A characterization of differential bundles in tangent categories (Q6617637)

A tangent category is a categorical abstraction of the tangent bundle construction for smooth manifolds and this structure on a category \(\mathbb{X}\) consists of a functor \(T : \mathbb{X}\to \mathbb{X}\), which plays the role of a ``tangent bundle'' construction for objects in \(\mathbb{X}\), along with various natural transformations involving \(T\). This notion is originally due to \textit{J. Rosický} [Diagrammes 12, JR 1--JR 11 (1984; Zbl 0561.18008)], but it was modified and the theory greatly extended by \textit{J. R. B. Cockett} and \textit{G. S. H. Cruttwell} [Appl. Categ. Struct. 22, No. 2, 331--417 (2014; Zbl 1304.18031)]. In that context, Cockett, J.R.B., Cruttwell develop the notion of differential bundle which, by work of \textit{B. MacAdam} [Appl. Categ. Struct. 29, No. 2, 285--310 (2021; Zbl 1467.18021)], generalizes the notion of smooth vector bundle to the abstract setting.\N\NThe paper presents a new description of differential bundles in an arbitrary tangent category and the motivation was the goal of classifying differential bundles in the Goodwillie tangent structure on the \(\infty\)-category of (differentiable) \(\infty\)-categories studied by \textit{K. Bauer} et al. in [``Tangent \(\infty\)-categories and Goodwillie calculus'', Preprint, \url{arxiv:2101.07819}]. The author provides a new characterization of differential bundles and show that, up to isomorphism, a differential bundle is determined by its projection map and zero section. Then, it is shown how these results can be used to quickly identify differential bundles in various tangent categories.