Goodwillie Approximations To Higher Categories

Goodwillie calculus involves the approximation of functors between higher categories by so-called polynomial functors. We show (under mild hypotheses) how to associate to a higher category C a Goodwillie tower, consisting of categories which are polynomial in an appropriate sense. These polynomial approximations enjoy universal properties with respect to polynomial functors out of C. Furthermore, we provide a classification of such Goodwillie towers in terms of the stabilization of C and the derivatives of the identity functor. In special cases this classification becomes very simple, allowing us to draw conclusions about the structure of the category C. As an example we give an application to Quillen's rational homotopy theory. In the sequel to this paper we work out consequences for the study of vn-periodic unstable homotopy theory and the Bousfield-Kuhn functors.