Timeline for alternative construction of the quotient group
Current License: CC BY-SA 2.5
19 events
| when toggle format | what | by | license | comment | |
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| Feb 8, 2010 at 0:58 | vote | accept | Martin Brandenburg | ||
| Feb 7, 2010 at 21:04 | answer | added | Emerton | timeline score: 3 | |
| Feb 7, 2010 at 20:18 | vote | accept | Martin Brandenburg | ||
| Feb 8, 2010 at 0:58 | |||||
| Feb 7, 2010 at 18:00 | answer | added | Bjorn Poonen | timeline score: 13 | |
| Feb 7, 2010 at 17:45 | answer | added | Tim Porter | timeline score: 7 | |
| Feb 7, 2010 at 17:07 | comment | added | Martin Brandenburg | @unknown: ah. indeed this yields another construction, but it's not explicit. we have to take a set $M$ of groups such that every homomorphic image of $G$ is isomorphic to some element of $M$, take the product of all these groups in $M$ and then take the equalizer of all endomorphisms of this product. @harry: I don't think that this has to do with the category-theoretic description of groups. | |
| Feb 7, 2010 at 17:01 | comment | added | Harry Gindi | @unknown: This is because you can encode groups as categories and morphisms as functors. This is a totally useless construction. | |
| Feb 7, 2010 at 16:48 | comment | added | user2734 | Please note that there is a discussion on this subject on pp. 57-58 of Mac Lane. In particular, on the top of p. 58 it is stated that the universal property of the quotient group can be obtained without cosets at all, using the AFT. (I did not check the details; if I do, I will send a complete answer. In any case, I suspect that it is the Representabilty Theorem of p. 122 of Mac Lane that is really required.) | |
| Feb 7, 2010 at 16:46 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Feb 7, 2010 at 16:38 | answer | added | gowers | timeline score: 2 | |
| Feb 7, 2010 at 16:32 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Feb 7, 2010 at 16:19 | comment | added | Tom Church | The question seems clear, and is a reasonable one. I doubt there is any good answer, but I'd love to see one if there is. | |
| Feb 7, 2010 at 16:18 | comment | added | Martin Brandenburg | yes the standard construction makes it clear. but I'm asking for another construction. if it exists. and when you don't know one or are not interested in this, just don't react to this silly question :-) | |
| Feb 7, 2010 at 16:16 | history | edited | Martin Brandenburg | CC BY-SA 2.5 |
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| Feb 7, 2010 at 16:13 | comment | added | Ben Webster♦ | Martin- I really have no idea what you're asking for in this question. The standard construction does make the universal property clear (obviously a map sending N to the identity send each coset a single point). | |
| Feb 7, 2010 at 16:12 | history | edited | Harry Gindi | CC BY-SA 2.5 |
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| Feb 7, 2010 at 16:10 | comment | added | Martin Brandenburg | the question is not about how one should develop the theory of the category of groups ... | |
| Feb 7, 2010 at 16:06 | comment | added | Harry Gindi | Uh.. You realize that the construction is necessary to prove that the category has quotients. It doesn't just fall out of thin air. | |
| Feb 7, 2010 at 15:54 | history | asked | Martin Brandenburg | CC BY-SA 2.5 |