Abstract: I will talk about an ongoing program (based mainly on joint work with Kristine Bauer and Matthew Burke) to develop various notions of differential geometry in the context of homotopy theory. Central to this program is a notion of “tangent category” defined originally by Rosicky, and studied in more detail by Cockett and Cruttwell. That concept uses certain categorical properties of the ordinary tangent bundle functor on the category of smooth manifolds as the basis for an abstract definition. Examples of tangent categories come from a variety of fields including algebraic geometry, commutative algebra, synthetic differential geometry, logic, and computer science.
With Bauer and Burke, we have extended these ideas to define tangent infinity-categories, and we have constructed a fundamental example: in place of smooth manifolds we use (certain kinds of) infinity-categories, and in place of the ordinary smooth tangent bundle we use Lurie’s tangent category construction. One of our main results is that this example encodes Goodwillie’s theory of functor calculus, making precise the analogy between Goodwillie calculus and ordinary differential calculus. Other examples of tangent infinity-categories arise from the theory of infinity-toposes. Future work involves identifying the analogues of vector bundles, connections, and curvature, and other concepts, in the context of these “homotopical” tangent categories.