Vector bundles and differential bundles in the category of smooth manifolds

TL;DR

This paper demonstrates that differential bundles in the category of smooth manifolds are equivalent to classical vector bundles, providing an algebraic characterization within tangent categories.

Contribution

It proves that differential bundles in smooth manifolds are exactly vector bundles, linking tangent categorical theory with classical differential geometry.

Findings

Differential bundles in smooth manifolds are vector bundles.

Provides an algebraic characterization of vector bundles.

Establishes models of tangent categorical algebraic theories.

Abstract

A tangent category is a category equipped with an endofunctor that satisfies certain axioms which capture the abstract properties of the tangent bundle functor from classical differential geometry. Cockett and Cruttwell introduced differential bundles in 2017 as an algebraic alternative to vector bundles in an arbitrary tangent category. In this paper, we prove that differential bundles in the category of smooth manifolds are precisely vector bundles. In particular, this means that we can give a characterisation of vector bundles that exhibits them as models of a tangent categorical essentially algebraic theory.