Definition 3

Let B be a topological space, $\mathscr{B}$ a basis of the topology of B$,\mathscr{F}= (\mathscr{F}(U), f_{UV})$ and $\mathscr{G}= (\mathscr{G}(U), g_{UV})$ presheaves on B relative to $\mathscr{B}$. A morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$ is called a projective system of mappings from $\mathscr{F}$ to $\mathscr{G}$.

In other words (E, III, p. 54), a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$ is a family $(\varphi_U)_{U\in\mathscr{B}}$ such that :

(MPF$_1$) For every open set U belonging to $\mathscr{B},\varphi_U$ is a mapping

from $\mathscr{F}(U)$ into $\mathscr{G}(U)$ ;

(MPF$_2$) For every pair $(U,V)$ of open sets belonging to $\mathscr{B}$ such that

$U\subset V$, one has $\varphi_U\circ f_{UV}=g_{UV}\circ \varphi_V$.

When $\mathscr{F}$ and $\mathscr{G}$ are sheaves, a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$ is also called a morphism of sheaves. If $\mathscr{F}$ and $\mathscr{G}$ are presheaves on B relative to $\mathscr{B}$, the morphisms of presheaves from $\mathscr{F}$ to $\mathscr{G}$ constitute a set denoted Mor($\mathscr{F};\mathscr{G}$). Instead of saying : “let $\varphi$ be a morphism of presheaves from $\mathscr{F}$ to $\mathscr{G}$”, one will often say “let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$ be a morphism of presheaves”.

Let $\mathscr{F},\mathscr{G},\mathscr{H}$ be presheaves on B relative to $\mathscr{B}$ and let $\varphi :\mathscr{F}\rightarrow \mathscr{G}$, $\psi :\mathscr{G}\rightarrow \mathscr{H}$ be morphisms of presheaves. The family $(\psi_U\circ \varphi_U)_{U\in\mathscr{B}}$ is a morphism of presheaves from $\mathscr{F}$ into $\mathscr{H}$ which is denoted by $\psi \circ \varphi$. The family (Id$_{\mathscr{F}(U)}$)$_{U\in\mathscr{B}}$ is a morphism of presheaves from $\mathscr{F}$ into itself which is denoted by Id$_{\mathscr{F}}$.

For a morphism of presheaves $\varphi = (\varphi_U):\mathscr{F}\rightarrow \mathscr{G}$ to be an isomorphism, it is necessary and sufficient that, for every open subset U of $\mathscr{B},\varphi_U$ be a bijection from $\mathscr{F}(U)$ onto $\mathscr{G}(U)$. It is equivalent to saying that there exists a morphism of presheaves $\psi :\mathscr{G}\rightarrow \mathscr{F}$ such that $\psi \circ \varphi =$ Id$_{\mathscr{F}}$ and $\varphi \circ \psi =$ Id$_{\mathscr{G}}$.

Let $\mathscr{F}$ and $\mathscr{G}$ be presheaves on B, relative to a base $\mathscr{B}$ of the topology of B, let $B'$ be an open subset of B and let $\mathscr{B}'$ be a base of the topology of $B'$ such that $\mathscr{B}'\subset \mathscr{B}$. Let $\varphi = (\varphi_U)_{U\in\mathscr{B}}$ be a morphism of presheaves from $\mathscr{F}$ into $\mathscr{G}$. Then $(\varphi_U)_{U\in\mathscr{B}'}$ is a morphism of presheaves from $\mathscr{F}|\mathscr{B}'$ into $\mathscr{G}|\mathscr{B}'$, which is denoted by $\varphi |\mathscr{B}'$. When $\mathscr{B}$ is the set of open subsets of B and $\mathscr{B}'$ the set of open subsets of $B',\varphi |\mathscr{B}'$ is a morphism of presheaves from $\mathscr{F}|B'$ into $\mathscr{G}|B'$ and is also denoted by $\varphi |B'$.