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  "body": "\n## Definition\n\nA *model category* is a [[category|A2DE66937D]] $\\mathcal{M}$ with all small [[limits|CHS5AFJGWE]] and [[colimits|BJS1CGTKQ5]], together with three classes of morphisms.\nThey are the [[weak equivalences|E2870TEDG7]] $W$, the *fibrations* $\\mathrm{Fib}$ and the *cofibrations* $\\mathrm{Cof}$.\nEach class contains the identities and is closed under composition.\nA morphism in $\\mathrm{Fib} \\cap W$ is a *trivial fibration*, and a morphism in $\\mathrm{Cof} \\cap W$ is a *trivial cofibration*.\nThe classes satisfy the following axioms.\n\n1. (two out of three) If $f$ and $g$ are composable and two of $f$, $g$, $g \\circ f$ are in $W$, then so is the third.\n2. (retracts) Each of $W$, $\\mathrm{Fib}$ and $\\mathrm{Cof}$ is closed under [[retracts|DXJH3DGXBC]] in the category of arrows of $\\mathcal{M}$.\n3. (lifting) Cofibrations have the left [[lifting property|AZK4RD8XV0]] with respect to trivial fibrations, and trivial cofibrations have the left lifting property with respect to fibrations.\n4. (factorization) Every morphism $f$ factors as $f = p \\circ i$ with $i$ a trivial cofibration and $p$ a fibration. It also factors as $f = q \\circ j$ with $j$ a cofibration and $q$ a trivial fibration.\n\n## Remarks\n\nSome authors ask only for finite limits and colimits, as Quillen did, and some ask the factorizations in axiom 4 to be functorial.\n\nExample: let $\\mathcal{C}$ be any category with all small limits and colimits, let $W$ be the isomorphisms, and let $\\mathrm{Fib}$ and $\\mathrm{Cof}$ be all morphisms.\nIsomorphisms satisfy two out of three and are closed under retracts.\nEvery square lifts against an isomorphism, which settles axiom 3, and $f = f \\circ \\id = \\id \\circ f$ gives axiom 4.\n\nNon-example: in the category of sets, let all three classes be all maps.\nThen the map $i \\colon \\set{0, 1} \\to \\set{0}$ is a cofibration and a trivial fibration.\nConsider the commutative square with $i$ on both vertical sides, $\\id$ on $\\set{0, 1}$ on top and $\\id$ on $\\set{0}$ below.\nA lift would be a map $h \\colon \\set{0} \\to \\set{0, 1}$ with $h \\circ i = \\id$, which does not exist, so axiom 3 fails.\n\nThe main example is the Quillen model structure on topological spaces: weak equivalences are weak homotopy equivalences, fibrations are Serre fibrations, and cofibrations are retracts of relative cell complexes.\nAn object $X$ is *fibrant* if $X \\to \\ast$ is a fibration and *cofibrant* if $\\emptyset \\to X$ is a cofibration.\n",
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