Lemma 19.9.1. Let $\mathcal{A}$ be an abelian category. Let
\[ \text{Cov} = \{ \{ f : V \to U\} \mid f\text{ is surjective}\} . \]
Then $(\mathcal{A}, \text{Cov})$ is a site, see Sites, Definition 7.6.2.
19.9 Embedding abelian categories
In this section we show that an abelian category embeds in the category of abelian sheaves on a site having enough points. The site will be the one described in the following lemma.
Proof. Note that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ is a set by our conventions about categories. An isomorphism is a surjective morphism. The composition of surjective morphisms is surjective. And the base change of a surjective morphism in $\mathcal{A}$ is surjective, see Homology, Lemma 12.5.14. $\square$
Let $\mathcal{A}$ be a pre-additive category. In this case the Yoneda embedding $\mathcal{A} \to \textit{PSh}(\mathcal{A})$, $X \mapsto h_ X$ factors through a functor $\mathcal{A} \to \textit{PAb}(\mathcal{A})$.
Lemma 19.9.2. Let $\mathcal{A}$ be an abelian category. Let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ be the site defined in Lemma 19.9.1. Then $X \mapsto h_ X$ defines a fully faithful, exact functor Moreover, the site $\mathcal{C}$ has enough points.
\[ \mathcal{A} \longrightarrow \textit{Ab}(\mathcal{C}). \]
Proof. Suppose that $f : V \to U$ is a surjective morphism of $\mathcal{A}$. Let $K = \mathop{\mathrm{Ker}}(f)$. Recall that $V \times _ U V = \mathop{\mathrm{Ker}}((f, -f) : V \oplus V \to U)$, see Homology, Example 12.5.6. In particular there exists an injection $K \oplus K \to V \times _ U V$. Let $p, q : V \times _ U V \to V$ be the two projection morphisms. Note that $p - q : V \times _ U V \to V$ is a morphism such that $f \circ (p - q) = 0$. Hence $p - q$ factors through $K \to V$. Let us denote this morphism by $c : V \times _ U V \to K$. And since the composition $K \oplus K \to V \times _ U V \to K$ is surjective, we conclude that $c$ is surjective. It follows that
\[ V \times _ U V \xrightarrow {p - q} V \to U \to 0 \]
is an exact sequence of $\mathcal{A}$. Hence for an object $X$ of $\mathcal{A}$ the sequence
\[ 0 \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(U, X) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(V, X) \to \mathop{\mathrm{Hom}}\nolimits _\mathcal {A}(V \times _ U V, X) \]
is an exact sequence of abelian groups, see Homology, Lemma 12.5.8. This means that $h_ X$ satisfies the sheaf condition on $\mathcal{C}$.
The functor is fully faithful by Categories, Lemma 4.3.5. The functor is a left exact functor between abelian categories by Homology, Lemma 12.5.8. To show that it is right exact, let $X \to Y$ be a surjective morphism of $\mathcal{A}$. Let $U$ be an object of $\mathcal{A}$, and let $s \in h_ Y(U) = \mathop{\mathrm{Mor}}\nolimits _\mathcal {A}(U, Y)$ be a section of $h_ Y$ over $U$. By Homology, Lemma 12.5.14 the projection $U \times _ Y X \to U$ is surjective. Hence $\{ V = U \times _ Y X \to U\} $ is a covering of $U$ such that $s|_ V$ lifts to a section of $h_ X$. This proves that $h_ X \to h_ Y$ is a surjection of abelian sheaves, see Sites, Lemma 7.11.2.
The site $\mathcal{C}$ has enough points by Sites, Proposition 7.39.3. $\square$
Remark 19.9.3. The Freyd-Mitchell embedding theorem says there exists a fully faithful exact functor from any abelian category $\mathcal{A}$ to the category of modules over a ring. Lemma 19.9.2 is not quite as strong. But the result is suitable for the Stacks project as we have to understand sheaves of abelian groups on sites in detail anyway. Moreover, “diagram chasing” works in the category of abelian sheaves on $\mathcal{C}$, for example by working with sections over objects, or by working on the level of stalks using that $\mathcal{C}$ has enough points. To see how to deduce the Freyd-Mitchell embedding theorem from Lemma 19.9.2 see Remark 19.9.5.
Remark 19.9.4. If $\mathcal{A}$ is a “big” abelian category, i.e., if $\mathcal{A}$ has a class of objects, then Lemma 19.9.2 does not work. In this case, given any set of objects $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A})$ there exists an abelian full subcategory $\mathcal{A}' \subset \mathcal{A}$ such that $\mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$ is a set and $E \subset \mathop{\mathrm{Ob}}\nolimits (\mathcal{A}')$. Then one can apply Lemma 19.9.2 to $\mathcal{A}'$. One can use this to prove that results depending on a diagram chase hold in $\mathcal{A}$.
Remark 19.9.5. Let $\mathcal{C}$ be a site. Note that $\textit{Ab}(\mathcal{C})$ has enough injectives, see Theorem 19.7.4. (In the case that $\mathcal{C}$ has enough points this is straightforward because $p_*I$ is an injective sheaf if $I$ is an injective $\mathbf{Z}$-module and $p$ is a point.) Also, $\textit{Ab}(\mathcal{C})$ has a cogenerator (details omitted). Hence Lemma 19.9.2 proves that we have a fully faithful, exact embedding $\mathcal{A} \to \mathcal{B}$ where $\mathcal{B}$ has a cogenerator and enough injectives. We can apply this to $\mathcal{A}^{opp}$ and we get a fully faithful exact functor $i : \mathcal{A} \to \mathcal{D} = \mathcal{B}^{opp}$ where $\mathcal{D}$ has enough projectives and a generator. Hence $\mathcal{D}$ has a projective generator $P$. Set $R = \mathop{\mathrm{Mor}}\nolimits _\mathcal {D}(P, P)$. Then One can check this is a fully faithful, exact functor. In other words, one retrieves the Freyd-Mitchell theorem mentioned in Remark 19.9.3 above.
\[ \mathcal{A} \longrightarrow \text{Mod}_ R, \quad X \longmapsto \mathop{\mathrm{Hom}}\nolimits _\mathcal {D}(P, X). \]
Remark 19.9.6. The arguments proving Lemmas 19.9.1 and 19.9.2 work also for exact categories, see [Appendix A, Buhler] and [1.1.4, BBD]. We quickly review this here and we add more details if we ever need it in the Stacks project. Let $\mathcal{A}$ be an additive category. A kernel-cokernel pair is a pair $(i, p)$ of morphisms of $\mathcal{A}$ with $i : A \to B$, $p : B \to C$ such that $i$ is the kernel of $p$ and $p$ is the cokernel of $i$. Given a set $\mathcal{E}$ of kernel-cokernel pairs we say $i : A \to B$ is an admissible monomorphism if $(i, p) \in \mathcal{E}$ for some morphism $p$. Similarly we say a morphism $p : B \to C$ is an admissible epimorphism if $(i, p) \in \mathcal{E}$ for some morphism $i$. The pair $(\mathcal{A}, \mathcal{E})$ is said to be an exact category if the following axioms hold
$\mathcal{E}$ is closed under isomorphisms of kernel-cokernel pairs,
for any object $A$ the morphism $1_ A$ is both an admissible epimorphism and an admissible monomorphism,
admissible monomorphisms are stable under composition,
admissible epimorphisms are stable under composition,
the push-out of an admissible monomorphism $i : A \to B$ via any morphism $A \to A'$ exist and the induced morphism $i' : A' \to B'$ is an admissible monomorphism, and
the base change of an admissible epimorphism $p : B \to C$ via any morphism $C' \to C$ exist and the induced morphism $p' : B' \to C'$ is an admissible epimorphism.
Given such a structure let $\mathcal{C} = (\mathcal{A}, \text{Cov})$ where coverings (i.e., elements of $\text{Cov}$) are given by admissible epimorphisms. The axioms listed above immediately imply that this is a site. Consider the functor
\[ F : \mathcal{A} \longrightarrow \textit{Ab}(\mathcal{C}), \quad X \longmapsto h_ X \]
exactly as in Lemma 19.9.2. It turns out that this functor is fully faithful, exact, and reflects exactness. Moreover, any extension of objects in the essential image of $F$ is in the essential image of $F$.
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