Timeline for Categorical proof subgroups of free groups are free?
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18 events
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| Feb 1, 2015 at 15:05 | vote | accept | Exterior | ||
| Jan 28, 2015 at 18:08 | comment | added | Arturo Magidin | (My comment was not in the linked question, but it is in this one ). | |
| Jan 28, 2015 at 18:04 | comment | added | Arturo Magidin | (continued) One would expect a truly "categorical proof" to be something you could do in the context of any variety of groups; but since the conclusion does not hold in most of them, I don't expect you would be able to find such a proof in the first place. | |
| Jan 28, 2015 at 18:02 | comment | added | Arturo Magidin | As I believe I said in math.SE, I don't think you can have a true "categorical proof" (though you can probably have proofs cast in the language of categories, or proofs that have a categorical flavor), because the fact is not true "categorically". If you look at the category of all groups in a variety of groups (which is a reflective subcategory of Groups and hence generally speaking well-behaved), the only varieties in which "subgroup of free is free" holds are the variety of all groups, all abelian groups, and all abelian groups of exponent $p$ (a prime). (cont) | |
| Jan 28, 2015 at 9:31 | answer | added | Qiaochu Yuan | timeline score: 12 | |
| Jan 28, 2015 at 3:57 | answer | added | Benjamin Steinberg | timeline score: 4 | |
| Jan 28, 2015 at 1:27 | comment | added | Qiaochu Yuan | Prospects for a categorical proof seem poor in that subgroups of free groups are not canonically free. I think freeness is in some sense a red herring and one should look for some other group-theoretic property equivalent to freeness but which makes no reference to a choice of free generators. Maybe cohomological dimension $1$? | |
| Jan 28, 2015 at 0:46 | comment | added | Benjamin Steinberg | arxiv.org/abs/1006.3833 is a version of my somewhat categorical proof with the language stripped out. I had it written with a more categorical language earlier on but removed usage of functors and explicit reference to a group being determined by its category of actions. | |
| Jan 27, 2015 at 23:50 | answer | added | Todd Trimble | timeline score: 12 | |
| Jan 27, 2015 at 23:04 | comment | added | Fernando Muro | What's the meaning of 'categorical proof'? | |
| Jan 27, 2015 at 22:55 | answer | added | Ben Webster♦ | timeline score: 9 | |
| Jan 27, 2015 at 22:09 | comment | added | Benjamin Steinberg | The idea is similar to the one we give in arxiv.org/abs/0812.0027 but tensor products of G-sets replace wreath products. | |
| Jan 27, 2015 at 22:07 | comment | added | Benjamin Steinberg | I have a sort of categorical proof that I never published. It uses that a group G is determined up to Isomorphism by its category of G-sets and it uses left Kan extension or tensor products. Also it verifies the universal mapping property. But at a certain moment you do need a Schreier transversal which uses choice. I can probably dig up the draft if someone wants it. I'd say that it makes categorical the part which can be but at some point a spanning tree or transversal is needed. | |
| Jan 27, 2015 at 22:01 | comment | added | Todd Trimble | We also had a long accompanying discussion of this nLab entry at the nForum: nforum.mathforge.org/discussion/4364/nielsenschreier-theorem | |
| Jan 27, 2015 at 21:59 | comment | added | Todd Trimble | See here: ncatlab.org/nlab/show/Nielsen-Schreier+theorem, where a topological proof and a groupoidal proof are given. It might help to read them in tandem. My impression is that you need some choice to do the proof, but I don't take that as ruling out a categorical proof. | |
| Jan 27, 2015 at 21:45 | comment | added | Pablo | Another example of such a category is that of Lie Algebras. | |
| Jan 27, 2015 at 21:43 | comment | added | Yemon Choi | There was some discussion of this in some posts at the n-category cafe, IIRC, but I don't have the links to hand | |
| Jan 27, 2015 at 21:35 | history | asked | Exterior | CC BY-SA 3.0 |