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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> Quillen equivalence </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7165/#Item_10" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</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/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </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> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#TwoOutOfThree'>2-out-of-3</a></li> <li><a href='#presentation_of_equivalence_of_categories'>Presentation of equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</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 <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> is a context in which we can do <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> or some generalization thereof; two model categories are ‘the same’ for this purpose if they are Quillen equivalent. For example, the classic version of homotopy theory can be done using either <a class="existingWikiWord" href="/nlab/show/topological+space">topological spaces</a> or <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a>. There is a model category of topological spaces, and a model category of simplicial sets, and they are Quillen equivalent.</p> <p>In short, Quillen equivalence is the right notion of <a class="existingWikiWord" href="/nlab/show/equivalence">equivalence</a> for <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> — and most importantly, this notion is weaker than <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>. The work of Dwyer–Kan, Bergner and others has shown that Quillen equivalent model categories <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</a> equivalent <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-category">(infinity,1)-categories</a>.</p> <h2 id="definition">Definition</h2> <p>Let <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">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/model+category">model categories</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>C</mi><mover><munder><mo>→</mo><mi>L</mi></munder><mover><mo>←</mo><mi>R</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) \;\colon\; C \stackrel{\overset{R}{\leftarrow}}{\underset{L}{\to}} D </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> with <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> <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <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>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">Ho C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mi>D</mi></mrow><annotation encoding="application/x-tex">Ho D</annotation></semantics></math> for the corresponding <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy categories</a>.</p> <p>By the discussion there, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">Ho C</annotation></semantics></math> may be regarded as obtained by first passing to the full <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> on <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant objects</a> and then <a class="existingWikiWord" href="/nlab/show/localization">inverting</a> <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>, and <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> (being a left Quillen adjoint) preserves weak equivalences between cofibrant objects. Thus, <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> induces a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mo>:</mo><mi>Ho</mi><mi>C</mi><mo>→</mo><mi>Ho</mi><mi>D</mi></mrow><annotation encoding="application/x-tex"> \mathbb{L} : Ho C \to Ho D </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy categories</a>, called its (total) left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>. Similarly (but dually), <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> induces a (total) right derived functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>:</mo><mi>Ho</mi><mi>D</mi><mo>→</mo><mi>Ho</mi><mi>C</mi></mrow><annotation encoding="application/x-tex">\mathbb{R} : Ho D \to Ho C</annotation></semantics></math>. See at <em><a href="homotopy+category+of+a+model+category#DerivedFunctors">homotopy category of a model category – derived functors</a></em> for more.</p> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</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">(L \dashv R)</annotation></semantics></math> is a <strong>Quillen equivalence</strong> if the following equivalent conditions are satisfied:</p> <ul> <li> <p>The total left <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{L} \colon Ho(C) \to Ho(D)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>;</p> </li> <li> <p>The total right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo lspace="verythinmathspace">:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{R} \colon Ho(D) \to Ho(C)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> of the <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a>;</p> </li> <li id="AdjunctOfWeakEquivalence"> <p>For every <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c \in C</annotation></semantics></math> and every <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d \in D</annotation></semantics></math>, 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>c</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \to R(d)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</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">C</annotation></semantics></math> precisely when the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo>→</mo><mi>d</mi></mrow><annotation encoding="application/x-tex">L(c) \to d</annotation></semantics></math> is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> <li> <p>The following two conditions hold:</p> <ol> <li> <p>The <a class="existingWikiWord" href="/nlab/show/derived+adjunction+unit">derived adjunction unit</a> is a weak equivalence, in that for every cofibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">c\in C</annotation></semantics></math>, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mover><mo>→</mo><mrow><msub><mi>η</mi> <mi>c</mi></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mi>fib</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c \overset{\eta_c}{\to} R(L(c)) \to R(L(c)^{fib})</annotation></semantics></math> (of the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> with a <a class="existingWikiWord" href="/nlab/show/fibrant+replacement">fibrant replacement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>L</mi><mo stretchy="false">(</mo><mi>c</mi><msup><mo stretchy="false">)</mo> <mi>fib</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(L(c) \stackrel{\simeq}{\to} L(c)^{fib})</annotation></semantics></math>) is a weak equivalence 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">C</annotation></semantics></math>,</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/derived+adjunction+counit">derived adjunction counit</a> is a weak equivalence, in that for every fibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">d\in D</annotation></semantics></math>, the composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mi>cof</mi></msup><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ϵ</mi> <mi>d</mi></msub></mrow></mover><mi>d</mi></mrow><annotation encoding="application/x-tex">L(R(d)^{cof}) \to L(R(d)) \overset{\epsilon_d}{\to} d</annotation></semantics></math> (of the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> with <a class="existingWikiWord" href="/nlab/show/cofibrant+replacement">cofibrant replacement</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><msup><mo stretchy="false">)</mo> <mi>cof</mi></msup><mover><mo>→</mo><mo>≃</mo></mover><mi>R</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(R(d)^{cof} \stackrel{\simeq}{\to} R(d))</annotation></semantics></math>) is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> </ol> </li> </ul> </div> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>Not every equivalence between homotopy categories of model categories lifts to a Quillen equivalence. An interesting <a class="existingWikiWord" href="/nlab/show/counterexample">counterexample</a> is given for instance by <a href="#DuggerShipley09">Dugger & Shipley (2009)</a>.</p> </div> <p>Here are further characterizations:</p> <div class="num_prop" id="InCaseTheRightAdjointCreatesWeakEquivalences"> <h6 id="proposition">Proposition</h6> <p>If in a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>𝒞</mi></mtd> <mtd><munderover><mo>⊥</mo><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></munderover></mtd> <mtd><mi>𝒟</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{\mathcal{C} &\underoverset{\underset{R}{\to}}{\overset{L}{\leftarrow}}{\bot}& \mathcal{D}}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <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> “<a class="existingWikiWord" href="/nlab/show/created+limit">creates</a> weak equivalences” (in that a morphism <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> 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 weak equivalence precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R(f)</annotation></semantics></math> is) then <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 \dashv R)</annotation></semantics></math> is a Quillen equivalence precisely already if for all cofibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d \in \mathcal{D}</annotation></semantics></math> the plain <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \overset{\eta}{\longrightarrow} R (L (d)) </annotation></semantics></math></div> <p>is a weak equivalence.</p> </div> <p>(e.g. <a href="#ErdalIlhan19">Erdal-Ilhan 19, Lemma 3.3.</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>Generally, <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 \dashv R)</annotation></semantics></math> is a Quillen equivalence precisely if</p> <ol> <li> <p>for every cofibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">d\in \mathcal{D}</annotation></semantics></math>, the “derived adjunction unit”, hence the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mover><mo>⟶</mo><mi>η</mi></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> d \overset{\eta}{\longrightarrow} R(L(d)) \overset{R(j_{L(d)})}{\longrightarrow} R(P(L(d))) </annotation></semantics></math></div> <p>(of the <a class="existingWikiWord" href="/nlab/show/adjunction+unit">adjunction unit</a> with image under <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> of any fibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>P</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L(d) \underoverset{\in W}{j_{L(d)}}{\longrightarrow} P(L(d))</annotation></semantics></math>) is a weak equivalence;</p> </li> <li> <p>for every fibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">c \in \mathcal{C}</annotation></semantics></math>, the “derived adjunction counit”, hence the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>c</mi></mrow><annotation encoding="application/x-tex"> L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c </annotation></semantics></math></div> <p>(of the <a class="existingWikiWord" href="/nlab/show/adjunction+counit">adjunction counit</a> with the image under <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> of any cofibrant replacement <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mrow><mo>∈</mo><mi>W</mi></mrow><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow></munderover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q R(c)\underoverset{\in W}{p_{R(c)}}{\longrightarrow} R(c)</annotation></semantics></math> is a weak equivalence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> </li> </ol> <p>Consider the first condition: Since <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> preserves the weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mrow><mi>L</mi><mo stretchy="false">(</mo><mi>d</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">j_{L(d)}</annotation></semantics></math>, by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> the composite in the first item is a weak equivalence precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is.</p> <p>Hence it is now sufficient to show that in this case the second condition above is automatic.</p> <p>Since <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> also reflects weak equivalences, the composite in item two is a weak equivalence precisely if its image</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) </annotation></semantics></math></div> <p>under <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.</p> <p>Moreover, assuming, by the above, that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\eta_{Q(R(c))}</annotation></semantics></math> on the cofibrant object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Q(R(c))</annotation></semantics></math> is a weak equivalence, then by <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> this composite is a weak equivalence precisely if the further composite with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>R</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Q(R(c)) \overset{\eta_{Q(R(c))}}{\longrightarrow} R(L(Q(R(c)))) \overset{R(L(p_{R(c))})}{\longrightarrow} R(L(R(c))) \overset{R(\epsilon)}{\longrightarrow} R(c) \,. </annotation></semantics></math></div> <p>But by the formula for <a class="existingWikiWord" href="/nlab/show/adjuncts">adjuncts</a>, this composite is the <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\dashv R)</annotation></semantics></math>-adjunct of the original composite, which is just <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">p_{R(c)}</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>ϵ</mi></mover><mi>c</mi></mrow><mrow><mi>Q</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow></mover><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ L(Q(R(c))) \overset{L(p_{R(c)})}{\longrightarrow} L(R(c)) \overset{\epsilon}{\longrightarrow} c }{ Q(R(C)) \overset{p_{R(c)}}{\longrightarrow} R(c) } \,. </annotation></semantics></math></div> <p>But <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>R</mi><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">p_{R(c)}</annotation></semantics></math> is a weak equivalence by definition of cofibrant replacement.</p> </div> <h2 id="properties">Properties</h2> <h3 id="TwoOutOfThree">2-out-of-3</h3> <p>Since <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalences of categories</a> enjoy the <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">2-out-of-3-property</a>, so do Quillen equivalences.</p> <h3 id="presentation_of_equivalence_of_categories">Presentation of equivalence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</h3> <p><a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched</a> Quillen equivalences between <a class="existingWikiWord" href="/nlab/show/combinatorial+model+categories">combinatorial model categories</a> present equivalences between the corresponding <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28infinity%2C1%29-categories">locally presentable (infinity,1)-categories</a>. And every equivalence between these is presented by a Zig-Zag of Quillen equivalences. See there for more details.</p> <h2 id="Examples">Examples</h2> <div class="num_example" id="TrivialQuillenEquivalence"> <h6 id="example">Example</h6> <p><strong>(trivial <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>)</strong></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>. Then the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <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> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> from <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> to itself:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><munderover><mrow><mphantom><mrow><msub><mrow></mrow> <mi>Qu</mi></msub></mrow></mphantom><msub><mo>≃</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mi>id</mi></munder><mover><mo>⟵</mo><mi>id</mi></mover></munderover><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{id}{\longrightarrow}} {\overset{id}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C} </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>From <a href="geometry+of+physics+--+categories+and+toposes#ComputationOfLeftRightDerivedFunctorsViaResolutions">this prop.</a> it is clear that in this case the <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕃</mi><mi>id</mi></mrow><annotation encoding="application/x-tex">\mathbb{L}id</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>id</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}id</annotation></semantics></math> both are themselves the <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on the <a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">homotopy category of a model category</a>, hence in particular are an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>.</p> </div> <p> <div class='num_prop' id='LeftBaseChangeQuillenEquivalence'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/left+base+change+Quillen+equivalence">left base change Quillen equivalence</a>)</strong> <br /></p> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mover><mo>⟶</mo><mrow><mo>∈</mo><mi mathvariant="normal">W</mi></mrow></mover><mi>T</mi></mrow><annotation encoding="application/x-tex">\phi \colon S \overset{ \in \mathrm{W} }{\longrightarrow} T</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</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>Then the <a href="slice+model+structure#BaseChangeQuillenAdjunction">left base change Quillen adjunction</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is a Quillen equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>T</mi></mrow></msub><munderover><mrow><mphantom><mrow><msub><mrow></mrow> <mi>Qu</mi></msub></mrow></mphantom><msub><mo>≃</mo> <mi>Qu</mi></msub></mrow><munder><mo>⟶</mo><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>ϕ</mi> <mo>!</mo></msub></mrow></mover></munderover><msub><mi>𝒞</mi> <mrow><mo stretchy="false">/</mo><mi>S</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}_{/T} \underoverset {\underset{\phi^*}{\longrightarrow}} {\overset{\phi_!}{\longleftarrow}} {\phantom{{}_{Qu}}\simeq_{Qu}} \mathcal{C}_{/S} </annotation></semantics></math></div> <p>if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> has this property:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast)</annotation></semantics></math> <em>The <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> (<a class="existingWikiWord" href="/nlab/show/base+change">base change</a>) of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> along any <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> is still a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a>.</em></p> </div> Notice that the property <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\ast)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is implied as soon as either:</p> <ul> <li> <p><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 class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper</a>, or</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/acyclic+fibration">acyclic fibration</a>, or</p> </li> <li> <p>both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a></p> </li> </ul> <p>(for the first this follows by definition; for the second by the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\phi^\ast</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a> by <a href="slice+model+structure#LeftBaseChangeQuillenAdjunction">this Prop.</a>; for the third by <a href="homotopy+pullback#HomotopyPullbackByOrdinaryPullback">this Prop.</a> on recognizing <a class="existingWikiWord" href="/nlab/show/homotopy+pullbacks">homotopy pullbacks</a>). <div class='proof'> <h6>Proof</h6> <p>Using the characterization of Quillen equivalences by derived adjuncts (<a href="#AdjunctOfWeakEquivalence">here</a>), the base change adjunction is a Quillen equivalence iff for</p> <ul> <li> <p>any <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">X \to S</annotation></semantics></math> in the slice over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is cofibrant 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> </li> <li> <p>and a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">p \colon Y \to T</annotation></semantics></math> in the slice over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> (i.e. <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> is a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</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> </li> </ul> <p>we have that</p> <p>(1) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mi>S</mi><msub><mo>×</mo> <mi>T</mi></msub><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to \phi^*(Y) = S \times_T Y</annotation></semantics></math> is a weak equivalence</p> <p>iff</p> <p>(2) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mo>!</mo></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\phi_!(X) \to Y</annotation></semantics></math> is a weak equivalence.</p> <p>But the latter morphism is the top composite in the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</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>X</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi><msub><mo>×</mo> <mi>T</mi></msub><mi>Y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>ϕ</mi></mrow></mover></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mo></mo><mrow><mo stretchy="false">(</mo><mi>pb</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mrow><mi>p</mi><mo>∈</mo><mi>Fib</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>S</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>ϕ</mi><mo>∈</mo><mi mathvariant="normal">W</mi></mrow></munder></mtd> <mtd><mi>T</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X &\longrightarrow& S \times_T Y &\overset{p^\ast \phi}{\longrightarrow}& Y \\ &\searrow& \big\downarrow &{}^{_{(pb)}}& \big\downarrow {}^{\mathrlap{p \in Fib}} \\ && S &\underset{\phi \in \mathrm{W} }{\longrightarrow}& T } </annotation></semantics></math></div> <p>Hence the <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a>-property says that (1) is equivalent to (2) if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">p^\ast \phi</annotation></semantics></math> is a weak equivalence.</p> <p>Conversely, taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \to \phi^\ast(X)</annotation></semantics></math> to be a weak equivalence (hence a <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi^\ast(X)</annotation></semantics></math>), <a class="existingWikiWord" href="/nlab/show/two-out-of-three">two-out-of-three</a> implies that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mo>!</mo></msub><mo>⊣</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\phi_! \dashv \phi^\ast)</annotation></semantics></math> is a Quillen equivalence, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mo>*</mo></msup><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">p^\ast \phi</annotation></semantics></math> is a weak equivalence.</p> <p></p> </div> </p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+reflection">Quillen reflection</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><strong>Quillen equivalence</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></p> </li> </ul> <h2 id="references">References</h2> <p>For standard references see at <em><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></em>.</p> <p>An example of an equivalence of <a class="existingWikiWord" href="/nlab/show/homotopy+categories">homotopy categories</a> of model categories which does not lift to a Quillen equivalence is in</p> <ul> <li id="DuggerShipley09"><a class="existingWikiWord" href="/nlab/show/Daniel+Dugger">Daniel Dugger</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>A curious example of triangulated-equivalent model categories which are not Quillen equivalent</em>, Algebraic & Geometric Topology 9 (2009) (<a href="http://homepages.math.uic.edu/~bshipley/dugger.shipley.curious.example.pdf">pdf</a>)</li> </ul> <p>The characterization of Quillen equivalences in the case that one of the adjoints creates equivalences appears for instance in</p> <ul> <li id="ErdalIlhan19">Mehmet Akif Erdal, Aslı Güçlükan İlhan, <em>A model structure via orbit spaces for equivariant homotopy</em>, Journal of Homotopy and Related Structures volume 14, pages 1131–1141 (2019) (<a href="https://arxiv.org/abs/1903.03152">arXiv:1903.03152</a>, <a href="https://doi.org/10.1007/s40062-019-00241-4">doi:10.1007/s40062-019-00241-4</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 14, 2023 at 14:17:30. 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