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  "body": "\n## Definition\n\nA morphism is a *regular epimorphism* if it is the [[coequalizer|DVSTWGVCSZ]] of some pair of morphisms.\nA category $\\mathcal{C}$ is *regular* if these three conditions hold.\n\n1. $\\mathcal{C}$ is [[finitely complete|PYSBNWCHCF]].\n2. For every morphism $f$ of $\\mathcal{C}$, the [[kernel pair|E70DQGM8X5]] of $f$ has a coequalizer.\n3. Regular epimorphisms are stable under [[pullback|Q05K04QJ71]]: the pullback of a regular epimorphism $e \\colon A \\to B$ along any $g \\colon C \\to B$ is a regular epimorphism.\n\n## Remarks\n\nA regular epimorphism is an [[epimorphism|RVXDSY04KG]].\nIn a regular category every morphism $f$ factors as $f = m \\circ e$ with $e$ a regular epimorphism and $m$ a monomorphism.\nHere $e$ is the coequalizer of the kernel pair of $f$, and $m$ is called the image of $f$.\n\nExample: the category of sets is regular.\nIt has all finite limits, and the kernel pair of $f \\colon A \\to B$ is the relation $\\set{(a, a') : f(a) = f(a')}$, whose coequalizer is the quotient of $A$ by it.\nRegular epimorphisms are the surjections, and a pullback of a surjection is a surjection.\n\nNon-example: the category of topological spaces satisfies clauses 1 and 2 but not clause 3.\nIts regular epimorphisms are the quotient maps, and a pullback of a quotient map need not be a quotient map.\n",
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