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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeAug 26th 2017

    I am confused by the Idea section at K-theory. It seems to say that we define an abelian group structure on the set of equivalence classes of objects of a stable (,1)(\infty,1)-category CC such that if axca\to x\to c is a fiber sequence then [x]=[a]+[c][x] = [a] + [c]. But surely the equivalence class of xx is not determined by those of aa and cc and the existence of such a fiber sequence! I always thought that [x]=[a]+[c][x] = [a] + [c] was a relation imposed on the abelian group generated by the equivalence classes of objects.

    This is a little unclear at Grothendieck group of stable infinity-categories too, although Grothendieck group of an abelian category says it as a relation the way I would expect.

    • CommentRowNumber2.
    • CommentAuthorMike Shulman
    • CommentTimeAug 26th 2017

    Anyway, the real questions I wanted to ask that led me to this page today are:

    1. If CC is a symmetric monoidal stable (,1)(\infty,1)-category (with tensor product exact in each variable), then its K-theory should inherit a ring structure. Where is a reference for this? The only nLab page I can find that’s related is K-theory of a bipermutative category, where the additive structure comes from another monoidal structure rather than from fiber sequences.

    2. If I understand correctly, K 0K_0 of a stable (,1)(\infty,1)-category CC is the universal recipient of a certain kind of map from the objects of CC. Is there a standard name for such a map in general? I.e. the map ob(C)K 0(C)ob(C) \to K_0(C) is the universal what? And similarly if CC is symmetric monoidal.

    3. If CC is a symmetric monoidal stable (,1)(\infty,1)-category in which all objects have duals, then the Euler characteristic of a dualizable object is, I believe, one of the maps considered in (2), and therefore factors through the K-theory. If true, this must be well-known; what is a reference?

    • CommentRowNumber3.
    • CommentAuthorDylan Wilson
    • CommentTimeAug 26th 2017
    1. Barwick’s “multiplicative structures on algebraic k-theory” works in the most generality (for Waldhausen infty-categories). Also Blumberg, Gepner, Tabuada if you just want stable infty-cats.
    2. I’ve heard these called “euler characteristic”.
    3. dunno
    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeAug 26th 2017

    Thanks Dylan! Now that you mention it, I think I have heard “Euler characteristic” for (2) also… which makes (3) even more obvious by terminological deduction… (-:

    • CommentRowNumber5.
    • CommentAuthorMarc Hoyois
    • CommentTimeAug 27th 2017

    A reference for 3 is Remark 6.6 in this paper: there is a morphism of E E_\infty-ring spectra from the nonconnective KK-theory of CC to the spectrum of endomorphisms of the unit in CC.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeAug 27th 2017

    Thanks Marc (although that’s a very hifalutin’ version!).

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeAug 27th 2017

    Mike, thanks for the alert, the entry K-theory is waiting for somebody with energy to bring it into better shape. (In the last months I worked on bringing topological K-theory into shape, no time and energy right now for more).

    On the other hand, re-reading the idea section now I don’t quite see which wording needs to be changed. But I am in a rush. Please feel free to edit!

    • CommentRowNumber8.
    • CommentAuthorMike Shulman
    • CommentTimeAug 28th 2017

    Okay, I made the fixes suggested in #1.

    • CommentRowNumber9.
    • CommentAuthorperezl.alonso
    • CommentTimeOct 19th 2023

    Since KO-theory is supposed to be related to the D-brane classification in type I string theory, and K-theory to D-branes in type II string theory, are there any hints that there exist other theories realizing a similar relationship but with quaternions (or even octonions)?

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2023

    So quaternionic K-theory is subsumed in KO, in degrees that are multiples of 4.