# Infinity-categories and Day convolution in Goodwillie calculus

Research paper by Michael Ching

Indexed on: 10 Jan '18Published on: 10 Jan '18Published in: arXiv - Mathematics - Algebraic Topology

#### Abstract

We prove two theorems about Goodwillie calculus and use those theorems to describe new models for Goodwillie derivatives of functors between pointed compactly-generated infinity-categories. The first theorem say that the construction of higher derivatives for spectrum-valued functors is a Day convolution of copies of the first derivative construction. The second theorem says that the derivatives of any functor can be realized as natural transformation objects for derivatives of spectrum-valued functors. Together these results allow us to construct an infinity-operad that models the derivatives of the identity functor on any pointed compactly-generated infinity-category. The derivatives of a functor between such infinity-categories then form a bimodule over the relevant infinity-operads.