:Ok(M)⟶M is an isotopy cofunctor, then the homotopy right Kan extension of F along the inclusion Ok(M)↪O(M) is also an isotopy cofunctor.