Waldhausen S-construction

Summary

The S-construction turns a Waldhausen category $\mathcal{C}$ into a simplicial category $S_\bullet \mathcal{C}$, whose $n$-simplices are chains of $n$ cofibrations together with chosen quotients. The K-theory space of $\mathcal{C}$ is the loop space of the realization of the weak equivalences $wS_\bullet \mathcal{C}$, and the K-groups are its homotopy groups. For an exact category with the isomorphisms as weak equivalences, it gives the same K-theory as Quillen's Q-construction. Iterating it gives the K-theory spectrum.