Max Blans: On the chain rule in Goodwillie calculus

Abstract.

Goodwillie calculus provides a way to approximate a functor by a tower of polynomial functors. The layers of this tower are controlled by stable objects, called the derivatives of the functor. Arone and Ching proved a chain rule which expresses the derivatives of a composite of two functors between pointed spaces or spectra in terms of the derivatives of the individual functors. The subject of this talk is recent joint work with Thomas Blom, in which we prove that the chain rule extends to a large class of presentable infinity-categories, confirming a conjecture of Lurie. Along the way I will explain how taking Goodwillie derivatives can be refined to a lax functor of \((\infty, 2)\)-categories.