Timeline for Characterize the category of rings
Current License: CC BY-SA 3.0
14 events
| when toggle format | what | by | license | comment | |
|---|---|---|---|---|---|
| Oct 23, 2014 at 18:01 | answer | added | Zhen Lin | timeline score: 12 | |
| Oct 23, 2014 at 16:26 | comment | added | Tom Leinster | @Paul Yes! That was why I assumed that I wasn't really answering Chris's question. As you say, it's kind of a tautological statement. | |
| Oct 23, 2014 at 16:23 | comment | added | Dimitri Chikhladze | Note that when the "characterization" is in the form of an embedding (Like Freid-Mitchel, or Barr's embedding of a regular category), essentially this embedding should be structure preserving. Such embeddings can indeed be viewed as characterizations of some sort since they allow working with an abstract category as if it were a specific one. | |
| Oct 23, 2014 at 15:02 | comment | added | Paul Taylor | @Tom, doesn't your observation work equally well with any algebraic theory in place of rings? It seems to be defining the thing essentially in terms of itself. | |
| Oct 23, 2014 at 14:54 | answer | added | Paul Taylor | timeline score: 8 | |
| Oct 23, 2014 at 13:48 | answer | added | Dimitri Chikhladze | timeline score: 7 | |
| Oct 23, 2014 at 13:33 | comment | added | Omar Antolín-Camarena | Oh, sorry, I hadn't seen the "(Sub)" at the beginning of the question. Ignore my previous comment. | |
| Oct 23, 2014 at 13:21 | comment | added | Omar Antolín-Camarena | What characterization of categories of modules do you mean? I thought the Mitchell embedding theorem just said that small Abelian categories embed into module categories, i.e., it says something about which small categories can be found inside modules categories (the answer is all Abelian ones), not about how to characterize entire categories of modules. Maybe there is some obvious trick I'm missing here. | |
| Oct 23, 2014 at 11:33 | comment | added | Tom Leinster | OK: the category of rings together with its forgetful functor to $\mathbf{Set}$ is terminal among all categories $C$ equipped with a functor $U: C \to \mathbf{Set}$ and a ring structure on the object $U \in [C, \mathbf{Set}]$! That's probably not the kind of thing you want, though... | |
| Oct 23, 2014 at 10:26 | comment | added | Chris Heunen | @Tom: preferably the former, but the latter would be an interesting step in that direction in its own right. | |
| Oct 23, 2014 at 10:25 | comment | added | Tom Leinster | Do you definitely want to characterize the category of rings, rather than the category of rings together with its forgetful functor to Set? Not that I know how to do either, but I think the two questions are significantly different in character. | |
| Oct 23, 2014 at 10:23 | comment | added | Chris Heunen | For example: the category of complex Hilbert spaces and bounded linear maps, see tac.mta.ca/tac/volumes/22/13/22-13abs.html. | |
| Oct 23, 2014 at 10:22 | comment | added | Chris Heunen | Though I'm primarily interested in rings, this could be made into a big list -- please feel free to add characterizations of other categories of well-known objects! | |
| Oct 23, 2014 at 10:21 | history | asked | Chris Heunen | CC BY-SA 3.0 |