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added to KK-theory brief remark and reference to relation to stable $\infty$-categories / triangulated categories
I had added some more references at KK-theory.
(I also expanded the Idea-section a little, but it’s still stubby and needs attention.)
Have been further adding bits and pieces to KK-theory. But still not really coherent.
:-) should KK-theory be a double letter theory now that we have letter theory?!
I think of KK theory rather as a bicycle theory.
$G$-theory mentioned in letter theory is in fact classical – I think it is the $K$-theory of the category of coherent sheaves over a scheme (there are two standard kinds of algebraic $K$-theories – one for locally free and another for coherent sheaves). I do not see the point to make the letter theories which do not exist from the list into the question mark links, it is misleading.
Which letter theories provably don’t exist?
You just added one more, G-theory. But, yeah, it’s an intentionally whimsical entry. I thought that’s clear enough.
But, yeah, it’s an intentionally whimsical entry.
Still, my point is why not listing nonexisting letter theories WITHOUT CREATING link. You know, false links clutter the list of all wanted entries (nonexisting entries which we seriously intend to create in future) with trash. (I know that the call to all entries list is now temporarily disabled but the lists for given category do list referred nonexisting entries)
Sure. No objection.
Spanier wrote about S-theory. Duality and S-theory. I guess it didn’t take off.
Thanks. Anyway, worthwhile to record: S-theory.
I have further worked on KK-theory and now it is beginning to look like a coherent entry, I think.
There is now the standard definition pretty much in full detail.
Then there is a section stating the universal characterization of KK as the additive, split-exact homotopy-invariant localization of C’Alg at the compact operators.
Also added a bunch more commented references.
Still just a start, but at least now it’s a decent start, I think.
Added a brief remark section Relation to operator K-cohomology, K-homology, twisted K-theory and a pointer to the article by Tu+Xu+LaurentGengoux on KK-theory of differentiable stacks. Also a smilar paragraph at groupoid convolution algebra.
I have re-organized and expanded the Idea-section at KK-theory.
added further commented references to KK-theory. Also added a section with some remarks on excision and the relation to E-theory.
I have added some pointers to presentatiosn of KK-groups in terms of spans/correspondences of spaces (“topological presentation”), in
Does anyone have more on this topic? It is tempting to wonder if this is pointing to a more general and more abstract statement.
Notably if we allow spans/correspondences of differentiable stacks and more generally allow the correspondence space to be a smooth groupoid, how far can we get with modelling KK-classes between the corresponding convolution algebras?
And once we talk about a kind of quotient on something like the (infinity,1)-category of spans $Span_1(Smooth \infty Grpd)$ in smooth infinity-groupoids in the first place, what might be a (more) elegant universal characterization?
It is tempting to think of the stabilization $Span_1(Smooth \infty Grpd) \to Stab(Span_1(Smooth \infty Grpd))$, but I don’t know.
Hm, now that I said this, it made me think that KK-theory should be closely related to motivic cohomology:
in both cases one considers an abelianization/stabilization of correspondences of the relevant spaces.
Hm. Looking around, I see that Grigory Garkusha has been writing about relating/identifying flavors of motivic cohomology and (algebraic) KK-theory, here.
That rings bells. Remembering that E-theory is not that distantly related to (strong) shape theory, and there is the Lurie form of Strong Shape theory, perhaps the place to look would be in the C*-algebra versions of strong shape but in a smooth setting. (I do not understand the motivic stuff enough to be more certain.)
Hmm, I see Garkusha has been working with Mike Prest who works in model theory. The latter appeared in the model theory/category theory thread, and is someone trying to develop a functorial model theory.
On that line (but a bit off thread) see here. The group in WIMCS working on this area is very active.
Meyer’s arXiv preprint Categorical aspects of bivariant K-theory (arXiv:math/0702145) (quoted in KK-theory) has in fact being published in K-theory and noncommutative geometry, pp. 1–40, edited by Guillermo Cortiñas, Joachim Cuntz, Max Karoubi, Ryszard Nest, Charles A. Weibel, Eur. Math. Soc, 2008
Yes, Urs, Grigory Garkusha has been working on a variant of motivic theory, so called K-motives; in addition to use of his own work, there is an ongoing collaboration he has with Ivan Panin, a strong algebraic geometer from Sankt Petersburg. Grigory asked me if I can help on a certain technical issue related to Barrat-Eccles operad, lifting certain tensor product to modules over that operad which are in the same time modules in the sense of certain spectral category (i.e. categories enriched over ring spectra). The tensor product structure is almost obvious, but up to certain natural isomorphisms, which seem to be a sort of $(\infty,1)$-distributive law. Work of Elmendorf and Mandell on permutative categories seem to touch a version of the same abstract problem in terms of a different version of the $E_\infty$-operad and mainly dwelling on $E_4$-truncation. I worked a bit on that problem in April 2012 but not enough afterwards to finish it; the question (in somewhat simplified form) is posted here: pdf and (as far as I remember the motivation) its solution would be a step in establishing a crucial property of an important spectral sequence which would be useful to attack a major open conjecture in algebraic geometry (on which I am not an expert). I hope to return to that problem soon (Grigory would also like to hear from anybody who has advices toward an effective solution to this problem (as it is motivated by computation, one needs to have explicit recipes).)
added some basic Examples
(had more in mind, but am being interrupted now…)
earlier today I had added to the references at KK-theory the note
This produces an enrichment $\mathbb{KK}$ of the KK category in $KU$-module spectra and a symmetric monoidal enriched functor
$\mathbb{KK} \to KU Mod$sending a $C^\ast$-algebra to its operator K-theory spectrum.
This induces some evident questions:
Does this spectrum-enrichment $\mathbb{KK}$ present a stable $\infty$-category structure?
To which extent is the functor $\mathbb{KK} \to KU Mod$ full and/or faithful?
If anyone has any further insights on this, I’d be most grateful for a hint.
I started to add to the section Triangulated and spectrum enriched strucure some of the infor provided on MO. Not done yet, but have to interrupt now.
a commenter on MO kindly points out that the old question of realizing KK-theory as the homotopy category of a stable $\infty$-category seems to finally have been answered:
(now Bunke, Engel & Land 2021 in the entry)
pointer
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