Timeline for K-theory, monoidal vs. exact

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Mar 14, 2018 at 19:44	comment	added	Tim Campion		To clarify: the comparison theorem that Grayson states (bottom of p. 11) says that the $Q$-construction agrees with the plus construction, so at first glance it appears that his comparison between $S^{-1} S$ and $QP$ proceeds indirectly through the plus construction. But it's actually the other way around: Grayson shows that $S^{-1} S = \Omega QP$ directly via the homotopy pullback square at the top of p.11, and separately shows that $S^{-1} S$ agrees with the plus construction via a homology computation. The theorem at the bottom of p. 11 is the concatenation of these results.
Jul 19, 2012 at 10:14	vote	accept	Simon Markett
Jul 18, 2012 at 20:47	comment	added	Dustin Clausen		mathoverflow.net/questions/1006/…
Jul 18, 2012 at 20:47	comment	added	Dustin Clausen		Maybe I should add -- the motivation for taking the group completion of iD is not, like, "well, the group completion of D is trivial, so lets try something else". Actually that's sort of backwards. The operation of group completing iD is exactly the homotopical analog of the usual Grothendieck approach to direct-sum K_0, so it doesn't need any further justification as a model for higher K-theory. On the other hand, the moral reason why the Q-construction models higher K-theory is more delicate -- I'd recommend reading the nice answers at [cont'd]
Jul 18, 2012 at 20:40	history	answered	Dustin Clausen	CC BY-SA 3.0