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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="factorization_systems">Factorization systems</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/factorization+system">factorization system</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">orthogonal</a>, <a class="existingWikiWord" href="/nlab/show/strict+factorization+system">strict</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+system">algebraic weak</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+factorization+system">enriched</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+factorization+system">reflective</a>, <a class="existingWikiWord" href="/nlab/show/stable+factorization+system">stable</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/factorization+system+over+a+subcategory">factorization system over a subcategory</a></p> <p><a class="existingWikiWord" href="/nlab/show/k-ary+factorization+system">k-ary factorization system</a>, <a class="existingWikiWord" href="/nlab/show/ternary+factorization+system">ternary factorization system</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/factorization+system+in+a+2-category">factorization system in a 2-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/factorization+system+in+an+%28%E2%88%9E%2C1%29-category">factorization system in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system+in+an+%28%E2%88%9E%2C1%29-category">orthogonal</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28epi%2C+mono%29+factorization+system">(epi, mono)</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n-epi%2C+n-mono%29+factorization+system">(n-epi, n-mono)</a>/<a class="existingWikiWord" href="/nlab/show/n-image">n-image</a></p> <p><a class="existingWikiWord" href="/nlab/show/Postnikov+system">Postnikov system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28eso%2C+fully+faithful%29+factorization+system">(eso, fully faithful)</a>,</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/%28eso%2Bfull%2C+faithful%29+factorization+system">(eso+full, faithful)</a>, <a class="existingWikiWord" href="/nlab/show/%28bo%2C+ff%29+factorization+system">(bo, ff)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28hyperconnected%2C+localic%29+factorization+system">(hyperconnected, localic)</a></li> </ul> </div></div> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#weak_factorization_systems'>Weak factorization systems</a></li> <li><a href='#orthogonal_factorization_systems'>Orthogonal factorization systems</a></li> <li><a href='#algebraic_weak_factorization_systems'>Algebraic weak factorization systems</a></li> <li><a href='#accessible_weak_factorization_systems'>Accessible weak factorization systems</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#ClosureProperties'>Closure properties</a></li> <li><a href='#retract_argument'>Retract argument</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>weak factorization system</em> on a <a class="existingWikiWord" href="/nlab/show/category">category</a> is a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{L},\mathcal{R})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> (“<a class="existingWikiWord" href="/nlab/show/projective+morphisms">projective morphisms</a>” and “<a class="existingWikiWord" href="/nlab/show/injective+morphisms">injective morphisms</a>”) such that 1) every morphism of the category factors as the composite of one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> followed by one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math>, and 2) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> are closed under having the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> against each other.</p> <p>If the liftings here are unique, then one speaks of an <em>orthogonal factorization system</em>. A classical example of an orthogonal factorization system is the <a class="existingWikiWord" href="/nlab/show/%28epi%2Cmono%29-factorization+system">(epi,mono)-factorization system</a> on the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> or in fact on any <a class="existingWikiWord" href="/nlab/show/topos">topos</a>.</p> <p>Non-orthogonal weak factorization systems are the key ingredient in <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>, which by definition carry a weak factorization system called (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{L} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a>,<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{R} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/acyclic+fibrations">acyclic fibrations</a>) and another one called (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{L} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/acyclic+cofibrations">acyclic cofibrations</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{R} =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>). Indeed most examples of non-orthogonal weak factorization systems arise in the context of model category theory. A key tool for constructing these, or verifying their existence, is the <em><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></em>.</p> <p>There are other <a class="existingWikiWord" href="/nlab/show/properties">properties</a> which one may find or impose on a weak factorization system, for instance <a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a>. There is also <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a> which one may find or impose, such as for <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+systems">algebraic weak factorization systems</a>. For more variants see at <em><a class="existingWikiWord" href="/nlab/show/factorization+system">factorization system</a></em>.</p> <h2 id="definition">Definition</h2> <h3 id="weak_factorization_systems">Weak factorization systems</h3> <div class="num_defn" id="WeakFactorizationSystem"> <h6 id="definition_2">Definition</h6> <p>A <strong>weak factorization system</strong> (WFS) on a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/pair">pair</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{L},\mathcal{R})</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> such that</p> <ol> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X\to Y</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> may be factored as the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> followed by one in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mover><mi>Z</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f\;\colon\; X \overset{\in \mathcal{L}}{\longrightarrow} Z \overset{\in \mathcal{R}}{\longrightarrow} Y \,. </annotation></semantics></math></div></li> <li> <p>The classes are closed under having the <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> against each other:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is precisely the class of morphisms having the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is precisely the class of morphisms having the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against every morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math>.</p> </li> </ol> </li> </ol> </div> <div class="num_defn" id="FunctorialFactorization"> <h6 id="definition_3">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/category">category</a>, a <strong><a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a></strong> of the morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>fact</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>⟶</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> fact \;\colon\; \mathcal{C}^{\Delta[1]} \longrightarrow \mathcal{C}^{\Delta[2]} </annotation></semantics></math></div> <p>which is a <a class="existingWikiWord" href="/nlab/show/section">section</a> of the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>→</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">d_1 \;\colon \;\mathcal{C}^{\Delta[2]}\to \mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In def. <a class="maruku-ref" href="#FunctorialFactorization"></a> we are using the following notation, see at <em><a class="existingWikiWord" href="/nlab/show/simplex+category">simplex category</a></em> and at <em><a class="existingWikiWord" href="/nlab/show/nerve+of+a+category">nerve of a category</a></em>:</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Delta[1] = \{0 \to 1\}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mn>1</mn><mo>→</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\Delta[2] = \{0 \to 1 \to 2\}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/ordinal+numbers">ordinal numbers</a>, regarded as <a class="existingWikiWord" href="/nlab/show/posets">posets</a> and hence as <a class="existingWikiWord" href="/nlab/show/categories">categories</a>. The <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Arr</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Arr(\mathcal{C})</annotation></semantics></math> is equivalently the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup><mo>≔</mo><mi>Funct</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]} \coloneqq Funct(\Delta[1], \mathcal{C})</annotation></semantics></math>, while <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>≔</mo><mi>Funct</mi><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo><mo>,</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[2]}\coloneqq Funct(\Delta[2], \mathcal{C})</annotation></semantics></math> has as objects pairs of composable morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. There are three injective functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\delta_i \colon [1] \rightarrow [2]</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>δ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\delta_i</annotation></semantics></math> omits the index <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> in its image. By precomposition, this induces <a class="existingWikiWord" href="/nlab/show/functors">functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>i</mi></msub><mo lspace="verythinmathspace">:</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>2</mn><mo stretchy="false">]</mo></mrow></msup><mo>⟶</mo><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">d_i \colon \mathcal{C}^{\Delta[2]} \longrightarrow \mathcal{C}^{\Delta[1]}</annotation></semantics></math>. Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">d_1</annotation></semantics></math> sends a pair of composable morphisms to their <a class="existingWikiWord" href="/nlab/show/composition">composition</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">d_2</annotation></semantics></math> sends a pair of composable morphisms to the first morphism;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">d_0</annotation></semantics></math> sends a pair of composable morphisms to the second morphism.</p> </li> </ul> </div> <div class="num_defn" id="FunctorialWeakFactorizationSystem"> <h6 id="definition_4">Definition</h6> <p>A weak factorization system, def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a>, is called a <strong>functorial weak factorization system</strong> if the factorization of morphisms may be chosen to be a <a class="existingWikiWord" href="/nlab/show/functorial+factorization">functorial factorization</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>fact</mi></mrow><annotation encoding="application/x-tex">fact</annotation></semantics></math>, def. <a class="maruku-ref" href="#FunctorialFactorization"></a>, i.e. such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>2</mn></msub><mo>∘</mo><mi>fact</mi></mrow><annotation encoding="application/x-tex">d_2 \circ fact</annotation></semantics></math> lands in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mn>0</mn></msub><mo>∘</mo><mi>fact</mi></mrow><annotation encoding="application/x-tex">d_0\circ fact</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math>.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Not all weak factorization systems are functorial, although most (including those produced by the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>, with due care) are. But all <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+systems">orthogonal factorization systems</a>, def. <a class="maruku-ref" href="#OrthogonalFactorizationSystem"></a>, automatically are functorial. An example of a weak factorization that is not functorial can be found in <a href="https://arxiv.org/abs/math/0106152">Isaksen 2001</a>.</p> </div> <h3 id="orthogonal_factorization_systems">Orthogonal factorization systems</h3> <div class="num_defn" id="OrthogonalFactorizationSystem"> <h6 id="definition_5">Definition</h6> <p>An <strong><a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">orthogonal factorization system</a></strong> (OFS) is a weak factorization system <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{L},\mathcal{R})</annotation></semantics></math>, def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a> such that the lifts of elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> against elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> are <em>unique</em>.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>While every OFS (def. <a class="maruku-ref" href="#OrthogonalFactorizationSystem"></a>) is a WFS (def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a>), the primary examples of each are different:</p> <p>A “basic example” of an OFS is <a class="existingWikiWord" href="/nlab/show/%28epi%2Cmono%29-factorization">(epi,mono)-factorization</a> in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> (meaning <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the collection of <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> that of <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>), while a “basic example” of a WFS is (mono, epi) in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math>. The superficial similarity of these two examples masks the fact that they generalize in very different ways.</p> <p>The OFS (epi, mono) generalizes to any <a class="existingWikiWord" href="/nlab/show/topos">topos</a> or <a class="existingWikiWord" href="/nlab/show/pretopos">pretopos</a>, and in fact to any <a class="existingWikiWord" href="/nlab/show/regular+category">regular category</a> if we replace “epi” with <a class="existingWikiWord" href="/nlab/show/regular+epimorphism">regular epi</a>. Likewise it generalizes to any <a class="existingWikiWord" href="/nlab/show/quasitopos">quasitopos</a> if we instead replace “mono” with <a class="existingWikiWord" href="/nlab/show/regular+monomorphism">regular mono</a>.</p> <p>On the other hand, saying that (mono,epi) is a WFS in <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is equivalent to the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>. A less loaded statement is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>,</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L,R)</annotation></semantics></math> is a WFS, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> is the class of inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↪</mo><mi>A</mi><mo>⊔</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A\hookrightarrow A\sqcup B</annotation></semantics></math> into a binary <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is the class of <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epis</a>. In this form the statement generalizes to any <a class="existingWikiWord" href="/nlab/show/extensive+category">extensive category</a>; see also <a class="existingWikiWord" href="/nlab/show/weak+factorization+system+on+Set">weak factorization system on Set</a>.</p> </div> <h3 id="algebraic_weak_factorization_systems">Algebraic weak factorization systems</h3> <p>An <a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+system">algebraic weak factorization system</a> enhances the <em>properties</em> of lifting and factorization to algebraic <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a>.</p> <h3 id="accessible_weak_factorization_systems">Accessible weak factorization systems</h3> <p>An <a class="existingWikiWord" href="/nlab/show/accessible+weak+factorization+system">accessible weak factorization system</a> is a wfs on a <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable category</a> whose factorization is given by an <a class="existingWikiWord" href="/nlab/show/accessible+functor">accessible functor</a>.</p> <h2 id="properties">Properties</h2> <h3 id="ClosureProperties">Closure properties</h3> <div class="num_prop" id="ClosuredPropertiesOfWeakFactorizationSystem"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{L},\mathcal{R})</annotation></semantics></math> be a weak factorization system, def. <a class="maruku-ref" href="#WeakFactorizationSystem"></a> on some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. Then</p> <ol> <li> <p>Both classes contain the class of <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> <li> <p>Both classes are closed under <a class="existingWikiWord" href="/nlab/show/composition">composition</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is also closed under <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a>.</p> </li> <li> <p>Both classes are closed under forming <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> (see remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/cobase+change">cobase change</a>”).</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (“<a class="existingWikiWord" href="/nlab/show/base+change">base change</a>”).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is closed under forming <a class="existingWikiWord" href="/nlab/show/products">products</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>We go through each item in turn.</p> <p><strong>containing isomorphisms</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>Iso</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{f}{\rightarrow}& X \\ {}_{\mathllap{\in Iso}}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{p}} \\ B &\underset{g}{\longrightarrow}& Y } </annotation></semantics></math></div> <p>with the left morphism an isomorphism, then a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> is given by using the <a class="existingWikiWord" href="/nlab/show/inverse">inverse</a> of this isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mrow></mrow> <mrow><mi>f</mi><mo>∘</mo><msup><mi>i</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></msup><mo>↗</mo></mrow><annotation encoding="application/x-tex">{}^{{f \circ i^{-1}}}\nearrow</annotation></semantics></math>. Hence in particular there is a lift when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">p \in \mathcal{R}</annotation></semantics></math> and so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">i \in \mathcal{L}</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p><strong>closure under composition</strong></p> <p>Given a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}} \\ {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>consider its <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> decomposition as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↘</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow &\searrow& \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}} \\ {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>Now the bottom commuting square has a lift, by assumption. This yields another <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> decomposition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{p_1}}_{\mathrlap{\in \mathcal{R}}} \\ \downarrow &\nearrow& \downarrow^{\mathrlap{p_2}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>and now the top commuting square has a lift by assumption. This is now equivalently a lift in the total diagram, showing that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub><mo>∘</mo><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1\circ p_1</annotation></semantics></math> has the right lifting property against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> and is hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math>. The case of composing two morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>. From this the closure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> follows since the latter is given by <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> of sequential composition and successive lifts against the underlying sequence as above constitutes a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a>, whence the extension of the lift to the colimit follows by its <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a>.</p> <p><strong>closure under retracts</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">i \in \mathcal{L}</annotation></semantics></math>, i.e. let there be a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_A \colon & A &\longrightarrow& C &\longrightarrow& A \\ & \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}} \\ id_B \colon & B &\longrightarrow& D &\longrightarrow& B } \,. </annotation></semantics></math></div> <p>Then for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>j</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ {}^{\mathllap{j}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& Y } </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>, it is equivalent to its <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite with that retract diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,. </annotation></semantics></math></div> <p>Now the pasting composite of the two squares on the right has a lift, by assumption,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>j</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0"><mi>i</mi></mpadded></msubsup></mtd> <mtd></mtd> <mtd><mo>↗</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& C &\longrightarrow& A &\longrightarrow& X \\ \downarrow^{\mathrlap{j}} && \downarrow^{\mathrlap{i}}_{\mathrlap{\in \mathcal{L}}} && \nearrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ B &\longrightarrow& D &\longrightarrow& B &\longrightarrow & Y } \,. </annotation></semantics></math></div> <p>By composition, this is also a lift in the total outer rectangle, hence in the original square. Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi></mrow><annotation encoding="application/x-tex">j</annotation></semantics></math> has the left lifting property against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">p \in \mathcal{R}</annotation></semantics></math> and hence is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p><strong>closure under pushout and pullback</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">p \in \mathcal{R}</annotation></semantics></math> and and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{{f^* p}}}\downarrow && \downarrow^{\mathrlap{p}} \\ Z &\stackrel{f}{\longrightarrow} & Y } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. We need to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow><annotation encoding="application/x-tex">f^* p</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> with respect to all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">i \in \mathcal{L}</annotation></semantics></math>. So let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X \\ {}^{\mathllap{i}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{\mathrlap{f^* p}}} \\ B &\stackrel{g}{\longrightarrow}& Z } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>. We need to construct a diagonal lift of that square. To that end, first consider the <a class="existingWikiWord" href="/nlab/show/pasting">pasting</a> composite with the pullback square from above to obtain the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X &\longrightarrow& X \\ {}^{\mathllap{i}}\downarrow && \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>By the right lifting property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, there is a diagonal lift of the total outer diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd> <mtd><msup><mrow></mrow> <mover><mrow><mo stretchy="false">(</mo><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo></mrow><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mi>g</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& X \\ \downarrow^{\mathrlap{i}} &{}^{\hat {(f g)}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{f g}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>By the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> this gives rise to the lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mover><mi>g</mi><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msup><mi>f</mi> <mo>*</mo></msup><mi>p</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>⟶</mo><mi>g</mi></mover></mtd> <mtd><mi>Z</mi></mtd> <mtd><mover><mo>⟶</mo><mi>f</mi></mover></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Z \times_f X &\longrightarrow& X \\ &{}^{\hat g} \nearrow& \downarrow^{\mathrlap{f^* p}} && \downarrow^{\mathrlap{p}} \\ B &\stackrel{g}{\longrightarrow}& Z &\stackrel{f}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>In order for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">\hat g</annotation></semantics></math> to qualify as the intended lift of the total diagram, it remains to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>i</mi></mpadded></msup></mtd> <mtd><msup><mrow></mrow> <mover><mi>g</mi><mo stretchy="false">^</mo></mover></msup><mo>↗</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\longrightarrow& Z \times_f X \\ \downarrow^{\mathrlap{i}} & {}^{\hat g}\nearrow \\ B } </annotation></semantics></math></div> <p>commutes. To do so we notice that we obtain two <a class="existingWikiWord" href="/nlab/show/cones">cones</a> with tip <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>:</p> <ul> <li> <p>one is given by the morphisms</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to Z \times_f X \to X</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{g}{\to} Z</annotation></semantics></math></li> </ol> <p>with universal morphism into the pullback being</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">A \to Z \times_f X</annotation></semantics></math></li> </ul> </li> <li> <p>the other by</p> <ol> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mover><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X \to X</annotation></semantics></math></li> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mi>g</mi></mover><mi>Z</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{g}{\to} Z</annotation></semantics></math>.</li> </ol> <p>with universal morphism into the pullback being</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>i</mi></mover><mi>B</mi><mover><mo>→</mo><mover><mi>g</mi><mo stretchy="false">^</mo></mover></mover><mi>Z</mi><msub><mo>×</mo> <mi>f</mi></msub><mi>X</mi></mrow><annotation encoding="application/x-tex">A \stackrel{i}{\to} B \stackrel{\hat g}{\to} Z \times_f X</annotation></semantics></math>.</li> </ul> </li> </ul> <p>The commutativity of the diagrams that we have established so far shows that the first and second morphisms here equal each other, respectively. By the fact that the universal morphism into a pullback diagram is <em>unique</em> this implies the required identity of morphisms.</p> <p>The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> <p><strong>closure under (co-)products</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>s</mi></msub><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mover><msub><mi>B</mi> <mi>s</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>ℒ</mi><msub><mo stretchy="false">}</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\{(A_s \overset{i_s}{\to} B_s) \in \mathcal{L}\}_{s \in S}</annotation></semantics></math> be a set of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math>. Since <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in the <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> are computed componentwise, their <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in this <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> is the universal morphism out of the coproduct of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\underset{s \in S}{\coprod} A_s</annotation></semantics></math> induced via its <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> by the set of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">i_s</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mover><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{s \in S}{\sqcup} A_s \overset{(i_s)_{s\in S}}{\longrightarrow} \underset{s \in S}{\sqcup} B_s \,. </annotation></semantics></math></div> <p>Now let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a>. This is in particular a <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a> under the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of objects, hence by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the coproduct, this is equivalent to a set of commuting diagrams</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left\{ \;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow && \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} \,. </annotation></semantics></math></div> <p>By assumption, each of these has a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">\ell_s</annotation></semantics></math>. The collection of these lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mrow><mo>{</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo>∈</mo><mi>ℒ</mi></mrow></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mi>ℓ</mi> <mi>s</mi></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>}</mo></mrow> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \left\{ \;\;\;\; \array{ A_s &\longrightarrow& X \\ {}^{\mathllap{i_s}}_{\mathllap{\in \mathcal{L}}}\downarrow &{}^{\ell_s}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ B_s &\longrightarrow& Y } \;\;\;\; \right\}_{s\in S} </annotation></semantics></math></div> <p>is now itself a compatible <a class="existingWikiWord" href="/nlab/show/cocone">cocone</a>, and so once more by the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the coproduct, this is equivalent to a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ℓ</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex">(\ell_s)_{s\in S}</annotation></semantics></math> in the original square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>A</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>i</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><msub><mi>ℓ</mi> <mi>s</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></msub></mrow></msup><mo>↗</mo></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∈</mo><mi>ℛ</mi></mrow></mpadded> <mpadded width="0"><mi>f</mi></mpadded></msubsup></mtd></mtr> <mtr><mtd><munder><mo>⊔</mo><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow></munder><msub><mi>B</mi> <mi>s</mi></msub></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \underset{s \in S}{\sqcup} A_s &\longrightarrow& X \\ {}^{\mathllap{(i_s)_{s\in S}}}\downarrow &{}^{(\ell_s)_{s\in S}}\nearrow& \downarrow^{\mathrlap{f}}_{\mathrlap{\in \mathcal{R}}} \\ \underset{s \in S}{\sqcup} B_s &\longrightarrow& Y } \,. </annotation></semantics></math></div> <p>This shows that the coproduct of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>s</mi></msub></mrow><annotation encoding="application/x-tex">i_s</annotation></semantics></math> has the left lifting property against all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">f\in \mathcal{R}</annotation></semantics></math> and is hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math>. The other case is <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a>.</p> </div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>Beware, in the situation of prop. <a class="maruku-ref" href="#ClosuredPropertiesOfWeakFactorizationSystem"></a>, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is not in general closed under all <a class="existingWikiWord" href="/nlab/show/colimits">colimits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>, and similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is not in general closed under all <a class="existingWikiWord" href="/nlab/show/limits">limits</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>. Also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is not in general closed under forming <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is not in general closed under forming <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>. However, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ℒ</mi><mo>,</mo><mi>ℛ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{L},\mathcal{R})</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/orthogonal+factorization+system">orthogonal factorization system</a>, def. <a class="maruku-ref" href="#OrthogonalFactorizationSystem"></a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is closed under all colimits and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℛ</mi></mrow><annotation encoding="application/x-tex">\mathcal{R}</annotation></semantics></math> is closed under all limits.</p> </div> <div class="num_remark" id="RetractsOfMorphisms"> <h6 id="remark_5">Remark</h6> <p>Here by a <em>retract</em> of a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{f}{\longrightarrow} Y</annotation></semantics></math> in some <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is meant a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as an object in the <a class="existingWikiWord" href="/nlab/show/arrow+category">arrow category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math>, hence a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mi>g</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A \stackrel{g}{\longrightarrow} B</annotation></semantics></math> such that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mrow><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{\Delta[1]}</annotation></semantics></math> there is a factorization of the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>g</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>g</mi><mo>⟶</mo><mi>f</mi><mo>⟶</mo><mi>g</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> id_g \;\colon\; g \longrightarrow f \longrightarrow g \,. </annotation></semantics></math></div> <p>This means equivalently that in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>A</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>B</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_A \colon & A &\longrightarrow& X &\longrightarrow& A \\ & \downarrow^{\mathrlap{g}} && \downarrow^{\mathrlap{f}} && \downarrow^{\mathrlap{g}} \\ id_B \colon & B &\longrightarrow& Y &\longrightarrow& B } \,. </annotation></semantics></math></div></div> <h3 id="retract_argument">Retract argument</h3> <div class="num_lemma" id="RetractArgument"> <h6 id="lemma">Lemma</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a>)</strong></p> <p>Consider a <a class="existingWikiWord" href="/nlab/show/composition">composite</a> <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>⟶</mo><mi>i</mi></mover><mi>A</mi><mover><mo>⟶</mo><mi>p</mi></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f \;\colon\; X\stackrel{i}{\longrightarrow} A \stackrel{p}{\longrightarrow} Y \,. </annotation></semantics></math></div> <p>Then:</p> <ol> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>.</p> </li> <li> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math>.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We discuss the first statement, the second is <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a>.</p> <p>Write the factorization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow && \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,. </annotation></semantics></math></div> <p>By the assumed <a class="existingWikiWord" href="/nlab/show/lifting+property">lifting property</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> there exists a diagonal filler <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> making a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>⟶</mo><mi>i</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>g</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Y</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X &\stackrel{i}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{g}}\nearrow& \downarrow^{\mathrlap{p}} \\ Y &= & Y } \,. </annotation></semantics></math></div> <p>By rearranging this diagram a little, it is equivalent to</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>p</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ & X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,. </annotation></semantics></math></div> <p>Completing this to the right, this yields a diagram exhibiting the required retract according to remark <a class="maruku-ref" href="#RetractsOfMorphisms"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>id</mi> <mi>X</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>id</mi> <mi>Y</mi></msub><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>Y</mi></mtd> <mtd><munder><mo>⟶</mo><mi>g</mi></munder></mtd> <mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>p</mi></munder></mtd> <mtd><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ id_X \colon & X &=& X &=& X \\ & {}^{\mathllap{f}}\downarrow && {}^{\mathllap{i}}\downarrow && {}^{\mathllap{f}}\downarrow \\ id_Y \colon & Y &\underset{g}{\longrightarrow}& A &\underset{p}{\longrightarrow}& Y } \,. </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">Model categories</a> provide many examples of weak factorization systems. In fact, most applications of WFS involve model categories or model-categorical ideas.</p> </li> <li> <p>The existence of certain <a class="existingWikiWord" href="/nlab/show/weak+factorization+system+on+Set">WFS on Set</a> is related to the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a>.</p> </li> <li> <p>See the <a class="existingWikiWord" href="/joyalscatlab/published/Weak+factorisation+systems">Catlab</a> for more examples.</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/retract+argument">retract argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+and+projective+morphisms">injective and projective morphisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+weak+factorization+system">algebraic weak factorization system</a></p> </li> </ul> <div> <p><strong>Algebraic model structures:</strong> <a class="existingWikiWord" href="/nlab/show/Quillen+model+structures">Quillen model structures</a>, mainly on <a class="existingWikiWord" href="/nlab/show/locally+presentable+categories">locally presentable categories</a>, and their constituent <a class="existingWikiWord" href="/nlab/show/categories+with+weak+equivalences">categories with weak equivalences</a> and <a class="existingWikiWord" href="/nlab/show/weak+factorization+systems">weak factorization systems</a>, that can be equipped with further algebraic structure and “freely generated” by small data.</p> <table><thead><tr><th>structure</th><th>small-set-generated</th><th>small-category-generated</th><th>algebraicized</th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+wfs">combinatorial wfs</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+wfs">accessible wfs</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+wfs">algebraic wfs</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/accessible+model+category">accessible model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/transfinite+construction+of+free+algebras">construction method</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></td><td style="text-align: left;">same as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/algebraic+small+object+argument">algebraic small object argument</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/joyalscatlab/published/HomePage">Joyal's CatLab</a>, <em><a class="existingWikiWord" href="/joyalscatlab/published/Weak+factorisation+systems">Weak factorisation systems</a></em></p> </li> <li id="Hirschorn"> <p>Philip S. Hirschhorn, <em>Model Categories and Their Localizations</em> (<a href="http://www.ams.org/bookstore?fn=20&arg1=whatsnew&item=SURV-99">AMS</a>, <a href="http://www.gbv.de/dms/goettingen/360115845.pdf">pdf toc</a>, <a href="http://www.maths.ed.ac.uk/~aar/papers/hirschhornloc.pdf">pdf</a>)</p> </li> <li> <p>Jiri Rosicky, Walter Tholen , <em>Factorization, Fibration and Torsion</em>, <a href="https://arxiv.org/abs/0801.0063">arxiv</a> (2007)</p> </li> </ul> <p>Introductory texts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Homotopy Theory</a></p> </li> <li id="Riehl2008"> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, <a href="https://math.jhu.edu/~eriehl/factorization.pdf"><em>Factorization Systems</em></a>, 2008</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Emily+Riehl">Emily Riehl</a>, Chapter 11 in: <em><a class="existingWikiWord" href="/nlab/show/Categorical+Homotopy+Theory">Categorical Homotopy Theory</a></em>, Cambridge University Press (2014) [<a href="https://doi.org/10.1017/CBO9781107261457">doi:10.1017/CBO9781107261457</a>, <a href="http://www.math.jhu.edu/~eriehl/cathtpy.pdf">pdf</a>]</p> </li> </ul> <p>The following dissertation section is entirely written after learning of <a href="#Riehl2008">Riehl (2008)</a> above, but has complementary examples and may dive deeper into some proofs:</p> <ul> <li id="Nuyts2020"><a class="existingWikiWord" href="/nlab/show/Andreas+Nuyts">Andreas Nuyts</a>, <em>Contributions to Multimode and Presheaf Type Theory, section 2.4: Factorization Systems</em>, <a href="https://lirias.kuleuven.be/retrieve/581985">PhD thesis</a>, KU Leuven, Belgium, 2020</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 17, 2024 at 16:39:09. 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