Timeline for Are bundle gerbes bundles of algebras?
Current License: CC BY-SA 3.0
8 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Nov 25, 2011 at 12:29 | vote | accept | Dmitri Pavlov | ||
| Aug 13, 2011 at 8:22 | comment | added | Theo Johnson-Freyd | @Dmitri: Oh, I see. I just generally assume that most categories do not have distinguished generators, and so didn't think to interpret your question that way. | |
| Aug 12, 2011 at 10:20 | answer | added | Konrad Waldorf | timeline score: 6 | |
| Aug 11, 2011 at 21:01 | comment | added | Dmitri Pavlov | @Theo: I am not sure that the algebra of global sections is defined only up to Morita equivalence. The bicategory of algebras is a fully faithful replete subbicategory of the bicategory of linear categories (with extra adjectives), and if objects of the latter are equipped with a choice of a generator then we can also construct a canonical inverse functor. So one way to reformulate my question is to ask whether the corresponding linear categories have canonical generators. | |
| Aug 11, 2011 at 20:28 | comment | added | Theo Johnson-Freyd | In bad situations some of these notions diverge. But I would expect that manifolds, say, live in the good-situation world. I have left this as a comment, rather than an answer, because I'm not sufficiently sure of its correctness. But if you think it sounds right, let me know. | |
| Aug 11, 2011 at 20:27 | comment | added | Theo Johnson-Freyd | There is a purely algebrogeometric construction that comes close to what you want, but I have no idea if it's equivalent. Given any symmetric monoidal category $\mathcal C$ (with extra adjectives if you want), e.g. the category of modules for your favorite commutative ring $\mathcal O(X)$, you can consider the 2-category of $\mathcal C$-modules. It is symmetric monoidal, and you can take the (twice-categorified) group of units therein. In good situations, the decategorification of this group is the classical Brauer group of $\mathcal O(X)$, and is equal to $\mathrm H^2(X,\mathbb G_m)$. | |
| Aug 11, 2011 at 20:24 | comment | added | Theo Johnson-Freyd | Almost certainly "the" algebra of global sections will only be defined up to Morita, right? What will be define is the category of global sections, which for any representing algebra is the category of modules of that algebra. Or have I misunderstood something? | |
| Aug 11, 2011 at 15:11 | history | asked | Dmitri Pavlov | CC BY-SA 3.0 |