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A constant sheaf A on X associated to an abelian group G is a sheaf such that $A_{x}$= G, for some fixed G and every x $\in$ X. (the stalks of A are equal to a constant group G)

For simplicity, let X= {p,q}.

I have read that we can define a constant sheaf as A associated with E= X $\times$ G= $\coprod_{x \in X} G$ (the etale space E), A:= sheaf of sections of $\pi$: E $\rightarrow$ X?

or as A associated with the space G $\oplus$ G; A:= sheaf of sectiosn of $\pi$: G$\oplus$G ->X? (not really sure about this one)

Can someone explain these two constructions. Are there any more ways to construct a compatible constant sheaf A given X and G?

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Your definition of a constant sheaf is not correct. It is not enough to ask for the stalks to be isomorphic. There are many counter examples : locally constant sheaves, locally free sheaves on a manifold...

The simplest definition of the constant sheaf with stalks $G$ is the sheaf of locally constant functions $U\rightarrow G$. In other words, this is the sheaf of continuous functions $U\rightarrow G$ where $G$ is endowed with the discrete topology.

As étalé space, this is indeed $E=\coprod_{x\in X}G=X\times G$ with the product topology (and $G$ the discrete topology). Indeed, the sections (on $U$) of the projection $E\rightarrow X$ can be written $s(x)=(x,f(x))$ for a function $f:U\rightarrow G$. Now, you can show that $s$ is continuous iff $f$ is continuous. So sections of the projection are in bijection with locally constant function $U\rightarrow G$.

But your notation $G\oplus G$ is misleading. As I understand it, this is only for $X=\{p,q\}$ a two points discrete space. But $G\oplus G$ is not the sheaf, instead we have $\Gamma(X,G)=G\oplus G$ because a continuous functions $f:X\rightarrow G$ is simply a pair $(f(p),f(q))$.