This is Exercise II.5 of Mac Lane and Moerdijk's, "Sheaves in Geometry and Logic [. . .]". According to Approach0, it is new to MSE.
The Details:
On p. 66, ibid. . . .
Definition 1: A sheaf of sets $F$ on a topological space $X$ is a functor $F:\mathcal{O}(X)^{{\rm op}}\to\mathbf{Sets}$ such that each open covering $U=\bigcup_iU_i, i\in I$, of open subsets of $U$ of $X$ yields an equaliser diagram
$$ FU\stackrel{e}{\dashrightarrow}\prod_{i\in I}FU_i\overset{p}{\underset{q}{\rightrightarrows}}\prod_{i,j\in I}(U_i\cap U_j),$$
where for $t\in FU,$ $e(t)=\{ t\rvert_{U_i}\mid i\in I\}$ and for a family $t_i\in FU_i$,
$$p\{ t_i\}=\{t_i\rvert_{(U_i\cap U_j)}\}\quad\text{ and }\quad q\{ t_i\}=\{t_j\rvert_{(U_i\cap U_j)}\}.$$
From p. 79, ibid. . . .
For any space $X$, a continuous map $p: Y\to X$ is called a space over $X$ or a bundle over $X$.${}^\dagger$
From p. 82, ibid. . . .
Definition 4: A covering map $p: \stackrel{\sim}{X} \to X$ is a continuous map between topological spaces such that each $x\in X$ has an open neighborhood $U$, with $x\in U \subset X$, for which $p^{-1}U$ is a disjoint union of open sets $U_i$, each of which is mapped homeomorphically onto $U$ by $p$.
On p. 88 ibid. . . .
A bundle $p: E \to X$ is said to be étale (or étale over $X$) when $p$ is a local homeomorphism in the following sense: To each $e\in E$ there is an open set $V$, with $e\in V\subset E$, such that $pV$ is open in $X$ and $p\rvert_V$ is a homeomorphism $V\to pV.$
From the exercise . . .
Definition: A sheaf $F$ on a locally connected space $X$ is locally constant if each point $x\in X$ has a basis of open neighborhoods $\mathcal{N}_x$ such that whenever $U, V \in\mathcal{N}_x$ with $U\subset V$, the restriction $FV\to FU$ is a bijection.
The Question:
Consider a sheaf $F$ on a locally connected space $X$. Prove that $F$ is locally constant iff the associated${}^{\dagger\dagger}$ étale space over $X$ is a covering.
Thoughts:
$(\Rightarrow)$ Let $F$ be a sheaf on a locally connected space $X$. Suppose, further, that $F$ is locally constant. Let $x\in X$. Then there is a basis $\mathcal{N}_x$ such that, for any $U, V\in\mathcal{N}_x$ with $U\subset V$, the restriction $FU\to FV$ is a bijection.
What do I do now?
Looking at the definition of a sheaf, I find myself a little lost here.
$(\Leftarrow)$ I'm completely lost here. I'm not sure I understand the definition of an étale space.
Further Context:
Related questions of mine include the following.
Please help :)
$\dagger$: I'm assuming $Y$ is also a topological space. Am I right?
$\dagger\dagger$: I'm assuming this is the associated bundle as described on page 82, ibid, that, by hypothesis of the question, happens to be étale.
Suggestion: first prove this for the constant sheaf, which corresponds to a trivial (disconnected) cover. This is the “local” version.
Side Note: $X$ must also be connected, else you can have different degree covers on different components and the sheaf still locally constant according to your definition.
Once you know the local version is true, Locally being the constant sheaf is the same as locally having trivial etale space which is definition of covering space (assuming $X$ connected to get evenly covered!). One way to go about proving that last part is to show the local version is functorial, which would imply that “local here corresponds to local there”, “locally (constant sheaf)” = “locally (trivial etale cover)”.