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

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

Which is the appropriate box to tick?

bookexists that does what you're looking for. A lot of these applications of homotopy theory to "quantum algebra" and noncommutative algebraic geometry arerelativelynew. It isn't very often that an introductory textbook comes out that covers the "bleeding edge", so to speak. You can try reading papers, but I doubt very much that you can find a book that covers the topic in the generality you're looking for. – Harry Gindi Mar 3 '10 at 19:03