Timeline for Does Morita theory hint higher modules for noncommutative ring?
Current License: CC BY-SA 4.0
11 events
| when toggle format | what | by | license | comment | |
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| Jul 21, 2020 at 18:53 | vote | accept | Student | ||
| Jul 21, 2020 at 18:51 | comment | added | Student | @ZhenLin one example I have in mind is the Tannakian reconstruction for (finite) groups, in which case one remembers the forgetful functor to (Vect). I never viewed that as a way to remember $\mathbb{C}[G]$. Thank you! | |
| Jul 21, 2020 at 13:26 | comment | added | Theo Johnson-Freyd | Switching to pointed categories is not trivial, though. One thing people sometimes mean when they say "Morita" is the bicategory of algebras an bimodules. The (lax) pointed version of this is algebras and pointed bimodules --- this is the same as the bicategory of pointed categories abstractly equivalent to $\mathrm{Mod}(R)$, and lax pointed functors which preserve all colimits. Its groupoid of objects is the groupoid of rings, but it has a lot more morphisms, so it is somewhere between homomrphisms are Morita maps. It is explored e.g. in the paper of Gwilliam and Scheimbauer. | |
| Jul 21, 2020 at 13:24 | comment | added | Theo Johnson-Freyd | @Student Yes, of course it is simply "using" the ring structure. Maybe if I had called it "$1_R$" rather than "$R_R$" you would have liked it better :) Seriously, if your goal is something that functorially recovers $R$ and works for all rings, then you will have to "use" the ring structure. | |
| Jul 21, 2020 at 13:21 | comment | added | Theo Johnson-Freyd | @QiaochuYuan Yes. | |
| Jul 20, 2020 at 17:29 | comment | added | Fernando Muro | @Student any ring is Morita equivalent to all its matrix rings, but a ring is seldom isomorphic to any of its matrix rings. | |
| Jul 20, 2020 at 10:31 | comment | added | Student | Thanks for all your comments! I'm aware that my question is not clear enough, and perhaps cannot be made precise. What I'm really hoping is a description of modules that is "implicit enough" and can recover the ring. Of course, this is highly subjective, so perhaps I should ask a slightly more concrete question: What kind of rings have that "Morita=Classical"? .. and try to see how it fails for the others. Alas, this isn't what I really hope for.. it's just an approximation. | |
| Jul 20, 2020 at 5:54 | comment | added | Qiaochu Yuan | This ought to work out to be a special case of a more general construction where an $E_n$-algebra can be recovered from the $E_{n-1}$-algebra structure on its category of modules, specialized to $n=0$ (an $E_0$-algebra is a pointed object). | |
| Jul 20, 2020 at 3:40 | comment | added | Zhen Lin | There's a further trick if you want to avoid explicitly using the ring structure of $R$ directly: instead of remembering the object $R_R$, you remember the forgetful functor $\mathbf{Mod} (R) \to \mathbf{Set}$. Of course, this amounts to remembering $R_R$ because it is the representing object for the forgetful functor... | |
| Jul 20, 2020 at 1:26 | comment | added | Student | Well.. this is slightly not what I expected.. since including $R_R$ in the data somehow uses the ring structure of $R$ directly.. and so no wonder it captures the ring structure :P But still thank you so much for your answer! | |
| Jul 19, 2020 at 23:32 | history | answered | Theo Johnson-Freyd | CC BY-SA 4.0 |