# Definition of algebraic K-theory space

Let $(C,wC)$ be a Waldhausen category. The algebraic K-theory space is the loop space of the classifying space of the simplicial pointed category $wS_*C$, i.e. of the topological realization of the bisimplicial set $N_*wS_*C$.

Is there a specific reason we want to consider $wS_*C$ instead of $S_*C$?

Moreover, why we want to consider the loop space instead of classifying space of the simplicial pointed category $wS_*C$ directly?

If you ignore w and consider just $S_* C$, that's equivalent to considering $w S_* C$, where $w$ is defined to consist of all the isomorphisms. The notation for that family $w$ is usually $i$.