Functor Calculus and Operads

Speaker: Gregory Arone (Virginia) Title: Part 1: operads, modules and the chain rule Abstract: Let F be a homotopy functor between the categories of pointed topological spaces or spectra. By the work of Goodwillie, the derivatives of F form a symmetric sequence of spectra ∂∗F . This symmetric sequence determines the homogeneous layers in the Taylor tower of F , but not the extensions in the tower. In these two talks we will explore the following question: what natural structure does ∂∗F possess, beyond being a symmetric sequence? Our ultimate goal is to describe a structure that is sufficient to recover the Taylor tower of F from the derivatives. Such a description could be considered an extension of Goodwillie’s classification of homogeneous functors to a classification of Taylor towers. By a theorem of Ching, the derivatives of the identity functor form an operad. In the first talk we will see that the derivatives of a general functor form a bimodule (or a right/left module, depending on the source and target categories of the functor) over this operad. Koszul duality for operads plays an interesting role in the proof. As an application we will show that the module structure on derivatives is exactly what one needs to write down a chain rule for the calculus of functors. Title: Part 2: beyond the module structure Abstract: The (bi)module structure on derivatives is not sufficient to recover the Taylor tower of a functor. In the second talk we will refine the structure as follows. We will see that there is a naturally defined comonad