A characterization of differential bundles in tangent categories

TL;DR

This paper provides a new characterization of differential bundles within tangent categories, simplifying their identification and understanding in the abstract setting of categorical tangent bundle theory.

Contribution

It introduces a novel characterization of differential bundles, showing they are determined by their projection map and zero section, facilitating easier recognition in tangent categories.

Findings

Differential bundles are characterized by their projection and zero section.

The new characterization simplifies identifying differential bundles.

Results apply across various tangent categories.

Abstract

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds. In that context, Cockett and Cruttwell develop the notion of differential bundle which, by work of MacAdam, generalizes the notion of smooth vector bundle to the abstract setting. Here we provide a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. We show how these results can be used to quickly identify differential bundles in various tangent categories.