Timeline for Bimodules in geometry
Current License: CC BY-SA 2.5
5 events
| when toggle format | what | by | license | comment | |
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| Oct 26, 2009 at 18:51 | comment | added | Reid Barton | Thanks, this must be Theorem 4.4.9 of DAG II. (It's not quite what I was hoping for, since it refers to objects of the category. I would like a characterization which only refers to the category viewed as an object of the (∞,2)-category of stable presentable (∞,1)-categories. But maybe that's too much to ask.) | |
| Oct 26, 2009 at 17:58 | comment | added | Dmitri Pavlov | I have a feeling that I might have seen something like this in one of Lurie's papers (DAG-I?), but I might be wrong on this issue. An abelian category is enriched over abelian groups, which correspond to spectra in the derived case, so our category should be enriched over spectra (looks like stable (∞,1)-category?). It should also be (homotopy) cocomplete and have some sort of (homotopy) generator. I guess one should look at the proof of the usual algebraic theorem to guess exact conditions that guarantee the existence of an equivalence. | |
| Oct 22, 2009 at 22:34 | comment | added | Reid Barton | Can you give a statement of Mitchell's theorem in the DAG case (or a reference)? | |
| Oct 22, 2009 at 21:18 | comment | added | Dmitri Pavlov | Most of the theories have some version of Eilenberg-Watts theorem (all (homotopy) cocontinuous functors come from bimodules). They usually also have some version of Mitchell's theorem that characterizes categories of modules. Combined together they give Morita equivalence theory that establishes an equivalence between the bicategory of rings, modules, and intertwiners and the appropriate bicategory of categories, functors, and natural transofrmations. But this is a universal phenomenon. It would be more interesting to find concrete applications in specific cases (like Ben Webster's examples). | |
| Oct 22, 2009 at 21:03 | history | answered | Reid Barton | CC BY-SA 2.5 |