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 $\mathbb C$.
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?