Michael Ching: Taylor towers and algebraic K-theory

Abstract: The Taylor tower of a functor (in the sense of Goodwillie's homotopy calculus) can be encoded via its symmetric sequence of  derivatives together with the action of a certain comonad on that symmetric sequence. The form of the comonad depends on the source and target categories of the functor. In the case of functors from based spaces to spectra the relevant comonad can be explicitly stated in terms of the little disc operads. I will describe the resulting action on the derivatives of Waldhausen's algebraic K-theory of spaces functor. This is joint work with Greg Arone.
Algebraic K-theory can also be viewed as a functor from (associative) ring spectra to spectra and there is a comonad for this situation too. I will explain preliminary attempts to understand the action on the derivatives in this case. This is based on work of Lindenstrauss and McCarthy on the Taylor tower for the algebraic K-theory of tensor algebras.