# 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?

## 1 Answer

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 $i$.

• Can you explain the equivalence? – 6666 Sep 17 '18 at 4:19
• It's part 2 of the corollary after lemma 1.4.1 of Waldhausen's paper "Algebraic K-theory of Spaces". – Dan Grayson Sep 17 '18 at 15:12
• Intuitively, the reason for the equivalence is that filtrations can "swallow" isomorphisms. I.e., if A_0 -> ... -> A_n is a chain of composable monomorphisms, and for some i, A_i -> A'_i is an isomorphism, then you can insert A'_i into the chain to get a chain whose length is 1 more. Such chains of length 1 more play a role in simplicial homotopies. – Dan Grayson Sep 17 '18 at 15:15
• For further intuition, see lemma 1.6.5, the swallowing lemma. – Dan Grayson Sep 17 '18 at 15:16
• PS: Another answer has been deleted, but it contained something worthwhile: if you don't use w, then you aren't studying what you want to study. – Dan Grayson Sep 17 '18 at 15:19