Title: Monads, nilpotency and Goodwillie calculus
Speaker: Michael Ching, Amherst
Abstract: Goodwillie's homotopy calculus provides a systematic sequence of "polynomial" approximations (which together are referred to as the Taylor tower) to suitable functors in homotopy theory. Of particular interest is the identity functor (on the category of based topological spaces) which, it turns out, has an interesting calculus. While not polynomial of any degree, the identity functor is "analytic" and its Taylor tower converges on connected nilpotent spaces (those with a nilpotent fundamental group acting nilpotently on the higher homotopy groups).
In this talk, I want to describe how the nth polynomial approximation to the identity functor can be given the structure of a monad. Algebras over this monad can be thought of as "n-nilpotent" spaces - in particular they are nilpotent in the above sense. I will also discuss analogous results for the identity functor on other categories. In the case of algebras over an operad (of spectra), the monad structures on the polynomial approximations to the identity are derived from work of Harper and Hess.
