Differential bundles as functors from free modules

TL;DR

This paper characterizes differential bundles in tangent categories as functors from a structure category, generalizing existing equivalences and establishing a functorial correspondence between differential functors and bundles.

Contribution

It generalizes the Garner-Leung equivalence to include lax functors and non-linear transformations, providing a functorial characterization of differential bundles.

Findings

Differential functors from the structure category to a tangent category correspond to differential bundles.

Evaluating a differential functor on the generating object yields a differential bundle.

The construction from bundles to functors is functorial and covers all bundles.

Abstract

This paper explores differential bundles in tangent categories, characterizing them as functors from a structure category. This is analogous to the actegory perspective of Garner and Leung, which we also use to describe the tangent categories of Rosick\'y, Cockett and Cruttwell. We generalize the Garner-Leung equivalence between tangent categories and Weil algebra actegories to include lax functors and non-linear natural transformations. The main result of this paper, is that differential functors between the structure category $\mathbb N^\bullet$ and a tangent category $\mathbb X$ are equivalent to differential bundles in $\mathbb X$. We obtain this result by showing that evaluating a differential functor on the generating object $\mathbb N^1$ of the structure category $\mathbb N^\bullet$ produces a differential bundle in a functorial way. Every differential bundle can be obtained this way. We show that obtaining such a functor from a bundle is a functorial construction. There are variations of these results for linear and additive morphisms of differential bundles.