1
$\begingroup$

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?

$\endgroup$
3
$\begingroup$

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 $i$.

$\endgroup$
8
  • $\begingroup$ Can you explain the equivalence? $\endgroup$ – 6666 Sep 17 '18 at 4:19
  • $\begingroup$ It's part 2 of the corollary after lemma 1.4.1 of Waldhausen's paper "Algebraic K-theory of Spaces". $\endgroup$ – Dan Grayson Sep 17 '18 at 15:12
  • $\begingroup$ 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. $\endgroup$ – Dan Grayson Sep 17 '18 at 15:15
  • $\begingroup$ For further intuition, see lemma 1.6.5, the swallowing lemma. $\endgroup$ – Dan Grayson Sep 17 '18 at 15:16
  • $\begingroup$ 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. $\endgroup$ – Dan Grayson Sep 17 '18 at 15:19

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.