Consider this definition of the Constant Sheaf in 'Introduction to Algebraic Geometry', Justin R. Smith:
So we have 2 groups here: the group $A$, which is the co-domain of the continuous functions, and the group of continuous functions $U \to A$, and that second group is what is assigned to the open sets in $V$ in sheaf theory.
I have several questions here:
1) what is the group operation in that second group (it cannot be composition) ? does it matter ?
2) what is the significance of the target of the continuous functions being a group ($A$) - let alone an abelian group - for the purpose of the Constant Sheaf definition ? why can't we just have any discrete set , like {0,1,2} for instance ?
3) what is the 'Constant' in Constant Sheaf referring to ? is it because the target of the continuous functions is some discrete set ?