Tangent infinity-categories and Goodwillie calculus

We make precise the analogy between Goodwillies 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 Rosicky). A tangent structure on an infinity-category X consists of an endofunctor on X, which 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 and that satisfy similar conditions. The tangent bundle functor in Goodwillie calculus is Luries tangent bundle for infinity-categories, introduced to generalize the cotangent complexes of Andre, Quillen and Illusie. We show that Luries 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 on the infinity-category of infinity-categories. 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 Goodwillies 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.