The category KK of bivariant operator K-theory (or possibly its E-theory variant) ought to be the homotopy category of something at least close to a stable infinity-category; notably in that it carries a by now well-known triangulated category structure.
What seems like a step in the direction of establishing such a stable $\infty$-category structure is in the note
- Michael Joachim, Stephan Stolz, An enrichment of KK-theory over the category of symmetric spectra Münster J. of Math. 2 (2009), 143–182 (pdf)
which produces
-
an enrichment $\mathbb{KK}$ of $KK$ in symmetric spectra, in fact in KU–module spectra;
-
a symmetric monoidal enriched functor $\mathbb{KK} \to \mathrm{KU} Mod$.
(A partial equivariant generalization of this is given by Mitchener in arXiv:0711.2152.)
This prompts some evident questions:
-
Does this enrichment exhibit a presentation of a stable $\infty$-category structure (or close)?
-
How far is that functor from being homotopy full and faithful?
Has anyone thought about this? What can one say?
(I see that Mahanta has a note arXiv:1211.6576 along these lines, but not sure yet if it helps with KK.)
Best Answer
There seems to be a mistake in the construction from "An enrichment..." See http://arxiv.org/pdf/1104.3441v1 page 3. That paper gives an alternative construction of a symmetric spectrum representing (equivariant) K-theory.
Concerning question 2: the induced functor $\mathrm{KK}\to\mathrm{Der}(\mathbf{K})$ is fully faithful and strongly monoidal on the bootstrap class of Rosenberg--Schochet (the localizing subcategory generated by $\mathbb C$). It cannot be fully faithful on all of $\mathrm{KK}$ because there are counterexamples to the Universal Coefficient Theorem in $\mathrm{KK}$ but not in $\mathrm{Der}(\mathbf{K})$.