Differential bundles in Goodwillie calculus

In joint work with Kristine Bauer and Matthew Burke, we have formalized the analogy between Goodwillie's functor calculus and the ordinary differential calculus of smooth manifolds. Both are examples of a tangent ∞-category. That abstract structure, introduced by Rosický in the 1-categorical setting and developed in more detail by Cockett and Cruttwell, is based on modest categorical properties of the tangent bundle functor in differential geometry. I will describe how we generalize their work to ∞-categories and how Goodwillie's theory fits into this framework. Our main result is that Goodwillie's Taylor tower can be completely recovered from a certain tangent ∞-category of ∞-categories.

This work makes it possible to investigate analogues in Goodwillie calculus of various aspects of ordinary differential geometry, and I will describe work in progress with Kaya Arro, where we identify the notions corresponding to smooth vector bundles. These turn out to be, more-or-less, cartesian and cocartesian fibrations of ∞-categories for which all the fibres are stable.