Differential bundles as functors from free modules

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 Rosický, 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.