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nLab > Latest Changes: K-theory of a symmetric monoidal infinity-category

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2014
    • (edited Apr 28th 2014)

    created a brief entry K-theory of a symmetric monoidal (∞,1)-category.

    In the course of this I have also split off a brief entry ∞-group completion from Grothendieck group and did some other cross-linking.

    (The collection of entries on algebraic K-theory and its variants that we have would deserve a serious clean-up….)

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 28th 2014

    In terms of generality, what’s the relation between this page, ordinary K-theory and algebraic K-theory? You say that the construction on this page

    …subsumes various other construction in algebraic K-theory.

    Does it subsume ordinary K-theory?

    And is there a nice abstract account of the differential case?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeApr 28th 2014
    • (edited Apr 28th 2014)

    Yes!

    That’s the content of BNV13, section 4.4, now briefly summarized at differential cohomology diagram – Smooth vector bundles and the e-invariant.

    So 𝒦\mathcal{K} applied to geometrically discrete symmetric monoidal ∞\infty-groupoids gives various instances of “algebraic” K-theory. The magic now is: applied to the smooth ∞\infty-groupoid Vect\mathbf{Vect} of complex vector bundles, then it produces a smooth spectrum whose shape is topological K-theory.

    In addition, its flat part is algebraic K-theory of the complex numbers. So the smooth spectrum 𝒦(Vect)\mathcal{K}(\mathbf{Vect}) “smoothly” interpolates between algebraic K-theory and the topological K-theory. The interpolation map is the points-to-pieces transform ♭→Π\flat \to \Pi of cohesion.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 29th 2014

    Impressive! Are there other good candidates to put there in the place of Vect\mathbf{Vect}?

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeApr 29th 2014
    • (edited Apr 29th 2014)

    Yes!

    We were all (Thomas and I at least, of course Thomas was the one who actually made it work!) very much motivated by finding the smooth version of Snaith’s theorem (me for the reasons discussed at motivic quantization: Snaith’s theorem may be read as exhibiting a “linearization and superposition principle” that is a higher analog of the one embodied by the complex numbers in quantum mechanics).

    So with BU(1) conn\mathbf{B}U(1)_{conn} the smooth 2-group of circle bundles with connection, write 𝕊[BU(1) conn]\mathbb{S}[\mathbf{B}U(1)_{conn}] for its smooth infinity-group E-infinity-ring (and the achievement here is to say what it means to do this smoothly) and write

    𝕊[BU(1) conn][b −1]∈Stab(Smooth∞Grpd) \mathbb{S}[\mathbf{B}U(1)_{conn}][b^{-1}] \in Stab(Smooth \infty Grpd)

    for the localization of that smooth E ∞E_\infty-ring at “the Bott element”. Then this is again differential refinement of KU to smooth spectra.

    This is BNV13, prop. 6.25. In a little moment I’ll have a survey of this added here.

    • CommentRowNumber6.
    • CommentAuthoradeelkh
    • CommentTimeApr 30th 2014

    What’s the relation between this and the work of Clark Barwick?

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeApr 30th 2014

    That’s a good question. I need to dig into that a bit more…

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014

    re #6:

    so apparently the relation is that Quillen’s Q, Waldhausen’s S, and then finally Barwick’s construction of K-theory of stable ∞\infty-categories factors through the K-theory of symmetric monoidal ∞\infty-catgeories by the ∞\infty-functor that universally splits exact sequences.

    As soon as I have more of a precise and citable statement here, I’ll add it to the entry.

    • CommentRowNumber9.
    • CommentAuthoradeelkh
    • CommentTimeMay 29th 2014
    • (edited May 29th 2014)

    Thanks. There is also the article

    where I believe they present the construction of the Bunke-Tamme paper in full generality. They also prove that algebraic K-theory, as a functor on the infinity-category of symmetric monoidal infinity-categories, is lax symmetric monoidal. I will add the reference (but I don’t feel confident enough on the topic to add any remarks about it in the text).

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeMay 29th 2014
    • (edited May 29th 2014)

    Thanks, yes, good point.

    Regarding my statement in #8: so it seems I won’t be able to provide a citation, what I have now is the word of somebody who should know and who says that he and somebody else who should really know did convince themselves of this. :-)

    By the way here on MO you see Tyler Lawson make almost the statement in #8, saying that the basic construction is the ∞\infty-group completion of monoidal structure and that the Quillen/Waldhausen/Barwick kind of construction are “souped up” versions of that, the souping-up being splitting of exact sequences. On the other hand, further down the comments (currently the last comment) you see that this important point is missed again.

    • CommentRowNumber11.
    • CommentAuthoradeelkh
    • CommentTimeJul 17th 2014
    • (edited Jul 17th 2014)

    Probably a stupid question: Given a scheme X, one can consider Perf(X) (perfect complexes) as a symmetric monoidal infinity-category, or just a stable infinity-category. Hence one can define the K-theory via the group completion of the associated E-infinity-monoid, or by applying the Waldhauden S-construction. I gather that the second definition gives the correct thing, as does the first in the affine case. Is it true that the first definition is wrong for non-affine schemes?

    • CommentRowNumber12.
    • CommentAuthoradeelkh
    • CommentTimeNov 24th 2014

    This is just to point out the new paper

    which considers an action of S^1 on the infinity-category of stable infinity-categories, defined somehow using Bott periodicity and the complex J-homomorphism, and shows that the functor defined by K-theory is invariant under this action.

    • CommentRowNumber13.
    • CommentAuthorUrs
    • CommentTimeNov 24th 2014

    Thanks. Haven’t looked at it yet (am on a slow connection not opening that pdf) so here just a question into the blue: is this S 1S^1 by any chance the SO(2)SO(2) that acts on dualizable objects in a symmetric monoidal (∞,2)(\infty,2)-category, such as, I imagine, would be formed by stable ∞\infty-categories?

    • CommentRowNumber14.
    • CommentAuthorDavidRoberts
    • CommentTimeNov 26th 2014

    @Urs #13: probably not, given the comment that David C highlighted on G+, namely that it is tricky to reconcile the different constructions: geometric and combinatorial. But one could hope that it turns out to be the same thing.

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