Kristine Bauer

Title: The free Lie algebras in Tebbe’s calculation of the derivatives of atomic functors
Abstract: A discrete module is a functor from finite pointed sets to chain complexes of R-modules. There are two ways to do functor calculus for discrete modules. The first is to find the Taylor tower in a way analogous to the Taylor series of functions of a real variable. A second approach is to something more akin to Lagrangian approximation. For functions of a real variable, f, the n-th Lagrangian approximation is the degree n polynomial function which agrees with f on n+1 point. In the case of a discrete module, F, one uses a left Kan extension to produce the best Lagrangian approximation to F. The quotient of successive Lagrangian polynomial functors are called atomic functors.
In her PhD thesis, Amelia Tebbe showed that the n-th derivatives of atomic functors, in the sense of Goodwillie calculus, involve products of free Lie algebras and simple cross effects. The goal of this talk is to present this calculation, and to ask the audience if this looks familiar.