Michael Ching: Tangent ∞-categories and Goodwillie calculus

ABSTRACT :

Goodwillie's calculus of functors is an area of homotopy theory developed in analogy with the ordinary differential calculus. The goal of this talk is to make that analogy more precise by describing a common category-theoretic foundation of which both Goodwillie's theory and the calculus of smooth manifolds are examples.

In the first part of the talk I will set out the theory of tangent categories developed first by Rosický, and then in more detail by Cockett and Cruttwell. The principal example here is the category of manifolds and smooth maps, though there are others coming from, for example, algebraic geometry and synthetic differential geometry. The only prerequisites for this part of the talk are a familiarity with the language of category theory, as well as some very basic knowledge about the tangent bundle of a smooth manifold.

In the second part of the talk, I will explain joint work with Kristine Bauer and Matthew Burke (both at Calgary) to bring Goodwillie calculus into a similar framework. My goal is to explain what the tangent bundle to a category is from this perspective, and how that tangent bundle admits much of the same underlying structure as the tangent bundle of a manifold. The technical foundation of this work is in Lurie's theory of ∞-categories, but the talk will be focused more on the intuition behind our construction, and on possibilities for transferring ideas from classical differential geometry into Goodwillie calculus.