Manifolds, K-theory and the calculus of functors

TL;DR

This paper explores the classification of the Taylor tower of functors from based spaces to spectra using comonad actions and operad module structures, with applications to algebraic K-theory.

Contribution

It establishes conditions under which the comonad action lifts to a module over the Koszul dual of the little L-discs operad, including for Waldhausen's algebraic K-theory.

Findings

The comonad action can be lifted to a module over the Koszul dual of the little L-discs operad.

The Taylor tower of Waldhausen's algebraic K-theory is classified by an action of the Koszul dual of the little 3-discs operad.

Abstract

The Taylor tower of a functor from based spaces to spectra can be classified according to the action of a certain comonad on the collection of derivatives of the functor. We describe various equivalent conditions under which this action can be lifted to the structure of a module over the Koszul dual of the little L-discs operad. In particular, we show that this is the case when the functor is a left Kan extension from a certain category of `pointed framed L-manifolds' and pointed framed embeddings. As an application we prove that the Taylor tower of Waldhausen's algebraic K-theory of spaces functor is classified by an action of the Koszul dual of the little 3-discs operad.