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1. • CommentRowNumber1.
   • CommentAuthorMike Shulman
   • CommentTimeAug 26th 2017
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   Author: Mike Shulman
   Format: MarkdownItexI 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 $(\infty,1)$-category $C$ such that if $a\to x\to c$ is a fiber sequence then $[x] = [a] + [c]$. But surely the equivalence class of $x$ is not determined by those of $a$ and $c$ and the existence of such a fiber sequence! I always thought that $[x] = [a] + [c]$ 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](https://ncatlab.org/nlab/show/Grothendieck+group#OfStableCategories) too, although [Grothendieck group of an abelian category](https://ncatlab.org/nlab/show/Grothendieck+group#of_an_abelian_category) says it as a relation the way I would expect.

   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 $(\infty,1)$-category $C$ such that if $a\to x\to c$ is a fiber sequence then $[x] = [a] + [c]$. But surely the equivalence class of $x$ is not determined by those of $a$ and $c$ and the existence of such a fiber sequence! I always thought that $[x] = [a] + [c]$ 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.

2. • CommentRowNumber2.
   • CommentAuthorMike Shulman
   • CommentTimeAug 26th 2017
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   Author: Mike Shulman
   Format: MarkdownItexAnyway, the real questions I wanted to ask that led me to this page today are: 1. If $C$ is a symmetric monoidal stable $(\infty,1)$-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, $K_0$ of a stable $(\infty,1)$-category $C$ is the universal recipient of a certain kind of map from the objects of $C$. Is there a standard name for such a map in general? I.e. the map $ob(C) \to K_0(C)$ is the universal what? And similarly if $C$ is symmetric monoidal. 3. If $C$ is a symmetric monoidal stable $(\infty,1)$-category in which all objects have duals, then the [Euler characteristic of a dualizable object](https://ncatlab.org/nlab/show/Euler+characteristic#of_an_object_in_a_symmetric_monoidal_category) 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?

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

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

   3. If $C$ is a symmetric monoidal stable $(\infty,1)$-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?

3. • CommentRowNumber3.
   • CommentAuthorDylan Wilson
   • CommentTimeAug 26th 2017
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   Author: Dylan Wilson
   Format: MarkdownItex1. 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

   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
4. • CommentRowNumber4.
   • CommentAuthorMike Shulman
   • CommentTimeAug 26th 2017
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   Author: Mike Shulman
   Format: MarkdownItexThanks 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... (-:

   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… (-:

5. • CommentRowNumber5.
   • CommentAuthorMarc Hoyois
   • CommentTimeAug 27th 2017
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   Author: Marc Hoyois
   Format: MarkdownItexA reference for 3 is Remark 6.6 in [this paper](https://arxiv.org/pdf/1511.03589.pdf): there is a morphism of $E_\infty$-ring spectra from the nonconnective $K$-theory of $C$ to the spectrum of endomorphisms of the unit in $C$.

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

6. • CommentRowNumber6.
   • CommentAuthorMike Shulman
   • CommentTimeAug 27th 2017
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   Author: Mike Shulman
   Format: MarkdownItexThanks Marc (although that's a very hifalutin' version!).

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

7. • CommentRowNumber7.
   • CommentAuthorUrs
   • CommentTimeAug 27th 2017
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   Author: Urs
   Format: MarkdownItexMike, 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!

   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!

8. • CommentRowNumber8.
   • CommentAuthorMike Shulman
   • CommentTimeAug 28th 2017
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   Author: Mike Shulman
   Format: MarkdownItexOkay, I made the fixes suggested in #1.

   Okay, I made the fixes suggested in #1.

9. • CommentRowNumber9.
   • CommentAuthorperezl.alonso
   • CommentTimeOct 19th 2023
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   Author: perezl.alonso
   Format: MarkdownItexSince [[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)?

   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)?

10. • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeOct 20th 2023
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    Author: Urs
    Format: MarkdownItexSo [[quaternionic K-theory]] is subsumed in KO, in degrees that are multiples of 4.

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