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

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


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

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