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So I do not understand that fully Urs’s post. First of all there is some confusion about the notation, “Write then $i:F\to A$ for the function that includes the fiber of $f$ over $pt_B$.” First of all $f$ is not defined yet, here but introduced as an arbitrary $f$ in the next sentence, I guess it is meant $c$, though I do not quite get the phrase “function which includes the fiber”. So I assume it is about $c$. Now $f$ is arbitrary, while a bit later it says “by construction of $f$”, what I do not understand again, as $f$ is taken to be arbitrary. Still, I think I can make sense up to that point. Now, what is the meaning of $c\circ f \simeq pt_B$ (these functions have different domains!) and what is the meaning of “equivalent” in function context. I understand that it is necessary, though not sufficient, condition that the map factors through $pt_B$ (correction, I am getting this now: the pullback condition makes it sufficient here as well!). Obstruction theory often extracts in fact the sufficient conditions as well, by induction on cells, which is an algorithmic and subtler procedure. But I would appreciate to see what survives really at what Urs calls “hi school level’ (though there is no hi school where students would understand this reasoning), it is appealing, but then I should not cheat with homotopy and see what equivalence etc. makes sense really with functions. Urs ?
Is it important that we branch this discussion off here? I would find it more convenient if we discussed this in one place, hence over at the $n$Cafe.
And, yes, that $f$ was a typo, as we also discussed already here at the Café. :-)
Of course the best thing would be to create stable trace of this discussion on the $n$Lab!
I have started making a quick note at obstruction. But I will have to stop doing this now and look into something else.
Urs, no, I can not correct typoses in cafe. I am not at home and have very difficult connection and even type some posts from kindle, what is VERY hard. Cafe is so difficult to communicate from kindle and even the sending makes lost of problems (like spurious lack of links, error messages and so on). Cafe is anyway pretty disfunctional, a post which I do here in 5 minutes takes me about half an hour in cafe and sometimes gets lost. Cafe is good reading occasionally, but never good for interaction, unless one is ready to spend few evenings for few messages.
And, yes, that f was a typo, as we also discussed already here at the Café. :-)
This proves my point about cafe vs forum: here you could correct it and over there you have typed huge block which stays uncorrected. (but what about illogical “by construction of $f$”)
Surely, I like the new nLab entry.
but what about illogical “by construction of f”
That’s another typo: “by construction of $F$”! :-)
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