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Sheaf

If you look up the word ''sheaf'' in the dictionary, you might find something like:
''A bundle of grain stalks laid lengthways and tied together after reaping.''
Allow me to tell you what a sheaf really is:

Definition: Let X be a topological space (typically a scheme). An abelian presheaf F on X consists of the following data:
- For each open set U of X, an abelian group F(U).
- For each inclusion of open sets V ⊆ U, a restriction morphism res_(V,U) : F(U) → F(V).
These restriction morphisms must satisfy:
- For every open set U of X, the map res_(U,U) : F(U) → F(U) is the identity map on F(U).
- If we have three open sets W ⊆ V ⊆ U, then we have res_(W,V) o res_(V,U) = res_(W,U).

If s ∈ F(U), then its restriction res_(V,U)(s) is often denoted s|_V.

Definition: A sheaf is a presheaf that satisfies the following two axioms:
- If { U_i } is an open covering of an open set U, and if s,t ∈ F(U) are such that s|_(U_i) = t|_(U_i) for each set U_i of the covering, then s = t.
- If { U_i } is an open covering of an open set U, and if for each i a section s_i ∈ F(U_i) is given such that for each pair U_i,U_j we have s_i|_(U_i ∩ U_j) = s_j|_(U_i ∩ U_j), then there is a section s ∈ F(U) such that s|_(U_i) = s_i for each i.

What you can see above is the poorly typeset definition of a sheaf as one knows it in mathematics, and then in particular algebraic geometry (admittedly known for its lengthy definitions). This is what ''real'' mathematics looks like, if there exists such a thing. It is nothing like what is being taught in high school. This misconception goes so deep that sometimes it makes me very sad.
Note: by giving the above definition I am not trying to make myself look smart or anyone else look dumb - I just want you to be able to see it. Understanding the definition is a matter of mathematical experience and not one of intellect; for now it is irrelevant.

Oh yeah, I made a new map.
Three puzzles, all with a somewhat creative idea.
Be sure to let me know when you break any of them :]
Some concepts were inspired by some of Fumbly Bumbly's latest maps.
This map is basically a meh Bottom Line.

Difficulty: maybe 4/10 all round.
Feel free to disagree with this and to let me know about it.

Of course I won't copystrike you for the music if you record your run :p
In other news: My puzzle game Sokoban: The RPG has been released on steam! Please consider checking it out!

Playthrough by 可砸的六度双音
Room 1 by bullfrog
Room 1 by Joric
Room 2 by Joric
Room 3 by Joric
Blind run by asears16
Blind run by Nock