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?