Abelian sheaves form a Grothendieck category

A morphism $\phi :\mathcal{F}\to \mathcal{G}$ of abelian sheaves is a family of group homomorphisms ${\phi }_U:\mathcal{F}(U)\to \mathcal{G}(U)$ over the open sets of $X$, compatible with restriction, and $\phi =\psi$ exactly when ${\phi }_U={\psi }_U$ for all $U$ [F1]. Hence $\mathrm{Hom}(\mathcal{F},\mathcal{G})$ is a subset of the product ${\prod }_U{\mathrm{Hom}}_{\mathbf{A}\mathbf{b}}(\mathcal{F}(U),\mathcal{G}(U))$ indexed by the set of opens, which is a set. Thus $\mathrm{A}\mathrm{b}(X)$ is locally small.

Let $({\mathcal{F}}_i{)}_{i\in I}$ be a family of abelian sheaves indexed by a set $I$ and let $P$ be the presheaf $U\mapsto {⨁}_i{\mathcal{F}}_i(U)$ with componentwise restriction maps. For an abelian sheaf $\mathcal{G}$, a presheaf morphism $P\to \mathcal{G}$ is exactly a family of morphisms ${\mathcal{F}}_i\to \mathcal{G}$ (the universal property of the direct sum of groups, applied over each open set), and [F2] converts presheaf morphisms $P\to \mathcal{G}$ into sheaf morphisms $aP\to \mathcal{G}$. So $\mathrm{Hom}(aP,\mathcal{G})\cong {\prod }_i\mathrm{Hom}({\mathcal{F}}_i,\mathcal{G})$, naturally in $\mathcal{G}$, and $aP$ is a coproduct of the family. Hence coproducts indexed by sets exist in $\mathrm{A}\mathrm{b}(X)$.

Filtered colimits of abelian groups are exact: by [F7] it suffices to treat $\mathbb{Z}$-modules, and $\mathbb{Z}\text{-}\mathbf{M}\mathbf{o}\mathbf{d}$ is a Grothendieck category [F8], hence satisfies AB5, so by [F6] every filtered colimit functor on it is exact. Thus for a small filtered diagram of short exact sequences $0\to A_i\to B_i\to C_i\to 0$ of abelian groups the colimit sequence $0\to {\mathrm{colim}}_i{A}_i\to {\mathrm{colim}}_i{B}_i\to {\mathrm{colim}}_i{C}_i\to 0$ is short exact.

Let $D$ be a small diagram in $\mathrm{A}\mathrm{b}(X)$. Because coequalizers exist in the abelian category $\mathrm{A}\mathrm{b}(X)$ [F1] and small coproducts exist by step 1.2, the coequalizer of the two canonical maps ${\coprod }_f{\mathcal{F}}_{f(0)}\rightrightarrows {\coprod }_i{\mathcal{F}}_i$ built from the diagram maps and the identities exists and satisfies the universal property of $\mathrm{colim}D$. Hence $\mathrm{A}\mathrm{b}(X)$ is cocomplete and satisfies AB3. [F1, step 1.2, construct]

For an open $U\subseteq X$ let $j_U:U\hookrightarrow X$ be the inclusion and let ${\mathbb{Z}}_U$ be the sheaf on $U$ associated with the constant presheaf with value $\mathbb{Z}$; write $G_U:=j_{U!}{\mathbb{Z}}_U$. For every abelian sheaf $\mathcal{F}$ on $X$, [F5] and the identification $j_U^{-1}\mathcal{F}=\mathcal{F}{\mathrm{\mid }}_U$ give ${\mathrm{Hom}}_X(G_U,\mathcal{F})\cong {\mathrm{Hom}}_U({\mathbb{Z}}_U,\mathcal{F}{\mathrm{\mid }}_U)$. A morphism ${\mathbb{Z}}_U\to \mathcal{F}{\mathrm{\mid }}_U$ corresponds, by the universal property of sheafification [F2], to a presheaf morphism out of the constant presheaf, which is exactly the data of an element $s\in \mathcal{F}(U)$ (the image of $1\in \mathbb{Z}$, the value of the constant presheaf on $U$, with compatibility forced by the restriction maps of the constant presheaf); conversely every $s\in \mathcal{F}(U)$ gives such a morphism by $n\mapsto n\cdot s{\mathrm{\mid }}_V$ over $V\subseteq U$. Hence ${\mathrm{Hom}}_X(G_U,\mathcal{F})\cong \mathcal{F}(U)$, naturally in $\mathcal{F}$. The coproduct $G:={\coprod }_U{G}_U$ over the set of all open subsets exists by step 1.2 and satisfies $\mathrm{Hom}(G,\mathcal{F})\cong {\prod }_U\mathcal{F}(U)$ naturally in $\mathcal{F}$. [F2, F5, step 1.2, construct]

Let $({\mathcal{F}}_i{)}_{i\in \mathcal{J}}$ be a small filtered diagram of abelian sheaves with objectwise colimit presheaf $P$, so that ${\mathrm{colim}}_i{\mathcal{F}}_i=aP$ by step 2.1 and step 1.2. Colimits of presheaves are computed objectwise, since a cocone on the diagram is exactly a compatible family of cocones over the open sets. Fix $x\in X$. An element of $P_x$ is represented by a pair $(i,s)$ with $s\in {\mathcal{F}}_i(W)$ for some open neighbourhood $W$ of $x$ [F9], and by [F10] applied to the filtered index categories $\mathcal{J}$ and ${\mathcal{N}}_x^{\mathrm{o}\mathrm{p}}$ two such pairs $(i_1,W_1,s_1)$ and $(i_2,W_2,s_2)$ have the same image in $P_x$ if and only if they can be refined to a common $(i_3,W_3,s_3)$; the same relation describes equality in ${\mathrm{colim}}_i({\mathcal{F}}_i{)}_x$, because $({\mathcal{F}}_i{)}_x={\mathrm{colim}}_{W\ni x}{\mathcal{F}}_i(W)$ [F9]. Both sides are therefore the filtered colimit of the same diagram $\mathcal{J}\times {\mathcal{N}}_x^{\mathrm{o}\mathrm{p}}\to \mathbf{A}\mathbf{b}$, and the canonical comparison is an isomorphism of abelian groups, compatible with the maps from the diagram. Composing with the bijection $(aP{)}_x\cong P_x$ of [F4] gives a natural isomorphism $({\mathrm{colim}}_i{\mathcal{F}}_i{)}_x\cong {\mathrm{colim}}_i({\mathcal{F}}_i{)}_x$: stalks of sheaves commute with filtered colimits. [F4, F9, F10, step 2.1, algebra]

Let $f\mathrm{\ne }g:\mathcal{F}\to {\mathcal{F}}_0$ be distinct morphisms of abelian sheaves. Since morphisms are their families of components [F1], $f-g\mathrm{\ne }0$ gives an open $U$ and a section $s\in \mathcal{F}(U)$ with $(f-g{)}_U(s)\mathrm{\ne }0$. Under the bijection of step 2.2 the pair $(U,s)$ corresponds to a morphism $h:G_U\to \mathcal{F}$; the coproduct universal property extends $h$ by zero on all other summands to a morphism $\bar{h}:G\to \mathcal{F}$; its composite with the injection $G_U\to G$ is $h$, so $(f-g)\circ \bar{h}\mathrm{\ne }0$, that is $f\circ \bar{h}\mathrm{\ne }g\circ \bar{h}$. By [F11] the singleton family $\{G\}$ is separating, so $G$ is a generator of $\mathrm{A}\mathrm{b}(X)$. [F1, F11, step 2.2]

Let $0\to {\mathcal{A}}_i\to {\mathcal{B}}_i\to {\mathcal{C}}_i\to 0$ be a small filtered diagram of short exact sequences of abelian sheaves, i.e. a short exact sequence of diagrams, and fix $x\in X$. The stalk sequences $0\to ({\mathcal{A}}_i{)}_x\to ({\mathcal{B}}_i{)}_x\to ({\mathcal{C}}_i{)}_x\to 0$ are exact by [F3], and by step 3.1 the stalk at $x$ of the colimit diagram is the filtered colimit of these exact sequences, which is short exact by step 1.3. Since exactness of sequences of sheaves is stalkwise [F3], the colimit sequence $0\to {\mathrm{colim}}_i{\mathcal{A}}_i\to {\mathrm{colim}}_i{\mathcal{B}}_i\to {\mathrm{colim}}_i{\mathcal{C}}_i\to 0$ is short exact. Thus every small filtered colimit functor on $\mathrm{A}\mathrm{b}(X)$ preserves short exact sequences; it is additive and preserves finite coproducts because it is a left adjoint (its right adjoint is the constant diagram functor) and preserves the zero object, and preservation of kernels and cokernels follows from the same stalkwise computation applied to the exact sequences $0\to \ker \to \cdot \to \mathrm{im}\to 0$. Hence every small filtered colimit functor on $\mathrm{A}\mathrm{b}(X)$ is exact. [F3, step 1.3, step 3.1, algebra]

By step 2.1 the category $\mathrm{A}\mathrm{b}(X)$ is cocomplete and abelian [F1], so the equivalence [F6] applies; step 4.1 shows that every small filtered colimit functor on it is exact, so $\mathrm{A}\mathrm{b}(X)$ satisfies AB5. [F1, F6, step 2.1, step 4.1]

The steps 1.1, 2.1, 5.1 and 3.2 show that $\mathrm{A}\mathrm{b}(X)$ is locally small, abelian, cocomplete, satisfies AB5 and has a generator; by [L1] the last three properties together with the abelian structure make it a Grothendieck category.