nLab
Waldhausen S-construction

Idea

The general procedure in K-theory is to assign a K-theory spectrum $K(C)$ to a stable (∞,1)-category $C$.

In practice these stable $(\mathrm{\infty },1)$-categories are usually presented by homotopical categories called Waldhausen categories.

The Waldhausen S-construction on a Waldhausen category $C\prime$ produces a simplicial set equivalent to the K-theory spectrum (see below) of the simplicial localization $C$ of $C\prime$: it is a concrete algorithm for computing K-theory spectra.

Definition

Recall from the definition at K-theory that the K-theory spectrum $K(C)$ of the (∞,1)-category $C$ is the diagonal simplicial set on the bisimplicial set $\mathrm{Core}(\mathrm{Func}({\Delta }^n,C))$ of sequences of morphisms in $C$ and equivalences between these (the core of the Segal space induced by $C$).

The Waldhausen S-construction mimics precisely this: for $C\prime$ a Waldhausen category for every integer $n$ define a simplicial set $S_{n}C\prime$ to be the nerve of the category whose

• objects are sequences $0\hookrightarrow A_{0,1}\hookrightarrow \cdots \hookrightarrow A_{0,n}$ of Waldhausen cofibrations;

  • together with choices of quotients $A_{ij}=A_{0,j}/A_{0,i}$, i.e. cofibration sequences $A_{0,i}\to A_{0,j}\to A_{ij}$

• morphisms are collections of morphisms $\{A_{i,j}\to B_{i,j}\}$ that commute with all diagrams in sight.

Then one finds that the realization of the bisimplicial set $S_{•}C\prime$ with respect to one variable is itself naturally a topological Waldhausen category. Therefore the above construction can be repeated to yield a sequence of topological categories $S_{•}^{(n)}C\prime$. The corresponding sequence of thick topological realizations is a spectrum

$$K(C{)}_n=\mid S_{•}^{(n)}C\prime \mid$$\mathbf{K}(C)_n = |S^{(n)}_\bullet C'|

this is the S-construction of the Waldhausen K-theory spectrum of $C\prime$.

(… roughly at least, need to polish this, see link below meanwhile…)

References

The Waldhausen S-construction is recalled for instance in section 1 of

• Paul D. Mitchener, Symmetric Waldhausen K-theory spectra of topological categories (pdf)