Tangent ∞-categories and Goodwillie calculus In 1984, Rosický introduced a notion of tangent category to capture the essential categorical properties of the tangent bundle functor on the category of smooth manifolds. In recent years, Cockett and Cruttwell have considerably expanded the theory, introducing abstract analogues of vector bundles, connections, and other ideas from differential geometry. As well as the prototypical case of smooth manifolds, examples of tangent categories arise from a range of subjects, including algebra, algebraic geometry, logic, and computer science. I'll talk about joint work with Kristine Bauer and Matthew Burke that extends Rosický's definition to ∞-categories, allowing for various new examples, such as derived smooth manifolds, E-infinity ring spectra, and ∞-topoi. Our main motivation was the construction of a tangent ∞-category which encodes Goodwillie's calculus of functors, and I'll explain how Tom's notions of n-excisive functor and Taylor tower arise naturally from the theory. If there's time, I'll also mention joint work with Kaya Arro, in which we identify the analogues of vector bundles in this context.

---

Last modified: Wed Apr 8 21:55:55 EDT 2026