Calculus III: Taylor Series

Abstract

We study functors from spaces to spaces or spectra that preserve weak homotopy equivalences. For each such functor we construct a universal $n$–excisive approximation, which may be thought of as its $n$–excisive part. Homogeneous functors, meaning $n$–excisive functors with trivial $\left(n-1\right)$–excisive part, can be classified: they correspond to symmetric functors of $n$ variables that are reduced and $1$–excisive in each variable. We discuss some important examples, including the identity functor and Waldhausen’s algebraic $K$–theory.