Exact category

Definition

Let $\mathcal{E}$ be an additive category. A kernel–cokernel pair is a pair of composable morphisms $A \xrightarrow{i} B \xrightarrow{p} C$ such that $i$ is a kernel of $p$ and $p$ is a cokernel of $i$. An exact structure on $\mathcal{E}$ is a class $\mathcal{S}$ of kernel–cokernel pairs, closed under isomorphism of pairs, whose members are called conflations. In a conflation $(i, p)$, the morphism $i$ is called an inflation and $p$ a deflation. The class $\mathcal{S}$ must satisfy:

1. For every object $A$, the identity $1_A$ is an inflation and a deflation.
2. A composite of two inflations is an inflation, and a composite of two deflations is a deflation.
3. For every inflation $i \colon A \to B$ and every morphism $f \colon A \to A'$, the pushout of $i$ along $f$ exists and is an inflation $A' \to B'$.
4. For every deflation $p \colon B \to C$ and every morphism $g \colon C' \to C$, the pullback of $p$ along $g$ exists and is a deflation $B' \to C'$.

An exact category is an additive category together with an exact structure.