It is the correct definition in the category of presheaves though. The point is, to actually talk about exactness categorically, one need a notion of image and cokernel of a morphism. To show that the category of sheaves contains such morphisms, one natural thing is to take the image of the morphism open-set-wise but this is actually only a presheaf and not a sheaf. To remedy this, one do the "sheafification" of this presheaf to get the cokernel or image sheaf in the category of sheaves. This is the reason why this definition does not work, because by shefifying the presheaf we have to make our definitions locally i.e. via stalks for example. and actually if you work on stalks level, then the sequence of sheaves is exact if and only if the induced sequence on the level of stalks is exact at every point!
@@fawzyhegab Oh, I know all of this. I'm just saying that Borcherds delivery of the fact that the "natural" definition of a exact sequence of sheaves doesn't work is hilarious.
I actually feel this is so motivating😀 The textbook I read jumps straight into the Etale space and I was completely drowned by the time I saw the exact sequence. I was wondering why people have to deal with that terrible Etale space (even when I saw the exact sequence fails on the sections, I didn't realize that's a motivation).
To confirm my understanding, I wrote the follows. If I misunderstand, I was wondering if someone could let me know. In the following, I inherit the notations in 15:26 and shows the details about the non exactness of the short exact sequence of sheaves. Let R denote reals and Z integers. The, the spaces are regarded as the quotient spaces, X_2 := R/2Z, X_1 := R/Z, X := R/Z. Then, F_2(X) is the set constructed by any continuous map f: X -> X_2 such that for any x and any integer n there is an integer m satisfying f(x + n) = f(x) + 2m. Similarly, F_1(X) is the set constructed by any continuous map f: X -> X_1 such that for any x and any integer n there is an integer m satisfying f(x + n) = f(x) + m. Then the non-surjection of the short exact sequence of sheaf is deduced from the fact that there is a map f in F_1(X) such that f is not in F_2(X). For example, the identity map f: X -> X_1 is the desired map which can't be written by the element of F_2(X).
I have the impression that that the space F_i(U) consists of the continuous section maps s_i for the natural projection pi_i from X to X_i, i.e. the composition pi_i s_i is identity.
I think the answer by Fan to the other comment actually addressed your question: if s is in F1(X), then pi_X composed with s should be id_X. If you take any "periodic functions on the circle", then when you project it back to X, it is at best a homeomorphism of X, rather than the identity map of X.
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"Any idiot can figure out what the definition of this should be"
30 seconds later
"Actually it is completely wrong."
Actually laughed out loud.
It is the correct definition in the category of presheaves though. The point is, to actually talk about exactness categorically, one need a notion of image and cokernel of a morphism. To show that the category of sheaves contains such morphisms, one natural thing is to take the image of the morphism open-set-wise but this is actually only a presheaf and not a sheaf. To remedy this, one do the "sheafification" of this presheaf to get the cokernel or image sheaf in the category of sheaves. This is the reason why this definition does not work, because by shefifying the presheaf we have to make our definitions locally i.e. via stalks for example. and actually if you work on stalks level, then the sequence of sheaves is exact if and only if the induced sequence on the level of stalks is exact at every point!
@@fawzyhegab Oh, I know all of this. I'm just saying that Borcherds delivery of the fact that the "natural" definition of a exact sequence of sheaves doesn't work is hilarious.
@@fawzyhegab And I didn't know any of this, thank you
The same here lol
I actually feel this is so motivating😀 The textbook I read jumps straight into the Etale space and I was completely drowned by the time I saw the exact sequence. I was wondering why people have to deal with that terrible Etale space (even when I saw the exact sequence fails on the sections, I didn't realize that's a motivation).
To confirm my understanding, I wrote the follows. If I misunderstand, I was wondering if someone could let me know.
In the following, I inherit the notations in 15:26 and shows the details about the non exactness of the short exact sequence of sheaves.
Let R denote reals and Z integers.
The, the spaces are regarded as the quotient spaces,
X_2 := R/2Z,
X_1 := R/Z,
X := R/Z.
Then, F_2(X) is the set constructed by any continuous map f: X -> X_2 such that for any x and any integer n there is an integer m satisfying f(x + n) = f(x) + 2m.
Similarly, F_1(X) is the set constructed by any continuous map f: X -> X_1 such that for any x and any integer n there is an integer m satisfying f(x + n) = f(x) + m.
Then the non-surjection of the short exact sequence of sheaf is deduced from the fact that there is a map f in F_1(X) such that f is not in F_2(X). For example, the identity map f: X -> X_1 is the desired map which can't be written by the element of F_2(X).
I have the impression that that the space F_i(U) consists of the continuous section maps s_i for the natural projection pi_i from X to X_i, i.e. the composition pi_i s_i is identity.
Around minute 16: Why is F1(X) a point? Isn't it equal to the infinite set of periodic functions on the circle?
I think the answer by Fan to the other comment actually addressed your question: if s is in F1(X), then pi_X composed with s should be id_X. If you take any "periodic functions on the circle", then when you project it back to X, it is at best a homeomorphism of X, rather than the identity map of X.
why are you always so implicit about everything :(