# Very good homogeneous functors in manifold calculus

## TL;DR

This paper characterizes very good homogeneous functors in manifold calculus as equivalent to functors from the fundamental groupoid of configuration spaces, linking manifold calculus with algebraic topology and representation theory.

## Contribution

It establishes an equivalence between very good homogeneous functors and functors from the fundamental groupoid of configuration spaces, providing new insights into their structure.

## Key findings

- Category of very good homogeneous functors is equivalent to functors from the fundamental groupoid of configuration space.
- When the configuration space is connected, these functors correspond to representations of its fundamental group.
- Introduces very good vector bundles, showing they form an abelian category equivalent to certain functors.

## Abstract

Let M be a smooth manifold, and let O(M) be the poset of open subsets of M. Let 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 O(M) to C is called very good if it sends isotopy equivalences to isomorphisms. In this paper we show that the category VGHF of such functors is equivalent to the category of contravariant functors from the fundamental groupoid of Conf(k, M) to C, where Conf(k, M) stands for the unordered configuration space of k points in M. As a consequence of this result, we show that the category VGHF is equivalent to the category of representations of the fundamental group of Conf(k, M) in C, provided that Conf(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.