The Stacks project

6.3 Presheaves

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 $\textit{PSh}(X)$.

The elements of the set $\mathcal{F}(U)$ are called the sections of $\mathcal{F}$ over $U$. For every $V \subset U$ the map $\rho ^ U_ V : \mathcal{F}(U) \to \mathcal{F}(V)$ is called the restriction map. We will use the notation $s|_ V := \rho ^ U_ V(s)$ if $s\in \mathcal{F}(U)$. This notation is consistent with the notion of restriction of functions from topology because if $W \subset V \subset U$ and $s$ is a section of $\mathcal{F}$ over $U$ then $s|_ W = (s|_ V)|_ W$ by the property of the restriction maps expressed in the definition above.

Another notation that is often used is to indicate sections over an open $U$ by the symbol $\Gamma (U, -)$ or by $H^0(U, -)$. In other words, the following equalities are tautological

\[ \Gamma (U, \mathcal{F}) = \mathcal{F}(U) = H^0(U, \mathcal{F}). \]

In this chapter we will not use this notation, but in others we will.

Definition 6.3.2. Let $X$ be a topological space. Let $A$ be a set. The constant presheaf with value $A$ is the presheaf that assigns the set $A$ to every open $U \subset X$, and such that all restriction mappings are $\text{id}_ A$.

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