Thomas Blom: A generalization of the Arone-Ching chain rule

Abstract.

The chain rule of Arone-Ching is a celebrated result in Goodwillie calculus and can be seen as a categorification of the chain rule from ordinary calculus. Given two functors from the category of spaces or spectra to itself, this chain rule describes how one can reconstruct the derivatives of the composite from the derivatives of the individual functors.

In this talk I will describe ongoing joint work with Max Blans, where we aim to give an alternative proof of the chain rule of Arone-Ching. This new approach has the advantage of working in much greater generality, where the category of spaces or spectra can be replaced by any suitable ∞-category. Moreover, we give a new construction of the derivatives that is lax monoidal. As a corollary, one obtains that the derivatives of any monad can be given the structure of an operad, a result that has long been believed to be true.

In the first half, I will give a brief introduction to Goodwillie calculus and motivate why such a generalization is desirable. In particular, I will discuss some motivation coming from the theory of operads. In the second half, I will explain the main ideas that go into our proof and end with a discussion of some open problems.