Differential bundles as functors from free modules
Florian Schwarz
TL;DR
The paper addresses how to categorize differential bundles in tangent categories by interpreting them as functors from the structure category $${\mathbb N^\bullet}$$ into a tangent category $\mathbb X$. Building on the Garner–Leung actegory framework, it generalizes the equivalence between tangent categories and Weil algebra actegories to lax functors and non-linear natural transformations. The central result shows an equivalence between differential bundles in $\mathbb X$ and differential functors ${\mathbb N^\bullet} \to \mathbb X$, with a constructive pathway: evaluating a differential functor on the generating object ${\mathbb N^1}$ yields a differential bundle, and conversely, every differential bundle induces a differential functor via a canonical Ind construction. The work provides a 1-categorical backbone for extensions to tangent infinity categories and connects differential bundles with classical differential-geometry notions such as vector bundles and connections, while also offering a foundation for exploring differential objects in dynamical-systems contexts. The results offer a versatile, functorial lens on differential geometry within a broad categorical setting, enabling robust translations between bundles, tangent structure, and actegory data.
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.