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The Stacks project

Definition 6.3.1. Let X be a topological space.

  1. A presheaf \mathcal{F} of sets on X is a rule which assigns to each open U \subset X a set \mathcal{F}(U) and to each inclusion V \subset U a map \rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V) such that \rho ^ U_ U = \text{id}_{\mathcal{F}(U)} and whenever W \subset V \subset U we have \rho ^ U_ W = \rho ^ V_ W \circ \rho ^ U_ V.

  2. A morphism \varphi : \mathcal{F} \to \mathcal{G} of presheaves of sets on X is a rule which assigns to each open U \subset X a map of sets \varphi : \mathcal{F}(U) \to \mathcal{G}(U) compatible with restriction maps, i.e., whenever V \subset U \subset X are open the diagram

    \xymatrix{ \mathcal{F}(U) \ar[r]^\varphi \ar[d]^{\rho ^ U_ V} & \mathcal{G}(U) \ar[d]^{\rho ^ U_ V} \\ \mathcal{F}(V) \ar[r]^\varphi & \mathcal{G}(V) }

    commutes.

  3. The category of presheaves of sets on X will be denoted \textit{PSh}(X).


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