2025/12/24 by Florian Schwarz, Schwarz, Florian
Computer Science · Mathematics · #(Secondary) #18A25 #18C10 #18D70 (Primary) #53C07 #Algebraic structures and combinatorial models #Category Theory (math.CT) #FOS: Mathematics #Homotopy and Cohomology in Algebraic Topology #Logic, programming, and type systems
paper · doi:10.48550/arxiv.2512.21147
openalex publication_date 2025/12/24 · openalex created_date 2025/12/26 · openalex updated_date 2026/07/28
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 N1 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.