Very good homogeneous functors in manifold calculus
Volume 158 / 2019
Abstract
Let $M$ be a smooth manifold, and let $\mathcal {O}(M)$ be the poset of open subsets of $M$. Let $\mathcal{C} $ be a category that has a zero object and all small limits. A homogeneous functor (in the sense of manifold calculus) of degree $k$ from $\mathcal{O}(M) $ to $\mathcal{C} $ is called very good if it sends isotopy equivalences to isomorphisms. In this paper we show that the category ${\rm VGHF}_k$ of such functors is equivalent to the category of contravariant functors from the fundamental groupoid of $F_k(M)$ to $\mathcal {C}$, where $F_k(M)$ stands for the unordered configuration space of $k$ points in $M$. As a consequence of this result, we show that the category ${\rm VGHF}_k$ is equivalent to the category of representations of $\pi _1(F_k(M))$ in $\mathcal {C}$, provided that $F_k(M)$ is connected. We also introduce a subcategory of vector bundles that we call very good vector bundles, and we show that it is abelian, and equivalent to a certain category of very good functors.
