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  "body": "\n## Definition\n\nLet $\\mathcal{C}$ be a [[category|A2DE66937D]], and let $E$ and $M$ be classes of morphisms of $\\mathcal{C}$.\nThe pair $(E, M)$ is an *orthogonal factorization system* if these three conditions hold.\n\n1. Both $E$ and $M$ contain all isomorphisms and are closed under composition.\n2. Every morphism $f$ of $\\mathcal{C}$ factors as $f = m \\circ e$ with $e \\in E$ and $m \\in M$.\n3. Every $e \\in E$ is orthogonal to every $m \\in M$: it has the [[left lifting property|AZK4RD8XV0]] with respect to $m$, with a unique filler.\n\n## Remarks\n\nConditions 1 to 3 imply that the factorization in clause 2 is unique up to a unique isomorphism.\nThey also imply that $E$ is exactly the class of morphisms orthogonal to all of $M$, and conversely.\nA *weak factorization system* asks instead for factorizations and for liftings without uniqueness.\nIt also asks that $E$ and $M$ be exactly the classes of maps with the left and the right lifting property against each other.\n\nExample: in the category of sets, let $E$ be the [[surjections|01RVCMSN5N]] and $M$ the [[injections|09NWETVTNY]].\nA function $f \\colon A \\to B$ factors as $A \\to f(A) \\to B$.\nFor orthogonality, take a square with $e \\colon A \\to B$ surjective, $m \\colon C \\to D$ injective, and $v \\circ e = m \\circ u$.\nDefine $d(b) = u(a)$ for any $a$ with $e(a) = b$; this does not depend on $a$, since $m$ is injective and $m(u(a)) = v(b)$.\nThen $d$ is the unique filler.\n\nNon-example: in sets, let $E$ and $M$ both be the class of all functions.\nConditions 1 and 2 hold, but condition 3 fails.\nFor $e \\colon \\emptyset \\to \\set{*}$ and $m \\colon \\set{0, 1} \\to \\set{*}$, the square with $v = \\id$ has two fillers $\\set{*} \\to \\set{0, 1}$.\n",
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