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I don't know the material, but from the sound of it, I think you can make this into a fine question for MO. Just make sure to provide some background. But rather than asking, "where does the proof use X?" I would ask, "Is the result true without X?":
Is every left fibration of simplicial sets a trivial Kan fibration?
I'm reading Lurie's Higher Topos Theory. Lemma 2.1.3.4 says blah blah blah, but I don't see where the proof uses this and that hypothesis. Can it be removed to strengthen the result?
You may also want to include something about (counter)examples you attempted to construct. If you also explain whether you think the result should be true without the hypothesis (something like "based on the intuition that contractibility gives you X, and a trivial Kan fibration is Y, it seems like you really need contractibility"), then I think it would be a great MO question.
In fact, I might even try to make the question more localized by turning your "I don't see why we need S_t to be contractible to extend f' as stated in the proof" into a question. (maybe that would be less localized; it doesn't really matter)
Looks like it worked. Also, it looks like MO just passed its 10,000-th post (that question was post number 10,002).
I agree that these type of questions are appropriate for MO.
I think this is not a good way to interpret "too localized." It's taken for granted that things of interest to mathematicians may not be of interest to a very large number of mathematicians. But that's still a fine use of the site.
I think Harry makes a very good point, the phrasing of questions about specific proposition can be too localized:
How to prove Proposition X in Y without Lemma Z?
Instead, they should be written to be self-contained (as Harry did):
How to prove that F is true without H? (I'm asking because I'm reading Y and it has proposition X and the proof seems to rely on lemma K which doesn't require H)
Right, I don't think that you're interpreting the phrase "too localized" in a way that will lead you to the right questions on the site. Obviously if you're asking a question abotu a book, it's good to provide enough context that people don't need to look at the book to know what you're talking about, but otherwise it's a great question. If someone mentions in a question that they're reading a book, and came to ask a question when they got stuck (as opposed to people who ask questions instead of reading the book), I say "Ah, here's someone who knows how to use MO."
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