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<title>Small object argument</title>
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<h1 id="firstHeading" class="firstHeading mw-first-heading">
<span id="openzim-page-title" class="mw-page-title-main"><span class="mw-page-title-main">Small object argument</span></span>
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<div id="mw-content-text" class="mw-body-content mw-content-ltr" lang="en" dir="ltr"><div class="mw-content-ltr mw-parser-output" lang="en" dir="ltr"><p>In mathematics, especially in <a href="Category_theory" title="Category theory">category theory</a>, Quillen’s <b>small object argument</b>, when applicable, constructs a factorization of a morphism in a functorial way. In practice, it can be used to show some class of morphisms constitutes a <a href="Weak_factorization_system" class="mw-redirect" title="Weak factorization system">weak factorization system</a> in the theory of model categories.
</p><p>The argument was introduced by Quillen to construct a model structure on the category of (reasonable) topological spaces.<sup id="cite_ref-1" class="reference"><a href="#cite_note-1"><span class="cite-bracket">[</span>1<span class="cite-bracket">]</span></a></sup> The original argument was later refined by Garner.<sup id="cite_ref-2" class="reference"><a href="#cite_note-2"><span class="cite-bracket">[</span>2<span class="cite-bracket">]</span></a></sup>
</p>
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<div class="mw-heading mw-heading2"><h2 id="Statement">Statement</h2></div>
<p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> be a category that has all small colimits. We say an object <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>x</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle x}</annotation>
</semantics>
</math></span><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"></span> in it is <a href="Compact_object_(mathematics)" title="Compact object (mathematics)">compact</a> with respect to an ordinal <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> if <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \operatorname {Hom} (x,-)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>Hom</mi>
<mo>⁡<!-- ⁡ --></mo>
<mo stretchy="false">(</mo>
<mi>x</mi>
<mo>,</mo>
<mo>−<!-- − --></mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \operatorname {Hom} (x,-)}</annotation>
</semantics>
</math></span><img src="./3a3e4cb5bb83df57fc0fefd1c81089c2ee69d7a7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.822ex; height:2.843ex;" alt="{\displaystyle \operatorname {Hom} (x,-)}" loading="lazy"></span> commutes with an <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>-filterted colimit. In practice, we fix <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span> and simply say an object is compact if it is so with respect to that fixed <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \omega }">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>ω<!-- ω --></mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle \omega }</annotation>
</semantics>
</math></span><img src="./48eff443f9de7a985bb94ca3bde20813ea737be8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.446ex; height:1.676ex;" alt="{\displaystyle \omega }" loading="lazy"></span>.
</p><p>If <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is a class of morphisms, we write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(F)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l(F)}</annotation>
</semantics>
</math></span><img src="./63886df39102582236ed492fbcc062535d53a514.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.243ex; height:2.843ex;" alt="{\displaystyle l(F)}" loading="lazy"></span> for the class of morphisms that satisfy the <a href="Left_lifting_property" class="mw-redirect" title="Left lifting property">left lifting property</a> with respect to <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span>. Similarly, we write <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(F)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r(F)}</annotation>
</semantics>
</math></span><img src="./7bfcf9329bb69fe6dee65af42a4b2c0578e0e524.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:4.599ex; height:2.843ex;" alt="{\displaystyle r(F)}" loading="lazy"></span> for the right lifting property. Then
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<p><strong class="theorem-name">Theorem</strong><span class="theoreme-tiret">—</span><sup id="cite_ref-3" class="reference"><a href="#cite_note-3"><span class="cite-bracket">[</span>3<span class="cite-bracket">]</span></a></sup><sup id="cite_ref-4" class="reference"><a href="#cite_note-4"><span class="cite-bracket">[</span>4<span class="cite-bracket">]</span></a></sup> Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> be a class of morphisms in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span>. If the source (domain) of each morphism in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>F</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle F}</annotation>
</semantics>
</math></span><img src="./545fd099af8541605f7ee55f08225526be88ce57.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.741ex; height:2.176ex;" alt="{\displaystyle F}" loading="lazy"></span> is compact, then each morphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
</semantics>
</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> admits a functorial factorization <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=p\circ i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
<mo>=</mo>
<mi>p</mi>
<mo>∘<!-- ∘ --></mo>
<mi>i</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle f=p\circ i}</annotation>
</semantics>
</math></span><img src="./b6bd91d8829ae4d3c1f0489a1b5f0caef343358d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:2.509ex;" alt="{\displaystyle f=p\circ i}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i,p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
<mo>,</mo>
<mi>p</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle i,p}</annotation>
</semantics>
</math></span><img src="./ee7620190d64d074f3bfdbb183982faef7908510.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.006ex; height:2.509ex;" alt="{\displaystyle i,p}" loading="lazy"></span> are in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(r(F)),r(F)}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>l</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
<mo>,</mo>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mi>F</mi>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle l(r(F)),r(F)}</annotation>
</semantics>
</math></span><img src="./7934f2f088aac95127328a902e705bcd21277c52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.734ex; height:2.843ex;" alt="{\displaystyle l(r(F)),r(F)}" loading="lazy"></span>.
</p>
</div>
<div class="mw-heading mw-heading2"><h2 id="Example:_presheaf">Example: presheaf</h2></div>
<p>Here is a simple example of how the argument works in the case of the category <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle C}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>C</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle C}</annotation>
</semantics>
</math></span><img src="./4fc55753007cd3c18576f7933f6f089196732029.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.766ex; height:2.176ex;" alt="{\displaystyle C}" loading="lazy"></span> of presheaves on some small category.<sup id="cite_ref-5" class="reference"><a href="#cite_note-5"><span class="cite-bracket">[</span>5<span class="cite-bracket">]</span></a></sup>
</p><p>Let <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle I}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>I</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle I}</annotation>
</semantics>
</math></span><img src="./535ea7fc4134a31cbe2251d9d3511374bc41be9f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.172ex; height:2.176ex;" alt="{\displaystyle I}" loading="lazy"></span> denote the set of monomorphisms of the form <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle K\to L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>K</mi>
<mo stretchy="false">→<!-- → --></mo>
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle K\to L}</annotation>
</semantics>
</math></span><img src="./6ebdd971c4df7ed7e82835f5ed921cf5475d596d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.263ex; height:2.176ex;" alt="{\displaystyle K\to L}" loading="lazy"></span>, <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle L}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>L</mi>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle L}</annotation>
</semantics>
</math></span><img src="./103168b86f781fe6e9a4a87b8ea1cebe0ad4ede8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.583ex; height:2.176ex;" alt="{\displaystyle L}" loading="lazy"></span> a quotient of a representable presheaf. Then <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle l(r(I))}">
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<mi>l</mi>
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<mo stretchy="false">)</mo>
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<annotation encoding="application/x-tex">{\displaystyle l(r(I))}</annotation>
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</math></span><img src="./0b3d4aa6c251dac38366c0f00d0efb3319bbb572.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.532ex; height:2.843ex;" alt="{\displaystyle l(r(I))}" loading="lazy"></span> can be shown to be equal to the class of monomorphisms. Then the small object argument says: each presheaf morphism <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f}">
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<mstyle displaystyle="true" scriptlevel="0">
<mi>f</mi>
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<annotation encoding="application/x-tex">{\displaystyle f}</annotation>
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</math></span><img src="./132e57acb643253e7810ee9702d9581f159a1c61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:1.279ex; height:2.509ex;" alt="{\displaystyle f}" loading="lazy"></span> can be factored as <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f=p\circ i}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
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<annotation encoding="application/x-tex">{\displaystyle f=p\circ i}</annotation>
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</math></span><img src="./b6bd91d8829ae4d3c1f0489a1b5f0caef343358d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.544ex; height:2.509ex;" alt="{\displaystyle f=p\circ i}" loading="lazy"></span> where <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>i</mi>
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<annotation encoding="application/x-tex">{\displaystyle i}</annotation>
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</math></span><img src="./add78d8608ad86e54951b8c8bd6c8d8416533d20.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:0.802ex; height:2.176ex;" alt="{\displaystyle i}" loading="lazy"></span> is a monomorphism and <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
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<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> in <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle r(I)=r(l(r(I))}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>r</mi>
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<mo>=</mo>
<mi>r</mi>
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<mi>l</mi>
<mo stretchy="false">(</mo>
<mi>r</mi>
<mo stretchy="false">(</mo>
<mi>I</mi>
<mo stretchy="false">)</mo>
<mo stretchy="false">)</mo>
</mstyle>
</mrow>
<annotation encoding="application/x-tex">{\displaystyle r(I)=r(l(r(I))}</annotation>
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</math></span><img src="./d2d564dc6e843e3077f7ab1ecdce89717cabb224.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:15.614ex; height:2.843ex;" alt="{\displaystyle r(I)=r(l(r(I))}" loading="lazy"></span>; i.e., <span class="mwe-math-element mwe-math-element-inline"><span class="mwe-math-mathml-inline mwe-math-mathml-a11y" style="display: none;"><math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">
<semantics>
<mrow class="MJX-TeXAtom-ORD">
<mstyle displaystyle="true" scriptlevel="0">
<mi>p</mi>
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<annotation encoding="application/x-tex">{\displaystyle p}</annotation>
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</math></span><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy"></span> is a morphism having the right lifting property with respect to monomorphisms.
</p>
<div class="mw-heading mw-heading2"><h2 id="Proof">Proof</h2></div>
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<p>For now, see.<sup id="cite_ref-6" class="reference"><a href="#cite_note-6"><span class="cite-bracket">[</span>6<span class="cite-bracket">]</span></a></sup> But roughly the construction is a sort of successive approximation.
</p>
<div class="mw-heading mw-heading2"><h2 id="See_also">See also</h2></div>
<ul><li><a href="Anodyne_extension" class="mw-redirect" title="Anodyne extension">Anodyne extension</a></li></ul>
<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>
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<div class="mw-references-wrap"><ol class="references">
<li id="cite_note-1"><span class="mw-cite-backlink"><b><a href="#cite_ref-1">^</a></b></span> <span class="reference-text">D. G. Quillen. Homotopical algebra. Lecture Notes in Mathematics, No. 43. Springer-Verlag,
Berlin, 1967</span>
</li>
<li id="cite_note-2"><span class="mw-cite-backlink"><b><a href="#cite_ref-2">^</a></b></span> <span class="reference-text">Richard Garner, Understanding the small object argument, Applied Categorical Structures 17 3 247-285 (2009) [arXiv:0712.0724, doi:10.1007/s10485-008-9137-4]</span>
</li>
<li id="cite_note-3"><span class="mw-cite-backlink"><b><a href="#cite_ref-3">^</a></b></span> <span class="reference-text"><a href="#CITEREFCisinski2023">Cisinski 2023</a>, Proposition 2.1.9.</span>
</li>
<li id="cite_note-4"><span class="mw-cite-backlink"><b><a href="#cite_ref-4">^</a></b></span> <span class="reference-text"><a href="#CITEREFRiehl2014">Riehl 2014</a>, Theorem 12.2.2.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFRiehl2014 (help)</span></span>
</li>
<li id="cite_note-5"><span class="mw-cite-backlink"><b><a href="#cite_ref-5">^</a></b></span> <span class="reference-text"><a href="#CITEREFCisinski2023">Cisinski 2023</a>, Example 2.1.11. Second method</span>
</li>
<li id="cite_note-6"><span class="mw-cite-backlink"><b><a href="#cite_ref-6">^</a></b></span> <span class="reference-text"><a href="#CITEREFRiehl2014">Riehl 2014</a>, § 12.2. and § 12.5.<span class="error harv-error" style="display: none; font-size:100%"> harvnb error: no target: CITEREFRiehl2014 (help)</span></span>
</li>
</ol></div></div>
<ul><li>Mark Hovey, Model categories, volume 63 of Mathematical Surveys and Monographs, American Mathematical Society, (2007),</li>
<li>Emily Riehl, Categorical Homotopy Theory, Cambridge University Press (2014) <a rel="nofollow" class="external autonumber" href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">[1]</a></li>
<li><style data-mw-deduplicate="TemplateStyles:r1238218222">
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</style><cite id="CITEREFCisinski2023" class="citation book cs1"><a href="Denis-Charles_Cisinski" title="Denis-Charles Cisinski">Cisinski, Denis-Charles</a> (2023). <a rel="nofollow" class="external text" href="https://cisinski.app.uni-regensburg.de/CatLR.pdf"><i>Higher Categories and Homotopical Algebra</i></a> <span class="cs1-format">(PDF)</span>. <a href="Cambridge_University_Press" title="Cambridge University Press">Cambridge University Press</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a>&nbsp;<bdi>978-1108473200</bdi>.</cite></li></ul>
<div class="mw-heading mw-heading2"><h2 id="Further_reading">Further reading</h2></div>
<ul><li><a rel="nofollow" class="external free" href="https://ncatlab.org/nlab/show/small+object+argument">https://ncatlab.org/nlab/show/small+object+argument</a></li></ul></div><!--htdig_noindex--><div><div class="zim-footer">
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