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.

中文翻译：

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光滑流形类别中的矢量束和微分束

切线范畴是配备自函子的范畴，该自函子满足某些公理，这些公理从经典微分几何中捕获切丛函子的抽象性质。Cockett 和 Cruttwell 在 2017 年引入了微分丛作为任意切线类别中向量丛的代数替代品。在本文中，我们证明了光滑流形范畴中的微分丛是精确的向量丛。特别是，这意味着我们可以给出向量丛的表征，将它们展示为切线分类本质代数理论的模型。