Tangent $\infty$-categories and Goodwillie calculus

Abstract

We make precise the analogy between Goodwillie's calculus of functors in homotopy theory and the differential calculus of smooth manifolds by introducing a higher-categorical framework of which both theories are examples. That framework is an extension to infinity-categories of the tangent categories of Cockett and Cruttwell (introduced originally by Rosick\'y). The basic data of a tangent infinity-category consist of an endofunctor, that plays the role of the tangent bundle construction, together with various natural transformations that mimic structure possessed by the ordinary tangent bundles of smooth manifolds. The role of the tangent bundle functor in Goodwillie calculus is played by Lurie's tangent bundle for infinity-categories, introduced to generalize the cotangent complexes of Andr\'e, Quillen and Illusie. We show that Lurie's construction admits the additional structure maps and satisfies the conditions needed to form a tangent infinity-category which we refer to as the Goodwillie tangent structure. Cockett and Cruttwell (and others) have started to develop various aspects of differential geometry in the abstract context of tangent categories, and we begin to apply those ideas to Goodwillie calculus. For example, we show that the role of Euclidean spaces in the calculus of manifolds is played in Goodwillie calculus by the stable infinity-categories. We also show that Goodwillie's n-excisive functors are the direct analogues of n-jets of smooth maps between manifolds; to state that connection precisely, we develop a notion of tangent (infinity,2)-category and show that Goodwillie calculus is best understood in that context.