The Stacks project

7.11 Injective and surjective maps of sheaves

Definition 7.11.1. Let $\mathcal{C}$ be a site, and let $\varphi : \mathcal{F} \to \mathcal{G}$ be a map of sheaves of sets.

  1. We say that $\varphi $ is injective if for every object $U$ of $\mathcal{C}$ the map $\varphi : \mathcal{F}(U) \to \mathcal{G}(U)$ is injective.

  2. We say that $\varphi $ is surjective if for every object $U$ of $\mathcal{C}$ and every section $s\in \mathcal{G}(U)$ there exists a covering $\{ U_ i \to U\} $ such that for all $i$ the restriction $s|_{U_ i}$ is in the image of $\varphi : \mathcal{F}(U_ i) \to \mathcal{G}(U_ i)$.

Lemma 7.11.2. The injective (resp. surjective) maps defined above are exactly the monomorphisms (resp. epimorphisms) of the category $\mathop{\mathit{Sh}}\nolimits (\mathcal{C})$. A map of sheaves is an isomorphism if and only if it is both injective and surjective.

Proof. Omitted. $\square$

Lemma 7.11.3. Let $\mathcal{C}$ be a site. Let $\mathcal{F} \to \mathcal{G}$ be a surjection of sheaves of sets. Then the diagram

\[ \xymatrix{ \mathcal{F} \times _\mathcal {G} \mathcal{F} \ar@<1ex>[r] \ar@<-1ex>[r] & \mathcal{F} \ar[r] & \mathcal{G}} \]

represents $\mathcal{G}$ as a coequalizer.

Proof. Let $\mathcal{H}$ be a sheaf of sets and let $\varphi : \mathcal{F} \to \mathcal{H}$ be a map of sheaves equalizing the two maps $\mathcal{F} \times _\mathcal {G} \mathcal{F} \to \mathcal{F}$. Let $\mathcal{G}' \subset \mathcal{G}$ be the presheaf image of the map $\mathcal{F} \to \mathcal{G}$. As the product $\mathcal{F} \times _\mathcal {G} \mathcal{F}$ may be computed in the category of presheaves we see that it is equal to the presheaf product $\mathcal{F} \times _{\mathcal{G}'} \mathcal{F}$. Hence $\varphi $ induces a unique map of presheaves $\psi ' : \mathcal{G}' \to \mathcal{H}$. Since $\mathcal{G}$ is the sheafification of $\mathcal{G}'$ by Lemma 7.11.2 we conclude that $\psi '$ extends uniquely to a map of sheaves $\psi : \mathcal{G} \to \mathcal{H}$. We omit the verification that $\varphi $ is equal to the composition of $\psi $ and the given map. $\square$

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