Timeline for Do non-associative objects have a natural notion of representation?
Current License: CC BY-SA 2.5
28 events
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| Feb 24, 2020 at 17:50 | answer | added | Tim Campion | timeline score: 2 | |
| Jul 18, 2016 at 8:59 | answer | added | goblin GONE | timeline score: 3 | |
| Feb 10, 2014 at 8:34 | vote | accept | Qiaochu Yuan | ||
| Feb 9, 2014 at 18:54 | answer | added | მამუკა ჯიბლაძე | timeline score: 2 | |
| Apr 13, 2010 at 2:21 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Apr 13, 2010 at 2:06 | comment | added | Qiaochu Yuan | @Harry: I'm afraid I don't understand what you mean by building algebras in this context, nor do I understand what you mean by delooping in this context. Could you give some more details? | |
| Apr 13, 2010 at 2:00 | comment | added | Harry Gindi | @Qiaochu: Since you can enrich semicategories etc, there is a notion of functoriality, so you can build algebras as well. Then in the spirit of the representation theory of rings without unity, just define a representation to be an additive semifunctor from the delooping into somewhere else. | |
| Apr 13, 2010 at 1:49 | comment | added | Tom Church | @Qiaochu: You phrased my point better than I could have; thanks! | |
| Apr 13, 2010 at 1:48 | comment | added | Qiaochu Yuan | @Tom: Yeah, I think I was a little confused when I wrote that. I want to say there's some kind of Tannakian construction for the category of group actions of a group, but now I don't actually know if that's true (and I don't think what I said was equivalent to that). @David: Yes, this seems like more or less the same as Vladimir's definition. I haven't made up my mind about it. @Tom: I see what you're saying. I've added a clarifying paragraph at the end. | |
| Apr 13, 2010 at 1:40 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Apr 13, 2010 at 1:36 | comment | added | François G. Dorais | One more for the quote board -- "putting dictionary before the horse" -- good one Tom! | |
| Apr 13, 2010 at 1:25 | answer | added | S. Carnahan♦ | timeline score: 13 | |
| Apr 13, 2010 at 1:16 | comment | added | Tom Church | Also, without picking a fight, I would like to quietly object to: "This is made precise in the sense that for any object A in a category C , the invertible morphisms A->A have a group structure." "Groups therefore come with a natural notion of representation: a functor out of G." "Lie algebras have a 'natural' notion of representation because we want the map from Lie groups to Lie algebras to be functorial." None of this is false, but it seems to be putting the dictionary before the horse. | |
| Apr 13, 2010 at 1:14 | comment | added | David Jordan | I imagine you came across this, which discusses in broad terms what are representations of non-associative algebras. projecteuclid.org/DPubS/Repository/1.0/… Probably any reasonable notion of (linear) representation of magmas would extend to a non-associative algebra, simply because linear maps can be added and scaled. Of course, if you study completely general magmas, there won't be much you can say... But they suggest various specializations. I didn't give this as an answer since I don't know anything about this topic. | |
| Apr 13, 2010 at 1:09 | comment | added | Tom Church | Question 2 seems to be false for groups (unless you make it trivial); why do you expect it to be true for magmas? | |
| Apr 12, 2010 at 23:12 | answer | added | Neel Krishnaswami | timeline score: 2 | |
| Apr 12, 2010 at 22:39 | comment | added | Qiaochu Yuan | Harry, I'm afraid I don't see the relevance of those links. | |
| Apr 12, 2010 at 22:11 | comment | added | Harry Gindi | ncatlab.org/nlab/show/semicategory and ncatlab.org/nlab/show/semifunctor | |
| Apr 12, 2010 at 21:52 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Apr 12, 2010 at 21:35 | comment | added | Qiaochu Yuan | @Steven: Yes, the words "associahedra" and "operad" certainly look relevant to the question from what I know about them, but I don't know enough to guess what notion of representation will fall out of such considerations. Of course, feel free to post an answer along these lines. | |
| Apr 12, 2010 at 21:30 | answer | added | Vladimir Dotsenko | timeline score: 9 | |
| Apr 12, 2010 at 21:28 | comment | added | Steven Gubkin | Well, maybe they are more like magmas in the category of topological spaces, which is not the same thing as a representation... | |
| Apr 12, 2010 at 21:27 | comment | added | Steven Gubkin | H-spaces are essentially continuous representations of magmas. Sometimes you require associativity up to a homotopy - these homotopies are then all nicely parametrized by the associahedra. Your condition about compatible morphisms from $M_n x M_m \to M_{n+m}$ is starting to look like you want some kind of operad structure. Just throwing some terminology out there for you to google and see if you think it is relevant. If this is the kind of thing you are after I could try to craft an actual answer to your question. | |
| Apr 12, 2010 at 20:42 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Apr 12, 2010 at 20:39 | comment | added | François G. Dorais | I can't think obvious non-associative structures to represent into. The free magma consists of finite binary trees with the operation that attaches two trees at a new root. Maybe thinking about this case might help... | |
| Apr 12, 2010 at 20:36 | history | edited | Qiaochu Yuan | CC BY-SA 2.5 |
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| Apr 12, 2010 at 20:18 | answer | added | Adam Gal | timeline score: 2 | |
| Apr 12, 2010 at 20:14 | history | asked | Qiaochu Yuan | CC BY-SA 2.5 |