Definitions of a constant sheaf 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 $\mathbb C$.

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

Here are my questions:


*

*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?

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

*Are $2$ and $3$ equivalent?

*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?
 A: *

*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.

*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).

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

*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...
