Assembly

Summary

Abstract. The goal of assembly is to approximate homotopy invariant functors from spaces to spectra by homotopy invariant and excisive functors from spaces to spectra. We show that there exists a best approximation, characterized by a universal property.

The Ordinary Assembly Map

We adopt a very category theoretic point of view in describing assembly maps. It has been formulated explicitly by Quinn in the appendix to [Q], and more implicitly in Quinn's thesis, in [QGF], in [And] and in articles of Waldhausen, e.g. [Wai], [Wa2]. See also [QAB]. Prom this point of view, the goal of assembly is: Given a homotopy invariant functor F from spaces to spectra, to approximate F from the left by an excisive homotopy invariant functor F%.

In this section, all spaces are homotopy equivalent to CW–spaces, all pairs of spaces are homotopy equivalent to CW–pairs, and all spectra are CW –spectra.

A functor F from spaces to spectra is homotopy invariant if it takes homotopy equivalences to homotopy equivalences. A homotopy invariant F is excisive if F(θ) is contractible and if F preserves homotopy pushout squares (alias homotopy cocartesian squares, see [Gol], [Go2]). The excision condition implies that F preserves finite coproducts, up to homotopy equivalence. Call F strongly excisive if it preserves arbitrary coproducts, up to homotopy equivalence.

If F is strongly excisive, then the functor π*F from spaces to graded abelian groups is a generalized homology theory—it has Mayer–Vietoris sequences, and satisfies the strong wedge axiom.