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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 $(\infty,1)$-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.

• CommentRowNumber2.
• CommentAuthorMike Shulman
• CommentTimeAug 26th 2017

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?

• 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 26th 2017

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 $C$.

• CommentRowNumber6.
• CommentAuthorMike Shulman
• CommentTimeAug 26th 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.