(no subject)

[leblon]

(У меня, кажется, дежа вю, ну да ладно.) Для моноидальной категории C есть понятие центра Дринфельда (это такая сплетенная моноидальная категория D, объекты которой, грубо говоря, - это объекты C, коммутирующие со всеми объекстами C). Моноидальная категория - это 2-категория с одним объектом. Можно придумать аналог центра Дринфельда для любой 2-категории. Этот конструкт имеет какое-то общепринятое название?

Comments

[kaledin-corpse.livejournal.com]

Depends. Here's the correct generalization: you takes the categories Fun(C^{\otimes n},C) (let's ignore the issue of what's a tensor product of abelian categories), put them into a cosimplicial category -- that is, a category cofibered over \Delta -- and take the category of global sections of the cofibration. This is an abelian category. The full subcategory spanned by cartesian sections is precisely the Drinfeld double. The trick is, you consider *all* sections, take the derived category, and only then you consider the full subcategory spanned by Cartesian sections. If you do it properly, you can derive a 124th proof of Deligne conjecture from this; I have been meaning to write this stuff down for several years already, but never got around to actually doing it.

An interesting case is when C is the category of endofunctors of some k-linear abelian B, k a field. Then the result of this procedure is just trivial! -- the dervied category of k-vector spaces. Sort like a higher version of the fact that a matrix algebra has no Hochschild cohomology.