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Whith (Kasparov, bivariant) KK-theory I am left with the nagging feeling that the theory is “more fundamental”, than has been made explicit, that there is a “more profound” universal characterization still to be uncovered.
This feeling is particularly driven by the characterization of certain KK-groups as abelianizations of correspondences of spaces, as recalled here. Since “most” $C^\ast$-algebras arise as topological/smooth groupoid convolution algebras an evident open question here seems to be the following:
Shouldn’t KK-theory have a neat characterization in terms of an abelianization/stabilization of correspondences of differentiable stacks? In particular if we allow at least the correspondence spaces themselves to be more general smooth groupoids, maybe?
Put this way, this seems to suggest another question:
Should KK-theory be thought of as an incarnation in topology/differential geometry of the same general principle which in algebraic geometry produces motivic cohomology?
Because in both cases one builds abelianizations of correspondences of the relevant “spaces”.
Looking around, I see that Grigory Garkusha recently seems to talking about something at least very similar sounding, here, though I still need to really absorb this.
Maybe here is another way to look at what I am after:
for $\mathbf{H}$ a cohesive infinity-topos and $n \in \mathbb{N}$, the (infinity,n)-category of spans $Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$ in $\mathbf{H}$ over the coefficient object for $n$-localized action functionals is – as recalled and discussed at nLab:prequantum field theory – the codomain for (topological) local prequantum field theories
$\exp(i S) : Bord_n \to Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1)) \,.$For $n=2$ we have some results (indicated/announced briefly at the end in the examples-section of Higher geometric prequantum field theory ) that show that the quantization of such a prequantum field theory wants to land in KK-theory, as a “geometric” improvement of the 2-category 2Mod of bare algebras and bare bimodules. In view of the partial characterization of KK-theory in terms of just equivalence classes of precisely such spans above, this makes me wonder:
might the quantization of $\exp(i S)$ be just the postcomposition with a kind of stabilization functor that sends spans/correspondences in $\mathbf{H}$ to their motivic/KK-theoretic abelianization?
For the case of discrete geometry, hence $\mathbf{H} = \infty Grpd \simeq L_{whe} sSet$, this idea or something close is appears in Baez, Hoffnung Walker (for 1-groupoids) and at least roughly also in Freed-Hopkins-Lurie-Teleman (for general $\infty$-groupoids). But I am after the geometric case here:
Doesn’t it look like KK-theory wants to be the answer to “What is the abelianization of spans of smooth groupoids?” ?
What is known? What can one say? What seems likely?
More specifically:
can the idempotent completion of the category of spans of higher stacks over $\mathbf{B}^n U(1)$ be thought of as analogous to the category of Chow motives?
(the fact that we are slicing over $\mathbf{B}^n U(1)$ already equips each correspondence with a cohomology class…)
Once upon a time it seems Jim Dolan suggested that $FinVect$ is something like the idempotent completion of suitably well behaved spans of discrete groupoids. What happened to this claim?
I just see that the above analogy, at least, between KK-theory and motivic cohomology is made also and already in
(very briefly on p. 1, briefly on p. 3).
Howver it seems that note just points out the analogy, without claiming to have more of a substantial relation. But maybe there is.
Urs, Alain Connes was also extensively publishing several years ago with motives in the world of noncommutative differential geometry, including with Matilde Marcolli etc.
Cf. NONCOMMUTATIVE GEOMETRY AND MOTIVES: THE THERMODYNAMICS OF ENDOMOTIVES, pdf with Consani and Marcolli.
Thanks for that pointer! I have added the reference here.
On the other hand, that is still just amplifying the analogy. The article that you refer to calls Fredholm-Hilbert bimodules as used in KK-theory “correspondences” because, yes, they clearly seem to play the same role as the actual correspondences in Chow motives.
But did anyone (maybe Connes later?) go further? We know for some special cases (such as between C*-algebras of actual functions on actual manifolds) that KK-classes are faithfully represented by actual geometric correspondence (as in the references here). But generally? Did anyone try to generalize this to convolution algebras of more general groupoids?
More precisely, what i am looking for is this: consider the category whose objects are smooth stacks, and whose morphisms from $X$ to $Y$ are diagrams of smooth stacks of the form
$\array{ && A \\ & \swarrow && \searrow \\ X && \swArrow_\phi && Y \\ & \searrow && \swarrow \\ && \flat \mathbf{B}^2U(1) }$equipped with the hopefully obvious composition law. The question I am asking is: can we apply some natural abelialization and/or completion operation to this category such as to represent KK-theory and/or a geometric analog of Chow motive theory?
Did Connes in his “extensive publishing” on this topic, as you say, ever consider this for $X$, $Y$ general Lie groupoids (instead of just smooth manifolds)? Or something in this direction?
By the way, if my little conjecture here is correct, that quantization of a local prequantum field theory (as discussed there)
$\exp(i S ) \colon Bord_n \to Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1))$is given just by postcomposition with a motive-style completion
$Span_n(\mathbf{H}, \flat \mathbf{B}^n U(1)) \to Motives_n(\mathbf{H})$then this would seem to be naturally suggestive of why one finds motivic structure in quantum field theory, such as motivic multiple zeta values in scattering amplitudes and the cosmic Galois group in renormalization.
So, how about the work of Arne Østvær ? He mixed general cubic sets with operator algebras to get a model category setup for operator algebras where KK-groups appear as one of the features. Groupoids should be special case of cubic sets…(and spaces of operator algebras) so it should be easy to accomodate…
So, how about the work of Arne Østvær ?
So I am looking at his text. He considers $\infty$-presheaves on noncommutative topological spaces (= $C^\ast Alg^{op}$) localized to satisfy some exactness property (he considers models both by cubical and by simplicial co-presheaves on $C^\ast Alg$). Then on the bottom of p. 71, in theorem 3.102 he recovers KK-groups as a certain derived hom between these localized $\infty$-presheaves.
This is nice, but It is well known (since Higson 87) that KK-theory (E-theory) is the additive split exact (exact) localization of $C^\ast Alg$ at the compact operators.
Maybe one can read Østvær’s description as a step towards understanding KK-theory in higher noncommutative topology. But what I am after here is different, I am talking about representing KK-theory by “geometric cocycles” given by spans of smooth groupoids (and, yes, then I want to pass to smooth higher groupoids).
Do you think Østvær’s work has any bearing on that? If so, please point me more precisely to a specific page and/or a specific theorem of his.
No, his work is basically just moding out by basic relations like the localization you mention. I thought that you want to have KK-group enrichement between stack-like objects, not that you want to use usual spans to get KK-groups themselves. Sorry if I mislead you.
Yes, I want spans to get the KK-groups. As in Chow motives.
The idea is that KK is just a completion of the form
$Span_1(SmoothGrp, \flat \mathbf{B}^2 U(1)) \to SmoothMotives_1(SmoothGrpd) \,.$I just heard a nice talk by Mahanta on how to regard KK-theory as noncommutative motives in a precise way. While the nLab is down, let me record this here:
In
it is shown that there is a universal functor $KK \longrightarrow NCC_{dg}$ from KK-theory to the category of noncommutative motives, which is the category of dg-categories and dg-profunctors up to homotopy between them. This is given by sending a C*-algebra to the dg-category of perfect complexes of (the unitalization of) its underlying associative algebra.
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