Why do my quantum group books avoid homotopical language?

Question

I am sitting on my carpet surrounded by books about quantum groups, and the only categorical concept they discuss are the representation categories of quantum groups.

Many notes closer to "Kontsevich stuff" discuss matters in a far more categorical/homotopical way, but they seem not to wish to touch the topic of quantum groups very much....

I know next to nothing about this matter, but it seems tempting to believe one could maybe also phrase the first chapters of the quantum group books in a language, where for example quasitriangulated qHopf algebras would just be particular cases of a general "coweak ${Hopf_\infty}$-algebra" (does such thing exist?). Probably then there should be some (${\infty}$,1) category around the corner and maybe some other person's inofficial online notes trying to rework such a picture in a dg Hopf algebra model category picture.

My quantum group books don't mention anything in such a direction (in fact ahead of a chapter on braided tensor categories the word 'category' does kind of not really appear at all). So either

1. [ ] I am missing a key point and just proved my stupidity to the public
2. [ ] There is a quantum group book that I have missed, namely ......
3. [ ] Such a picture is boring and/or wrong for the following reason .....

Which is the appropriate box to tick?

Answer 1

The question is rather rambling and it is more about not so well-defined appetites (do you have a more conrete motivation?).

There is one thing which however makes full sense and deserves the consideration. Namely it has been asked what about higher categorical analogues of (noncommutative noncocommutative) Hopf algebras. This is not a trivial subject, because it is easier to do resolutions of operads than more general properads. Anyway the infinity-bialgebras are much easier than the Hopf counterpart. There is important work of Umble and Saneblidze in this direction (cf. arxiv/0709.3436). The motivating examples are however rather different than quantum groups, coming from rational homotopy theory, I think.

Similarly, there is no free Hopf algebra in an obvious sense what makes difficult to naturally interpret deformation complexes for Hopf algebras (there is a notion called free Hopf algebra, concerning something else). Boris Shoikhet, aided with some help from Kontsevich, as well as Martin Markl have looked into this.

Another relevant issue is to include various higher function algebras on higher categorical groups, enveloping algebras of higher Lie algebras (cf. baranovsky (pdf) or arxiv 0706.1396 version), usual quantum groups, examples of secondary Steenrod algebra of Bauese etc. into a unique natural higher Hopf setting. I have not seen that.

The author of the question might also be interested in a monoidal bicategorical approach to general Hopf algebroids by Street and Day.

Answer 2

There is certainly a natural homotopical analog of braided tensor category, namely a stable $E_2$ category (ie an $E_2$ object of the $\infty$-category of dg categories, or if you prefer or stable $(\infty,1)$-categories). Such things can be defined using Lurie's DAG I (for stable) and III (for $E_2$). Rather than trying to define versions of Hopf algebras, you can talk about fiber functors on such. In fact the general Tannakian pattern discussed in other MO posts generalizes from the symmetric monoidal setting to the braided setting -- i.e. given a braided category (say in this homotopical sense) you can define a "Spectrum", consisting of $E_2$ functors to modules over various $E_3$-algebras (which are $E_2$ categories). This defines a kind of object that you can call an $E_3$ stack (stack on $E_3$ algebras). [If you work in a nonderived setting there's no difference between $E_3$ and commutative.] This has an underlying usual stack. Anyway I learned all this from John Francis, who's been working on developing $E_n$-algebraic geometry...

anyway that was a digression, the point is you can talk about $E_2$ categories with a good fiber functor and use that as the definition of an $\infty$-quantum group..

As for other interactions, there is a very significant interplay between homotopy theory and quantum groups in the work in progress of Gaitsgory and Lurie on "quantum geometric Langlands". This was the topic of the 2008 Talbot workshop (see here). There are some related notes also by Gaitsgory and Lurie here. One awesome idea is the use of the $E_2$ perspective to explain WHY quantum groups relate to local systems on configuration spaces of points (Drinfeld-Kohno theorem and its generalizations) and in fact use it to prove the Kazhdan-Lusztig equivalence between quantum groups and affine Lie algebras. This would be a perfect topic for your dreamed-of book --- for now I'd make do with a paper or even course notes!!