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  "body": "\n## Summary\n\nA category is exact in the sense of Barr, or Barr-exact, when it is a [[regular category|2SM3JRJXA4]] with effective equivalence relations.\nThis means that every internal equivalence relation is the kernel pair of some morphism.\nSo quotients by equivalence relations exist and behave as in sets.\nEvery elementary topos, every abelian category, the category of sets and every variety of algebras such as groups or rings are Barr-exact.\nThis notion is different from the exact categories of Quillen used in algebraic K-theory.\n\n## Explanations\n\n### L1 human\n\nIn sets, any equivalence relation on $X$ is the relation \"has the same image\" for the quotient map $X \\to X / {\\sim}$.\nA Barr-exact category is one where this holds for every equivalence relation expressed inside the category.\nFor groups, an equivalence relation that is a subgroup of $G \\times G$ is the relation \"same coset\" for a normal subgroup, and the quotient group provides the map.\n",
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