Example 6.15.6. Let $F : \mathcal{C} \to \textit{Sets}$ be a type of algebraic structures. Let $X$ be a topological space. Suppose that for every $x \in X$ we are given an object $A_ x \in \mathop{\mathrm{Ob}}\nolimits (\mathcal{C})$. Consider the presheaf $\Pi $ with values in $\mathcal{C}$ on $X$ defined by the rule $\Pi (U) = \prod _{x \in U} A_ x$ (with obvious restriction mappings). Note that the associated presheaf of sets $U \mapsto F(\Pi (U)) = \prod _{x \in U} F(A_ x)$ is a sheaf by Example 6.7.5. Hence $\Pi $ is a sheaf of algebraic structures of type $(\mathcal{C} , F)$. This gives many examples of sheaves of abelian groups, groups, rings, etc.

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