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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.
Anyway, the real questions I wanted to ask that led me to this page today are:
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.
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.
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?
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… (-:
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 $C$.
Thanks Marc (although that’s a very hifalutin’ version!).
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!
Okay, I made the fixes suggested in #1.
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