Definitions of a constant sheaf

Question

I am trying to understand local systems and the first step is to understand constant sheaves. I am confused, because I see different definitions used. For symplicity, I will ask about sheaves over $X$ with values in $\mathbb C$.

1. The constant presheaf is the sheaf $U \mapsto \mathbb C$ with restriction maps the identity. Its sheafification is called the constant sheaf.
2. A sheaf is called a constant sheaf, if it is isomorphic to the constant sheaf
   (See on stacksproject)
3. The sheaf of locally constant functions is called a constant sheaf
4. The constant sheaf is a sheaf on X whose stalks are all equal to $\mathbb C$.
   (Wikipedia)

Here are my questions:

1. First of all, 4. does not seem right to me, as it would imply that all line bundles are constant sheafs. Did I understand this correctly?

2. Is the constant presheaf already a sheaf on reasonable spaces, for example on the interval?

3. Are $2$ and $3$ equivalent?

4. And most importantly, how can I show that a sheaf is a constant sheaf? My approach so far was to construct an isomorphism of sheaves to the sheaf whose restriction maps are all the identity. But as I know now, this was not an isomorphism of sheaves, only of pre-sheaves.

I am rather new here, would it be better to ask the questions separately?

Answer 1

1. For sheaves of $\mathcal{O}_X$-modules, there are two different notions stalks and fibers, which should not be mixed up. Whereas a line bundle has fiber isomorphic to $\mathcal{C}$, its stalks are in fact much bigger (isomorphic to the local rings of $\mathcal{O}_X$). So no, line bundle are not constant sheaves and definition 4. is right.

2. No, the constant presheaf is not a sheaf on an interval. In fact, if $X=[0,1]$ and $U=(0.1,0.2)\cup (0.8,0.9)$, the constant presheaf $\mathcal{P}$ has obviously $\mathcal{P}(U)=\mathbb{C}$ whereas its sheafification has $\mathcal{P}^+(U)=\mathbb{C}^2$ as you can easily see with definition 3. Actually, the constant presheaf is a sheaf iff the space is irreducible. (So this may holds for schemes, but not for Hausdorff topological spaces unless it is trivial).

3. Weird question, definition 3. only gives an example of a constant sheaf.

4. This is often a hard problem. It might uses some advanced notions (monodromy...). At least, if you can construct a global section of your sheaf $\mathcal{F}$, you then have a morphism from the constant sheaf to $\mathcal{F}$ and you can now check that this morphism is an isomorphism on stalks. But constructing a global section might be very hard...