The Stacks project

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

  1. $\mathcal{E}$ is closed under isomorphisms of kernel-cokernel pairs,

  2. for any object $A$ the morphism $1_ A$ is both an admissible epimorphism and an admissible monomorphism,

  3. admissible monomorphisms are stable under composition,

  4. admissible epimorphisms are stable under composition,

  5. 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

  6. 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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