# Classification of polynomial functors in manifold calculus, Part I

## TL;DR

This paper introduces a classification framework for polynomial functors on manifolds, linking them to linear functors on associated topological spaces, advancing the understanding of manifold calculus.

## Contribution

It provides a novel classification of polynomial functors in manifold calculus via linear functors from a constructed topological space.

## Key findings

- Polynomial functors of degree <= k are classified by linear functors from O(X(k,M)) to C.
- Defines a new topological space X(k,M) associated with a manifold M.
- Establishes a correspondence between polynomial functors and linear functors in a model category.

## Abstract

For a smooth manifold M, we define a topological space X(k,M), and show that polynomial functors O(M)--> C of degree <= k from the poset of open subsets of M to a simplicial model category can be classified be a version of linear functors from O(X(k,M)) to C.