Definition 6.30.1. Let $X$ be a topological space. Let $\mathcal{B}$ be a basis for the topology on $X$.

A

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

*morphism $\varphi : \mathcal{F} \to \mathcal{G}$ of presheaves of sets on $\mathcal{B}$*is a rule which assigns to each element $U \in \mathcal{B}$ a map of sets $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ compatible with restriction maps.

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Comment #3218 by Kevin Buzzard on

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