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  "body": "\n## Definition\n\nLet $\\mathcal{E}$ be an [[additive category|B2VP96NJSK]].\nA *kernel–cokernel pair* is a pair of composable morphisms $A \\xrightarrow{i} B \\xrightarrow{p} C$ such that $i$ is a [[kernel|S417YNEJ6M]] of $p$ and $p$ is a [[cokernel|H8W38933NC]] of $i$.\nAn *exact structure* on $\\mathcal{E}$ is a class $\\mathcal{S}$ of kernel–cokernel pairs, closed under isomorphism of pairs, whose members are called *conflations*.\nIn a conflation $(i, p)$, the morphism $i$ is called an *inflation* and $p$ a *deflation*.\nThe class $\\mathcal{S}$ must satisfy:\n\n1. For every object $A$, the identity $1_A$ is an inflation and a deflation.\n2. A composite of two inflations is an inflation, and a composite of two deflations is a deflation.\n3. For every inflation $i \\colon A \\to B$ and every morphism $f \\colon A \\to A'$, the [[pushout|Q699KKMVJD]] of $i$ along $f$ exists and is an inflation $A' \\to B'$.\n4. For every deflation $p \\colon B \\to C$ and every morphism $g \\colon C' \\to C$, the [[pullback|Q05K04QJ71]] of $p$ along $g$ exists and is a deflation $B' \\to C'$.\n\nAn *exact category* is an additive category together with an exact structure.\n\n## Remarks\n\nThis is Quillen's notion, in the streamlined form of Keller.\nInflations are also called admissible monomorphisms, and deflations admissible epimorphisms.\n\nExample: every [[abelian category|Y9CZ3MTNF1]] with the class of all short exact sequences is an exact category.\nEvery morphism has a kernel and a cokernel, pushouts and pullbacks exist, and pushouts of monomorphisms are monomorphisms while pullbacks of epimorphisms are epimorphisms.\nExample: every additive category with the class of split exact sequences $A \\to A \\oplus C \\to C$ is an exact category.\n\nNon-example: in the category of abelian groups, the class of short exact sequences whose three terms are finite groups fails clause 1.\nThe identity of $\\Z$ is not an inflation, because the only kernel–cokernel pair $\\Z \\xrightarrow{1} \\Z \\to 0$ that contains it has an infinite term.\n",
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