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12 2 1.1 11 6.9 11.4 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> monoidal Quillen adjunction </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/2060/#Item_8" 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> <h4 id="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></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='#Recognition'>Recognition of monoidal Quillen adjunctions</a></li> <li><a href='#LiftToAdjunctionOnMonoids'>Lift to an adjunction on monoids</a></li> <li><a href='#LiftToQuillenAdjunctionOnMonoids'>Lift to a Quillen equivalence on monoids</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>The concept of <em>monoidal Quillen adjunction</em> is a lift of the concept of <em><a class="existingWikiWord" href="/nlab/show/strong+monoidal+adjunctions">strong monoidal adjunctions</a></em> (<a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> for which the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> so that the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> is, canonically, a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a>) from the context of plain <a class="existingWikiWord" href="/nlab/show/categories">categories</a> to that of <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>.</p> <h2 id="definition">Definition</h2> <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">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> <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a>, a <strong>lax monoidal Quillen adjunction</strong></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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D </annotation></semantics></math></div> <p>is</p> <ul> <li> <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> between the underlying <a class="existingWikiWord" href="/nlab/show/model+categories">model categories</a>;</p> </li> <li> <p>equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> on <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> with respect to the underlying <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></p> </li> <li> <p>such that the induced structure of an <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal functor</a> on <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> satisfies:</p> <ol> <li> <p>for all cofibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>∈</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">x,y \in D</annotation></semantics></math> the oplax monoidal transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tilde\nabla_{x,y} : L(x \otimes y) \to L(x) \otimes L(y) </annotation></semantics></math></div> <p>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>for some (hence any) cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><msub><mover><mi>I</mi><mo stretchy="false">^</mo></mover> <mi>D</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">q : \hat I_D \stackrel{\simeq}{\to} I_D</annotation></semantics></math> of the monoidal unit object 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>, 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><msub><mover><mi>I</mi><mo stretchy="false">^</mo></mover> <mi>D</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mi>e</mi><mo stretchy="false">˜</mo></mover></mover><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde e}{\to} I_C </annotation></semantics></math></div> <p>with the oplax monoidal counit 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> </ol> </li> </ul> <p>This is called a <strong>strong monoidal Quillen adjunction</strong> if <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 a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a>. In this case the first condition above on <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 vacuous, and the second becomes vacuous if the unit object of <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> is cofibrant.</p> <p>If a monoidal Quillen adjunction is also a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> it is called a <strong>monoidal Quillen equivalence</strong>.</p> <h2 id="properties">Properties</h2> <h3 id="Recognition">Recognition of monoidal Quillen adjunctions</h3> <div class="un_theorem"> <h6 id="theorem">Theorem</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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a> and let <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> be equipped with the strcuture of a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a>.</p> <p>Then the following two conditions are sufficient for <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> to be a lax monoidal Quillen adjunction:</p> <ol> <li> <p>for some (hence any) cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>:</mo><msub><mover><mi>I</mi><mo stretchy="false">^</mo></mover> <mi>D</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">q : \hat I_D \stackrel{\simeq}{\to} I_D</annotation></semantics></math> of the unit object 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>, the composite morphism</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><msub><mover><mi>I</mi><mo stretchy="false">^</mo></mover> <mi>D</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mi>L</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow></mover><mi>L</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>D</mi></msub><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mover><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> L(\hat I_D) \stackrel{L(q)}{\to} L(I_D) \stackrel{\tilde i}{\to} I_C </annotation></semantics></math></div> <p>is a weak equivalence, (wher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>i</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde i</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><msub><mi>I</mi> <mi>D</mi></msub><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i : I_D \to R(I_C)</annotation></semantics></math>);</p> </li> <li> <p>the unit object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">I_D</annotation></semantics></math> <em>detects weak equivalences</em> in that for every weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> between fibrant objects the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo stretchy="false">(</mo><mi>Q</mi><msub><mi>I</mi> <mi>D</mi></msub><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D^{\Delta^{op}}(Q I_D, f)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>s in the category of <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s 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> is an equivalence of Kan complexes, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Q</mi><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">Q I_D</annotation></semantics></math> a cofibrant resolution in the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>D</mi> <mi>Reedy</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup></mrow><annotation encoding="application/x-tex">D^{\Delta^{op}}_{Reedy}</annotation></semantics></math>.</p> </li> </ol> </div> <p>This is proposition 3.16 in (<a href="#SchwedeShipley">SchwedeShipley</a>).</p> <h3 id="LiftToAdjunctionOnMonoids">Lift to an adjunction on monoids</h3> <p>We discuss how a monoidal Quillen adjunction <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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> induces, under mild conditions, an adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D)</annotation></semantics></math> on the corresponding <a class="existingWikiWord" href="/nlab/show/categories+of+monoids">categories of monoids</a>. In the following section we discuss how this is itself a Quillen adjunction</p> <p>The <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">R : C \to D</annotation></semantics></math> induces (as described there) a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R : Mon(C) \to Mon(D)</annotation></semantics></math> on monoids (which by slight abuse of notation we denote by the same symbol). Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow></msub><mo>:</mo><mi>R</mi><mi>X</mi><mo>⊗</mo><mi>R</mi><mi>Y</mi><mo>→</mo><mi>R</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\nabla_{X,Y} : R X \otimes R Y \to R(X \otimes Y)</annotation></semantics></math> for the lax monoidal structure on <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>. This induces canonically the structure of a <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal functor</a> (as described there) on the left adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>:</mo><mi>D</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">L : D \to C</annotation></semantics></math>. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>L</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mi>X</mi><mo>⊗</mo><mi>L</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\tilde\nabla : L(X \otimes Y) \to L X \otimes L Y</annotation></semantics></math> for this oplax structure.</p> <p>While <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> will not extend to a functor on the <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> unless <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 a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> there is nevertheless an adjoint <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup></mrow><annotation encoding="application/x-tex">L^{mon}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R : Mon(C) \to Mon(D)</annotation></semantics></math>.</p> <p>As described at <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a>, if <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> has countable <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>s preserved by the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>, then we have a <a class="existingWikiWord" href="/nlab/show/free+functor">free functor</a>/<a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a></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>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>C</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (F \dashv U) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> over the object <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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C, \otimes)</annotation></semantics></math>.</p> <div class="num_prop" id="AdjunctionOnMonoids"> <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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> be a pair of adjoint functors between monoidal categories where <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 a lax monoidal functor 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> has all small colimits.</p> <p>Then the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R : Mon(C) \to Mon(D)</annotation></semantics></math> has a left adjoint</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L^{mon} : Mon(D) \to Mon(C) </annotation></semantics></math></div> <p>given by forming the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mo>:</mo><mi>B</mi><mo>↦</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><msub><mi>F</mi> <mi>C</mi></msub><mi>L</mi><msub><mi>F</mi> <mi>D</mi></msub><mi>B</mi><mover><mo>→</mo><mo>→</mo></mover><msub><mi>F</mi> <mi>C</mi></msub><mi>L</mi><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L^{mon} : B \mapsto \lim_{\to} (F_C L F_D B \stackrel{\to}{\to} F_C L B) </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math> of the following two morphisms</p> <ul> <li> <p>the first one is the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>C</mi></msub><mo>∘</mo><mi>L</mi></mrow><annotation encoding="application/x-tex">F_C \circ L</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">adjunction counit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>D</mi></msub><msub><mi>U</mi> <mi>D</mi></msub><mi>B</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex"> F_D U_D B \to B</annotation></semantics></math>;</p> </li> <li> <p>the second is the unique <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>-monoid morphism that restricts to the <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>-morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><msub><mi>F</mi> <mi>D</mi></msub><mi>B</mi><mo>≃</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mi>L</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mrow></mover><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mo stretchy="false">(</mo><mi>L</mi><mi>B</mi><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>≃</mo><msub><mi>F</mi> <mi>C</mi></msub><mi>L</mi><mi>B</mi></mrow><annotation encoding="application/x-tex"> L F_D B \simeq \coprod_{n \in \mathbb{N}} L( B^{\otimes n}) \stackrel{\coprod \tilde \nabla}{\to} \coprod_{n \in \mathbb{N}} (L B)^{\otimes n} \simeq F_C L B </annotation></semantics></math></div> <p>which is componentwise given by the <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal structure</a> on <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> induced by the lax monoidal structure on <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> </li> </ul> </div> <p>This is considered on p. 305 of (<a href="#SchwedeShipley">SchwedeShipley</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>To see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L^{mon} \dashv R)</annotation></semantics></math> first notice that a morphism of monoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> L^{mon} X \to Y </annotation></semantics></math></div> <p>is by the definition of coequalizer a morphism of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><msub><mi>F</mi> <mi>C</mi></msub><mi>L</mi><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : F_C L X \to Y</annotation></semantics></math> satisfying a condition. By the free property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>F</mi> <mi>C</mi></msub><mi>L</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">F_C L X</annotation></semantics></math> this in turn is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>:</mo><mi>L</mi><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f_1 : L X \to Y</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">C</annotation></semantics></math> which by <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 morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>:</mo><mi>X</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">\tilde f_1 : X \to R Y</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">C</annotation></semantics></math>. So we need to show that the condition satisfied by <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 precisely the condition that makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tilde f_1</annotation></semantics></math> a morphism of monoids in 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>X</mi><mo>⊗</mo><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub><mo>⊗</mo><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>R</mi><mi>Y</mi><mo>⊗</mo><mi>R</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>R</mi><mi>Y</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \otimes X &\stackrel{\tilde f_1 \otimes \tilde f_1}{\to}& R Y \otimes R Y \\ \downarrow && \downarrow \\ && R ( Y \otimes Y) \\ \downarrow && \downarrow \\ X &\stackrel{\tilde f_1}{\to}& R Y } </annotation></semantics></math></div> <p>commutes. We insert the definition of the <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\tilde f_1</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/lax+natural+transformation">lax naturality</a> square of <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> to get</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><mo>⊗</mo><mi>X</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>R</mi><mi>L</mi><mi>X</mi><mo>⊗</mo><mi>R</mi><mi>L</mi><mi>X</mi></mtd> <mtd><mover><mo>→</mo><mrow><mi>R</mi><msub><mi>f</mi> <mn>1</mn></msub><mo>⊗</mo><mi>R</mi><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>R</mi><mi>Y</mi><mo>⊗</mo><mi>R</mi><mi>Y</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>=</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>L</mi><mi>X</mi><mo>⊗</mo><mi>L</mi><mi>Y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>R</mi><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mrow><msub><mover><mi>f</mi><mo stretchy="false">˜</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd></mtd> <mtd><mi>R</mi><mi>Y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ X \otimes X &\to& R L X \otimes R L X &\stackrel{R f_1 \otimes R f_1}{\to}& R Y \otimes R Y \\ \downarrow && \downarrow &=& \downarrow \\ && R(L X \otimes L Y) &\stackrel{R f_1}{\to}& R ( Y \otimes Y) \\ \downarrow && && \downarrow \\ X & &\stackrel{\tilde f_1}{\to}&& R Y } \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of the left/bottom composite is</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>X</mi><mo>⊗</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><mi>Y</mi></mrow><annotation encoding="application/x-tex"> L(X\otimes X) \to L X \stackrel{f_1}{\to} Y </annotation></semantics></math></div> <p>while the adjunct of the top/right composite is that of the diagonal, which is</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>X</mi><mo>⊗</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mover><mi>L</mi><mi>X</mi><mo>⊗</mo><mi>L</mi><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>f</mi> <mn>1</mn></msub></mrow></mover><mi>Y</mi><mo>⊗</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X \stackrel{f_1 \otimes f_1}{\to} Y \otimes Y \to Y \,. </annotation></semantics></math></div> <p>This in turn is by the definition 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> in terms of its components equal to</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>X</mi><mo>⊗</mo><mi>X</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mover><mi>L</mi><mi>X</mi><mo>⊗</mo><mi>L</mi><mi>X</mi><mover><mo>→</mo><mrow><msub><mi>f</mi> <mn>2</mn></msub></mrow></mover><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L(X \otimes X) \stackrel{\tilde \nabla}{\to} L X \otimes L X \stackrel{f_2}{\to}Y \,. </annotation></semantics></math></div> <p>The coequalizer property says indeed precisely that these two adjuncts are equal.</p> </div> <div class="num_lemma" id="LemmaOnNaturalIso"> <h6 id="lemma">Lemma</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mo>∘</mo><msub><mi>F</mi> <mi>D</mi></msub><mo>≃</mo><msub><mi>F</mi> <mi>C</mi></msub><mo>∘</mo><mi>L</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L^{mon} \circ F_D \simeq F_C \circ L \,. </annotation></semantics></math></div></div> <p>This is considered on p. 305 of (<a href="#SchwedeShipley">SchwedeShipley</a>).</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>On a monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mi>F</mi><mi>K</mi><mo>→</mo><mi>F</mi><mi>L</mi><mi>K</mi></mrow><annotation encoding="application/x-tex"> L^{mon} F K \to F L K </annotation></semantics></math></div> <p>is defined as a coequalizing morphism of monoids</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mi>L</mi><mi>F</mi><mi>K</mi><mo>→</mo><mi>F</mi><mi>L</mi><mi>K</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F L F K \to F L K \,. </annotation></semantics></math></div> <p>This in turn is given by a morphism 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> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>F</mi><mi>K</mi><mo>→</mo><mi>F</mi><mi>L</mi><mi>K</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L F K \to F L K \,. </annotation></semantics></math></div> <p>Take this to be given componentwise by the oplax counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>e</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde e</annotation></semantics></math>.</p> <p>This does coequalize then: for one route is</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><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mi>L</mi><mo stretchy="false">(</mo><mi>K</mi><mo>⊗</mo><mi>K</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mover><mi>L</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>L</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L( (K) \otimes (K) ) \to L(K \otimes K) \stackrel{\tilde \nabla}{\to} L(K) \otimes L(K) </annotation></semantics></math></div> <p>and the other</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mover><mo>∇</mo><mo stretchy="false">˜</mo></mover></mover><mi>L</mi><mi>K</mi><mo>⊗</mo><mi>L</mi><mi>K</mi><mover><mo>→</mo><mi>Id</mi></mover><mi>L</mi><mi>K</mi><mo>⊗</mo><mi>L</mi><mi>K</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> K( (K) \otimes (K) ) \stackrel{\tilde \nabla}{\to} L K \otimes L K \stackrel{Id}{\to} L K \otimes L K \,. </annotation></semantics></math></div></div> <h3 id="LiftToQuillenAdjunctionOnMonoids">Lift to a Quillen equivalence on monoids</h3> <p>We now describe how the adjunction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L^{mon} \dashv R)</annotation></semantics></math> established above becomes a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> for the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a>s on the categories of monoids, transferred along the <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">forgetful</a>/<a class="existingWikiWord" href="/nlab/show/free+functor">free functor</a> adjunction</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><msub><mi>F</mi> <mi>C</mi></msub><mo>⊣</mo><msub><mi>U</mi> <mi>C</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>C</mi></mrow><annotation encoding="application/x-tex"> (F_C \dashv U_C) : Mon(C) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} C </annotation></semantics></math></div> <p>and how it becomes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> 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">(L \dashv R)</annotation></semantics></math> is a monoidal Quillen eqivalence.</p> <p>See <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">model structure on monoids</a>.</p> <div class="un_assumptio"> <h6 id="assumption">Assumption</h6> <p>We assume for this section that the monoidal model category <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> <ul> <li> <p>is <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a>;</p> </li> <li> <p>is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> </li> <li> <p>satisfies the <a class="existingWikiWord" href="/nlab/show/monoid+axiom+in+a+monoidal+model+category">monoid axiom in a monoidal model category</a>.</p> </li> </ul> </div> <p>Then by (<a href="SchwedeShipleyAlgebras">SchwedeShipleyAlgebras</a>) the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids+in+a+monoidal+model+category">model structure on monoids in a monoidal model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math> exists.</p> <p>Notice also that by cofibrant generation every cofibrant object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F \dashv U)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cell+object">cell object</a>.</p> <p>=–</p> <div class="un_theorem" id="LiftedQuillenAdjunction"> <h6 id="theorem_2">Theorem</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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> be a lax monoidal Quillen adjunction between <a class="existingWikiWord" href="/nlab/show/monoidal+model+categories">monoidal model categories</a> with cofibrant unit obects.</p> <p>Then also the adjunction</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><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mrow><msup><mi>L</mi> <mi>mon</mi></msup></mrow></mover></mover><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> (L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L^{mon}}{\leftarrow}}{\underset{R}{\to}} Mon(D) \,, </annotation></semantics></math></div> <p>from <a href="#AdjunctionOnMonoids">above</a> is a Quillen adjunction between the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">model structures on monoids</a>.</p> <p>If the forgetful functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">U_C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">U_D</annotation></semantics></math> <a href="#CreatedModelStructure">create</a> model structures on monoids, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L^{mon} \dashv R)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> 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">(L \dashv R)</annotation></semantics></math> is.</p> </div> <p>This is theorem 3.12 in (<a href="#SchwedeShipley">SchwedeShipley</a>). Its proof uses the following technical lemmas.</p> <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><mo>:</mo><mi>C</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>D</mi></mrow><annotation encoding="application/x-tex">(L \dashv R) : C \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} D</annotation></semantics></math> be a monoidal Quillen adjunction between monoidal model categories with cofibrant unit objects.</p> <p>Suppose the adjunction</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><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mo>:</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (L^{mon} \dashv R) : Mon(C) \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} Mon(D) </annotation></semantics></math></div> <p>described above exists (just as an adjunction, not yet assumed to be a Quillen adjunction).</p> <div class="num_lemma" id="VeryFirstTechnicalLemma"> <h6 id="lemma_2">Lemma</h6> <p>The morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><msub><mi>I</mi> <mi>D</mi></msub><mo>→</mo><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> L^{mon} I_D \to I_C </annotation></semantics></math></div> <p>induced by the oplax counit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>e</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>L</mi><msub><mi>I</mi> <mi>D</mi></msub><mo>→</mo><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\tilde e : L I_D \to I_C</annotation></semantics></math> of the oplax monoidal functor is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> of monoids.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>We have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">I_D</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">I_C</annotation></semantics></math> are the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(D)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math>, respectively. Because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup></mrow><annotation encoding="application/x-tex">L^{mon}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, it preserves these initial objects, so that there is <em>some</em> isomorphism as claimed. It is hence sufficient to show that the oplax counit induces a morphism of monoids at all, by the universal property of the initial object it will be an isomorphism.</p> <p>It is clear that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>n</mi></munder><msup><mi>μ</mi> <mi>n</mi></msup><msup><mover><mi>e</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>:</mo><mi>F</mi><mi>L</mi><msub><mi>I</mi> <mi>D</mi></msub><mo>→</mo><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex"> \coprod_n \mu^n {\tilde e}^{\otimes n} : F L I_D \to I_C </annotation></semantics></math></div> <p>is a morphism of monoids, because</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><mo stretchy="false">(</mo><mi>L</mi><mi>I</mi><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo>⊗</mo><mo stretchy="false">(</mo><mi>L</mi><mi>I</mi><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>μ</mi> <mi>I</mi> <mi>k</mi></msubsup><msup><mover><mi>e</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>⊗</mo><mi>k</mi></mrow></msup><mo>⊗</mo><msubsup><mi>μ</mi> <mi>I</mi> <mrow><mi>n</mi><mo>−</mo><mi>k</mi></mrow></msubsup><msup><mover><mi>e</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>⊗</mo><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mi>k</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mover></mtd> <mtd><mi>I</mi><mo>⊗</mo><mi>I</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>L</mi><mi>I</mi><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>μ</mi> <mi>I</mi> <mi>n</mi></msubsup><msup><mover><mi>e</mi><mo stretchy="false">˜</mo></mover> <mi>n</mi></msup></mrow></mover></mtd> <mtd><mi>I</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (L I)^{\otimes k} \otimes (L I)^{\otimes (n-k)} &\stackrel{\mu_I^k {\tilde e}^{\otimes k} \otimes \mu_I^{n-k}{\tilde e}^{\otimes (n-k)}}{\to}& I \otimes I \\ \downarrow && \downarrow \\ (L I)^{\otimes n} &\stackrel{\mu^n_I {\tilde e}^{n}}{\to}& I } </annotation></semantics></math></div> <p>commutes. So we have to show that this morphism coequalizes the two morphisms in the definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">L^{mon} I_D</annotation></semantics></math>. By the same argument as in the <a href="#AdjunctionOnMonoids">above proof</a> this is equivalent to showing 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><msub><mi>I</mi> <mi>C</mi></msub><mo>⊗</mo><msub><mi>I</mi> <mi>C</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mi>e</mi><mo>⊗</mo><mi>e</mi></mrow></mover></mtd> <mtd><mi>R</mi><msub><mi>I</mi> <mi>C</mi></msub><mo>⊗</mo><mi>R</mi><msub><mi>I</mi> <mi>C</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>C</mi></msub><mo>⊗</mo><msub><mi>I</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>I</mi> <mi>C</mi></msub></mtd> <mtd><mover><mo>→</mo><mi>e</mi></mover></mtd> <mtd><mi>R</mi><msub><mi>I</mi> <mi>C</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ I_C \otimes I_C &\stackrel{e \otimes e}{\to}& R I_C \otimes R I_C \\ \downarrow && \downarrow \\ && R (I_C \otimes I_C) \\ \downarrow && \downarrow \\ I_C &\stackrel{e}{\to}& R I_C } </annotation></semantics></math></div> <p>commutes. This follows from the unitality of the <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</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>.</p> </div> <div class="num_lemma" id="OneTechnicalLemma"> <h6 id="lemma_3">Lemma</h6> <p>For every monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \in Mon(D)</annotation></semantics></math> which is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F \dashv U)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/cell+object">cell object</a>, 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>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub><mo>:</mo><mi>L</mi><mi>B</mi><mo>→</mo><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi></mrow><annotation encoding="application/x-tex"> \chi_B : L B \to L^{mon} B </annotation></semantics></math></div> <p>to the morphism underlying the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>R</mi><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi></mrow><annotation encoding="application/x-tex">B \to R L^{mon} B</annotation></semantics></math> is a weak equivalence.</p> </div> <p>This is proposition 5.1 in (<a href="#SchwedeShipley">SchwedeShipley</a>).</p> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>We first show this for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>=</mo><msub><mi>I</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">B = I_D</annotation></semantics></math> the tensor unit 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>, which in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(D)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a>s:</p> <ul> <li> <p>We claim hat the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">adjunction unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>I</mi> <mi>C</mi></msub><mo>→</mo><mi>R</mi><msup><mi>L</mi> <mi>mon</mi></msup><msub><mi>I</mi> <mi>D</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>R</mi><mo stretchy="false">(</mo><msub><mi>I</mi> <mi>C</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">I_C \to R L^{mon} I_D \stackrel{\simeq}{\to} R(I_C)</annotation></semantics></math> is the lax monoidal unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> of <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>To see this, use that by the <a href="#VeryFirstTechnicalLemma">previous lemma</a> 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>-<a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>→</mo><mi>R</mi><msup><mi>L</mi> <mi>mon</mi></msup><mi>I</mi><mo>→</mo><mi>R</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">I \to R L^{mon} I \to R I</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>I</mi><mo>→</mo><msup><mi>L</mi> <mi>mon</mi></msup><mi>I</mi><mover><mo>→</mo><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>n</mi></msub><msup><mi>μ</mi> <mi>n</mi></msup><msup><mover><mi>e</mi><mo stretchy="false">˜</mo></mover> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup></mrow></mover><mi>I</mi></mrow><annotation encoding="application/x-tex">L I \to L^{mon} I \stackrel{\coprod_n \mu^n {\tilde e}^{\otimes n}}{\to} I</annotation></semantics></math>. Here the first morphism factors through the single power of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>I</mi></mrow><annotation encoding="application/x-tex">L I</annotation></semantics></math>, hence this is indeed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>e</mi><mo stretchy="false">˜</mo></mover><mo>:</mo><mi>L</mi><msub><mi>I</mi> <mi>D</mi></msub><mo>→</mo><msub><mi>I</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">\tilde e : L I_D \to I_C</annotation></semantics></math>.</p> <p>Therefore by the axioms on monoidal Quillen adjunctions 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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>I</mi></msub></mrow><annotation encoding="application/x-tex">\chi_I</annotation></semantics></math> is a weak equivalence.</p> </li> </ul> <p>We now proceed from this by induction over the cells of the <a class="existingWikiWord" href="/nlab/show/cell+object">cell object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>.</p> <p>So assume now that we have already shown that on some cell object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\chi_B</annotation></semantics></math> is a weak equivalence. We want to deduce then that that after forming a new monoid <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> by cell attachment, i.e. by a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</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>F</mi><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F K &\to& F K' \\ \downarrow && \downarrow \\ B &\to& P } </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">K \to K'</annotation></semantics></math> a cofibration 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>, also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>P</mi></msub><mo>:</mo><mi>L</mi><mi>P</mi><mo>→</mo><msup><mi>L</mi> <mi>mon</mi></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">\chi_P : L P \to L^{mon} P </annotation></semantics></math> is a weak equivalence.</p> <p>Notice that since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup></mrow><annotation encoding="application/x-tex">L^{mon}</annotation></semantics></math> is left adjoint also</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><msup><mi>L</mi> <mi>mon</mi></msup><mi>F</mi><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>L</mi> <mi>mon</mi></msup><mi>F</mi><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>L</mi> <mi>mon</mi></msup><mi>P</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ L^{mon} F K &\to& L^{mon} F K' \\ \downarrow && \downarrow \\ L^{mon} B &\to& L^{mon} P } </annotation></semantics></math></div> <p>is a pushout in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math>, and by the natural isomorphism from <a href="#LemmaOnNaturalIso">the above lemma</a> so is</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>F</mi><mi>L</mi><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mi>L</mi><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msup><mi>L</mi> <mi>mon</mi></msup><mi>P</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F L K &\to& F L K' \\ \downarrow && \downarrow \\ L^{mon} B &\to& L^{mon} P } \,. </annotation></semantics></math></div> <ul> <li> <p>We claim that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is cofibrant and that we can without restriction assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">K'</annotation></semantics></math> to be cofibrant 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> <p>The first statement follows from an inductive application of the construction of pushouts as discussed at <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> in the section <a href="http://ncatlab.org/nlab/show/category+of+monoids#FreeMonoids">free monoids</a>. For the second statement notice that since <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 left adjoint and preserves pushouts 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>, we have that <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 also the pushout of the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>F</mi><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></munder><mi>K</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>P</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>=</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>F</mi><mi>K</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mi>K</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>F</mi><mi>B</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>F</mi><mi>B</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd></mtd> <mtd><mo>→</mo></mtd> <mtd></mtd> <mtd><mi>P</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ F B &\to& F( B \coprod_K K' ) \\ \downarrow && \downarrow \\ B &\to& P } \right) = \left( \array{ && F K &\to& F K' \\ && \downarrow && \downarrow \\ F B &\to& F B \\ \downarrow &&&& \downarrow \\ B && \to && P } \right) \,. </annotation></semantics></math></div> <p>Since cofibrations are preserved by the Quillen left adjoint <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> and under pushout, it follows that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></msub><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B \coprod_K K'</annotation></semantics></math> is cofibrant if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">K \to K'</annotation></semantics></math> is a cofibration. So <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>B</mi><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>K</mi></msub><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">B \to B \coprod_K K'</annotation></semantics></math> can be used in place of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>K</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">K \to K'</annotation></semantics></math>.</p> </li> </ul> <p>Notice that this means that our pushout square is in fact a <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a> square (as discussed there). In particular a weak equivalence of these pushout diagrams will induce a weak equivalence of the pushouts, so that is what we will establish.</p> <p>We now proceed as in <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> in the section <a href="http://ncatlab.org/nlab/show/category+of+monoids#FreeMonoids">free monoids</a> for getting the following statement about the object underlying <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> <p>This <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/colimit">colimit</a> of a sequence of cofibrations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>≃</mo><munder><mi>lim</mi> <mo>→</mo></munder><mo stretchy="false">(</mo><mi>B</mi><mo>:</mo><mo>=</mo><msub><mi>P</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>P</mi> <mn>1</mn></msub><mo>↪</mo><msub><mi>P</mi> <mn>2</mn></msub><mo>↪</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P \simeq \lim_{\to} ( B := P_0 \hookrightarrow P_1 \hookrightarrow P_2 \hookrightarrow \cdots ) </annotation></semantics></math></div> <p>such that each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo lspace="0em" rspace="thinmathspace">hookrightarow</mo><msub><mi>P</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">P_{n-1} \hookrightarow P_n</annotation></semantics></math> is a pushout 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> of a particular cofibration <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Q</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>K</mi><mo>,</mo><mi>K</mi><mo>′</mo><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>B</mi><mo>⊗</mo><mi>K</mi><mo>′</mo><msup><mo stretchy="false">)</mo> <mrow><mo>⊗</mo><mi>n</mi></mrow></msup><mo>⊗</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">Q_n(K,K', B) \hookrightarrow (B \otimes K')^{\otimes n} \otimes B</annotation></semantics></math></p> <p>By the coresponding disccussion of these pushouts under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup></mrow><annotation encoding="application/x-tex">L^{mon}</annotation></semantics></math> it follows that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">L^{mon} P</annotation></semantics></math> is the colimit of a sequence of cofibrations betwen objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">R_n</annotation></semantics></math> that are pushouts of these particular cofibrations.</p> <p>And the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">\chi_P</annotation></semantics></math> respects all that and sends</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mrow><msub><mi>P</mi> <mi>n</mi></msub></mrow></msub><mo>:</mo><mi>L</mi><msub><mi>P</mi> <mi>n</mi></msub><mo>→</mo><msup><mi>L</mi> <mi>mon</mi></msup><msub><mi>R</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex"> \chi_{P_n} : L P_n \to L^{mon} R_n </annotation></semantics></math></div> <p>at each stage of the cell attachments. So it is sufficient to show that the three components of these maps on the pushout squares are weak equivalences. Since we showed above that our pushout squares are actually <a class="existingWikiWord" href="/nlab/show/homotopy+pushout">homotopy pushout</a> squares, this will imply that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>P</mi></msub></mrow><annotation encoding="application/x-tex">\chi_P</annotation></semantics></math> is a weak equivalence.</p> <p>This again works by proceeding as in <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> in the section <a href="http://ncatlab.org/nlab/show/category+of+monoids#FreeMonoids">free monoids</a>.</p> </div> <div class="un_lemma" id="AnotherTechnicalLemma"> <h6 id="lemma_4">Lemma</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>C</mi></msub></mrow><annotation encoding="application/x-tex">U_C</annotation></semantics></math> <a href="#CreatedModelStructure">creates</a> the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math> and the unit 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> is cofibrant, then a cofibrant <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>-monoid is also cofibrant as an object 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> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This is once more a consequence of the lemma on pushouts at at <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> in the section <a href="http://ncatlab.org/nlab/show/category+of+monoids#FreeMonoids">free monoids</a>.</p> </div> <p>We have now collected all prerequisites and turn to the proof of the <a href="#LiftedQuillenAdjunction">theorem about lifted Quillen adjunctions</a>.</p> <div class="proof"> <h6 id="proof_of_the_theorem">Proof of the theorem</h6> <p>That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>L</mi> <mi>mon</mi></msup><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(L^{mon} \dashv R)</annotation></semantics></math> is a Quillen adjunction is clear, as the <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">model structure on monoids</a> has fibrations and acyclic fibrations those in the underlying category, and these are preserved by <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>So the essential statement is that it is a Quillen equivalence of <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.</p> <p>First notice that since by assumption the <a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">model structure on monoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(D)</annotation></semantics></math> is <a href="#CreatedModelStructure">created</a> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>U</mi> <mi>D</mi></msub></mrow><annotation encoding="application/x-tex">U_D</annotation></semantics></math> it follows by definition that the cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/retract">retract</a> of a <a class="existingWikiWord" href="/nlab/show/cell+object">cell object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(D)</annotation></semantics></math>. Then the <a href="#OneTechnicalLemma">above lemma</a> asserts that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub><mo>:</mo><mi>L</mi><mi>B</mi><mo>→</mo><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi></mrow><annotation encoding="application/x-tex"> \chi_B : L B \to L^{mon} B </annotation></semantics></math></div> <p>is a weak equivalence.</p> <p>To prove the theorem, we have to show for every cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>∈</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \in Mon(D)</annotation></semantics></math> and fibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Y \in Mon(C)</annotation></semantics></math> that a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>R</mi><mi>Y</mi></mrow><annotation encoding="application/x-tex">B \to R Y</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>Mon</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(D)</annotation></semantics></math> (hence its underlying morphism 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>) precisely if its adjunct <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">L^{mon} B \to Y</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>Mon</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(C)</annotation></semantics></math> (hence its underlying morphism 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> <p>By definition of <a class="existingWikiWord" href="/nlab/show/adjunct">adjunct</a> we have that</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>B</mi><mo>→</mo><mi>R</mi><mi>Y</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>B</mi><mo>→</mo><mi>R</mi><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi><mo>→</mo><mi>R</mi><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (B \to R Y) = ( B \to R L^{mon} B \to R Y) \,. </annotation></semantics></math></div> <p>By the <a href="#AnotherTechnicalLemma">second lemma above</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is cofibrant also 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>. Therefore, since <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 between <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>, the right hand is a weak equivalence precisely if its <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</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>L</mi><mi>B</mi><mover><mo>→</mo><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow></mover><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex"> L B \stackrel{\chi_B}{\to} L^{mon} B \to Y </annotation></semantics></math></div> <p>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>. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>χ</mi> <mi>B</mi></msub></mrow><annotation encoding="application/x-tex">\chi_B</annotation></semantics></math> is a weak equivalence, this is the case precisely if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mi>mon</mi></msup><mi>B</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">L^{mon}B \to Y</annotation></semantics></math> is a weak equivalence.</p> </div> <h2 id="examples">Examples</h2> <div class="num_example" id="Stabilization"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/stabilization">stabilization</a> adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo>⊣</mo><msup><mi>Ω</mi> <mn>∞</mn></msup><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><msup><mi>Ω</mi> <mn>∞</mn></msup></mrow></munder><mover><mo>⟵</mo><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover></munderover><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spaces</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \left( \Sigma^\infty(-)_+ \dashv \Omega^\infty \right) \;\colon\; Ho(Spectra) \underoverset {\underset{\Omega^\infty}{\longrightarrow}} {\overset{\Sigma^\infty(-)}{\longleftarrow}} {\bot} Ho(Spaces) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/classical+homotopy+category">classical homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spaces</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Spaces)</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Spectra)</annotation></semantics></math> is a monoidal adjunction, since the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Σ</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">\Sigma^\infty(-)_+</annotation></semantics></math> (forming the <a class="existingWikiWord" href="/nlab/show/suspension+spectrum">suspension spectrum</a> of a space after freely <a class="existingWikiWord" href="/nlab/show/pointed+topological+space">adjoining a basepoint</a>) is <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal</a> with respect to forming <a class="existingWikiWord" href="/nlab/show/product+topological+spaces">product topological spaces</a> and forming <a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a>, respectively. Hence this is a <a class="existingWikiWord" href="/nlab/show/monoidal+adjunction">monoidal adjunction</a>.</p> <p>In fact this is the <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> of what is even a <a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></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><msubsup><mi>Σ</mi> <mi>orth</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub><mo>⊣</mo><msubsup><mi>Ω</mi> <mi>orth</mi> <mn>∞</mn></msubsup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>OrthSpec</mi> <mi>stable</mi></msub><munderover><mrow></mrow><munder><mo>⟶</mo><mrow><msubsup><mi>Ω</mi> <mi>orth</mi> <mn>∞</mn></msubsup></mrow></munder><mover><mo>⟵</mo><mrow><msubsup><mi>Σ</mi> <mi>orth</mi> <mn>∞</mn></msubsup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mo>+</mo></msub></mrow></mover></munderover><msub><mi>Top</mi> <mi>Quillen</mi></msub></mrow><annotation encoding="application/x-tex"> (\Sigma^\infty_{orth}(-)_+ \dashv \Omega^\infty_{orth}) \;\colon\; OrthSpec_{stable} \underoverset {\underset{\Omega_{orth}^\infty}{\longrightarrow}} {\overset{\Sigma_{orth}^\infty(-)_+}{\longleftarrow}} {} Top_{Quillen} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure on topological spaces</a> and the stable <a class="existingWikiWord" href="/nlab/show/model+structure+on+orthogonal+spectra">model structure on orthogonal spectra</a> (<a href="Introduction+to+Stable+homotopy+theory+--+1-2#StableMonoidalQuillenSuspensionSpectrumFunctor">this cor.</a>) which implies (strong) modality of the derived functors on homotopy categories (<a href="Introduction+to+Stable+homotopy+theory+--+1-2#StrongMonoidalDerivedFunctorFromStrongMonoidalQuillenAdjunction">this prop.</a>).</p> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math></p> <ul> <li> <p>Examples arise in the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a>. See there for details.</p> </li> <li> <p>The quivalence between <a class="existingWikiWord" href="/nlab/show/module+spectra">module spectra</a> and <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> is exhibited by monoidal Quillen equivalences. See <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a> for details.</p> </li> </ul> <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/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><strong>monoidal Quillen adjunction</strong></p> </li> </ul> <h2 id="references">References</h2> <p>The notion of strong monoidal Quillen adjunction is def. 4.2.16 in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Model Categories</em> Mathematical Surveys and Monographs, Volume 63, AMS (1999) (<a href="https://www.math.rochester.edu/people/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>)</li> </ul> <p>The lax monoidal version is considered as definition 3.6 of</p> <ul> <li id="SchwedeShipley"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Equivalences of monoidal model categories</em> , Algebr. Geom. Topol. 3 (2003), 287–334 (<a href="http://arxiv.org/abs/math.AT/0209342">arXiv:math.AT/0209342</a>)</li> </ul> <p>The statements involving pushouts along free monoid morphisms are discussed in lemma 6.2 of</p> <ul id="SchwedeShipleyAlgebras"> <li><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Algebras and modules in monoidal model categories</em> Proc. London Math. Soc. (2000) 80(2): 491-511 (<a href="http://www.math.uic.edu/~bshipley/monoidal.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 31, 2019 at 14:34:42. 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