Linearized Waldhausen $K$-Theory

Question

In Waldhausen's foundational $A(X)$ paper (in Springer LNM 1126) there are some brief remarks on p. 400 about how to define the "linearized" $K$-theory of a space $X$ using abelian group objects in the category of spaces over $X$ rather than abelian group objects in the category of $G$-spaces.

I need to understand these remarks, and would be happy to hear from anybody who already understands them.

Answer 1

Let me make some guesses, although I worry I'll only say some formal things you already know (or that are wrong).

One description of the Waldhausen K-theory of a pointed connected space $X$ is that it is the algebraic K-theory of the group algebra $\mathbb{S}[\Omega X]$ of its loop space over the sphere spectrum. Module spectra over $\mathbb{S}[\Omega X]$ have the following equivalent descriptions:

• Representations of $\Omega X$ on spectra.
• Families of spectra over $X$ (by a kind of Koszul duality).

What Waldhausen seems to mean by "linearization" is passing from the sphere spectrum to $\mathbb{Z}$, by which I mean $\mathbb{Z}$ the Eilenberg-MacLane spectrum. Now we want to look at the algebraic K-theory of the group algebra $\mathbb{Z}[\Omega X]$. A more familiar avatar of this, passing through stable Dold-Kan, is chains $C_{\bullet}(\Omega X)$. Module spectra over this group algebra have the following equivalent descriptions:

• Representations of $\Omega X$ on $\mathbb{Z}$-module spectra.
• Representations of $\Omega X$ on chain complexes over $\mathbb{Z}$ (by stable Dold-Kan).
• Under a connectivity hypothesis, representations of $\Omega X$ on simplicial abelian groups (by ordinary Dold-Kan).
• Under a connectivity hypothesis, representations of $\Omega X$ on topological abelian groups (by a suitably monoidal version of the equivalence between simplicial sets and topological spaces).
• Under a connectivity hypothesis, families of topological abelian groups over $X$, or abelian groups in families of spaces over $X$ (by Koszul duality again).