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As I write this comment, the question has attracted 4 votes to close - I personally don't have an active wish to see it stay open, but perhaps it could benefit from some discussion here. It is not clear to me that what the questioner intends by "isomorphism" is the same as the technical category-theoretic notion, but perhaps I have misread or misunderstood.
I think it is a bad sign that the most popular answer is a joke (admittedly a good one).
Sadly for me, I think I would like to see the question closed, and I would like for the last close vote to be cast by someone other than me. I don't know quite what to make of this...
Isn't this a good candidate for mathoverflow.tea, or whatever the sister site is that was being discussed? Did anything happen with that?
Update: once I realized that I had voted to close essentially the same question on math.SE, I found my resolve and voted to close this question.
I was a bit surprised to see some familiar names among those who closed the MU question. That brings forth a delicate ethical issue: since there is apparently a substantial overlap between people with voting privileges on both sites and it only takes a few such users to close a question, ought they not to refrain from duplicating their votes? For otherwise, they would be imposing their will at both MO and MU, which can result in mathematical questions disliked by a few powerful users with strong opinions having literally nowhere else to go and lead to very bad feelings on the substantial part of the community that is unable to affect the process.
On the closing itself: the MU closing message states that
It's impossible to objectively answer this question; questions of this type are too open ended and usually lead to confrontation and argument.
I don't see any evidence of this happening at either MO or MU. It is true that the question gathered only 1 up signed vote at MO as of this writing, but it seems disingenuous to close a question not even 6 hours old that has already attracted 17 reasonable, and even insightful, answers. In fact, it looks remarkably like another attempt by the OBN to make everyone else to toe the line. (The MO stated reason for closing is "off topic").
No, it's not. The reason fishing expeditions are so hard to close is precisely because they're so easy to answer and because everyone wants to contribute. The fact that many of those answers are interesting is, if anything, evidence that the fishing expedition is fishing successfully.
On the other hand I would've been perfectly happy with a question of the form "what are concrete, striking examples of isomorphisms to describe to students," which is the question that some people ended up answering anyway.
@VP: I don't find your argument against voting to close on the two sites all that compelling. I certainly don't use the same standards to vote to close an MO question as a math.SE question. (For example, there was a question about adding fractions on math.SE that was closed as "off-topic". I left a comment questioning that: surely this is part of "math at all levels", which is the stated purpose of math.SE. Moreover, befitting the much wider scope of math.SE, I vote to close much more rarely on that site.)
One reason to vote to close a question on both sites -- which you need not agree with, but I think is certainly ethically defensible -- is that it is simply too broad to be appropriate for any SE-engine driven Q&A site. These sites are designed to answer specific, focused questions. (Some people want to also ask broader questions under the "community-wiki" label, and that's a defensible position too. It seems that the reaction to this should be to be doing exactly what we are doing: to (i) vote our preferences on the matter, (ii) discuss why we feel a certain way, and (iii) try to keep an open mind.)
In particular, I have come to regard use of the word "favorite" as a sign that the question is too broad to be useful. It basically just invites any and all answers and allows no real justification for the voting up and down of answers. Why are we ranking people's favorite isomorphisms?
As Qiaochu remarks, there is a kernel of a good question in here. Namely, the OP seems to be interested in teaching the concept of isomorphism and is looking for examples of successful pedagogy in this area. Again, the word "favorite" works against this, because it doesn't make sense to justify your favorite things ("And why exactly do you like raindrops on roses?"), and indeed a lot of people are answering with things like "I don't know whether this would be a good example for students, but I have always liked..." I think the question would be worthy of reopening after being edited to emphasize the pedagogical aspect and enjoining responders to explain why their answer would be helpful for students.
@alex_o: fair enough. I should've said instead that it is very hard to ask a good "fishing expedition" question, and the quality of the answers to such a question does not necessarily correlate with the quality of the question itself. I agree that some questions which might fall under this category can be quite interesting, but it's important to keep in mind that many of these questions were asked when the standards of the site were more lax, and some of them have recently been closed because they are attracting bad or repetitive answers.
Let me suggest taking the concept of fishing expedition to its logical extreme: "what is your favorite mathematical statement?" I am sure you will agree that this is too general to be a sensible MO question (although possibly I am wrong), even though it might generate a wealth of interesting answers. If we can agree on this, then we can agree that some level of specificity is necessary for such a question. I currently believe that this question is not sufficiently specific. If you can convince me that this is not true, I will vote to reopen.
I also do not understand your last comment. The questions may seem similar to you, but the question I suggested is geared towards a specific purpose: pedagogy. That gives a basis for voting on answers: vote up answers that would be good to present to students, and vote down answers that would not be good to present to students. (For example, I personally have an answer I'm dying to add to the question (except that I am not convinced that the question is worth reopening); it is my "favorite," but it is in no way a suitable example for beginning students.)
@Gil: Isn't that what web logs are for...?
I do not think the mission of the site needs to change, and I would prefer if such open-ended questions were not asked on mathoverflow.net. I agree with Harry Gindi that a blog is fine place for open-ended questions for the readers, and I think JSE has had a reasonable amount of success with that on quomodocumque.wordpress.com.
If Gil Kalai is worried about not getting enough readers answering questions on his blog, then it might be a good idea to post material on his blog that attracts more readers that are inclined to answer his questions. Mathoverflow.net has a large readership, but I do not think most of them are here for navel-gazing, since we put some effort into making sure that such introspective questions are rare. It is not in good faith to change the scope of the site simply to take advantage of the fact that there are a large number of readers who might have answers to a particular question. As Ryan Budney said, we run the risk of alienating mathematicians who can contribute on a concrete mathematical level.
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