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?
