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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> model structure on dg-algebras </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/356/#Item_21" 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="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#general'>General</a></li> <li><a href='#on_connective_dgcalgebras'>On connective dgc-algebras</a></li> <ul> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#SullivanAlgebras'>Cofibrations and Sullivan algebras</a></li> <li><a href='#HomComplexes'>Simplicial hom-complexes</a></li> <li><a href='#relation_to_simplicial_sets'>Relation to simplicial sets</a></li> <li><a href='#relation_to_cosimplicial_commutative_algbras'>Relation to cosimplicial commutative algbras</a></li> <li><a href='#PreservationOfWeakEquivalencesUnderPushout'>Preservation of weak equivalences under pushout</a></li> <li><a href='#change_of_scalars'>Change of scalars</a></li> <li><a href='#CommVsNoncomm'>Commutative vs. non-commutative dg-algebras</a></li> </ul> </ul> <li><a href='#Unbounded'>Unbounded dg-algebras</a></li> <ul> <li><a href='#GradingsAndConventions'>Gradings and conventions</a></li> <li><a href='#ForUnboundedDGAlgebrasDefinition'>Definition</a></li> <li><a href='#properties_2'>Properties</a></li> <ul> <li><a href='#Properness'>Properness</a></li> <li><a href='#derived_tensor_product'>Derived tensor product</a></li> <li><a href='#SimplicialHomObjects'>Derived hom-functor</a></li> <li><a href='#DerivedCopowering'>Derived copowering over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math></a></li> <li><a href='#DerivedPowering'>Derived powering over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math></a></li> <ul> <li><a href='#claim'>Claim</a></li> </ul> <li><a href='#PathObjectsForUnboundedCommutative'>Path objects</a></li> <li><a href='#RelationToAInfinityAlgebras'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math>-algebra spectra</a></li> <li><a href='#RelationToEInfinityAlgebras'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔼</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{E}_\infty</annotation></semantics></math>-algebras</a></li> </ul> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#on_connective_dgcalgebras_2'>On connective dgc-algebras</a></li> <li><a href='#on_noncommutative_dgalgebras'>On non-commutative dg-algebras</a></li> <li><a href='#on_unbounded_dgalgebras'>On unbounded dg-algebras</a></li> <li><a href='#more'>More</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>A <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on a category of <a class="existingWikiWord" href="/nlab/show/differential+graded+algebras">differential graded algebras</a> or more specifically on a <a class="existingWikiWord" href="/nlab/show/category+of+differential+graded-commutative+algebras">category of differential graded-commutative algebras</a> tends to present an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebras</a>.</p> <p>For dg-algebras bounded in negative or positive degrees, the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> asserts that their model category structures are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to the corresponding <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+algebras">model structure on (co)simplicial algebras</a>. This case plays a central role in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>.</p> <p>The case of model structures on unbounded dg-algebras may be thought of as induced from this by passage to the <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> modeled on formal duals of the bounded dg-algebras. This is described at <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a>.</p> <h2 id="general">General</h2> <p>The category of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s is that of <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>s in a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>. Accordingly general results on 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> apply.</p> <p>Below we spell out special cases, such as restricting to <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> when working over a <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, or restricting to non-negatively graded cochain dg-algebras.</p> <h2 id="on_connective_dgcalgebras">On connective dgc-algebras</h2> <p>We discuss the projective model structure on <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+algebras">differential non-negatively graded-commutative algebras</a>. This was originally introduced in <a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76</a> as a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> for <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a>‘s approach to <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>.</p> <h3 id="definition">Definition</h3> <div class="num_defn" id="dgcCochainAlgebrasInNonNegativeDegrees"> <h6 id="definition_2">Definition</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, write</p> <div class="maruku-equation" id="eq:CategoryOfdgcAlgebras"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Categories</mi></mrow><annotation encoding="application/x-tex"> dgcAlg^{\geq 0}_{k} \;\in\; Categories </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">unital</a> <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+algebras">differential graded-commutative algebras</a> over <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> in non-negative degrees, equivalently the category of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\geq 0}(k)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a> in non-negative degrees, equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a>.</p> </div> <p>(<a href="#GelfandManin96">Gelfand-Manin 96, V.3.1</a>)</p> <div class="num_example"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/initial+object">initial</a> and <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a>)</strong></p> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">dgcAlg^{\geq 0}_{k}</annotation></semantics></math> <a class="maruku-eqref" href="#eq:CategoryOfdgcAlgebras">(1)</a>:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> is the ground field algebra <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>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> is the zero algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> (which is indeed a unital algebra).</p> </li> </ol> </div> <p>(Beware that this is incorrectly stated in <a href="#GelfandManin96">Gelfand-Manin 96, p. 335</a>)</p> <p>More generally:</p> <div class="num_example"> <h6 id="example_2">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/coproducts">coproducts</a> and <a class="existingWikiWord" href="/nlab/show/products">products</a>)</strong></p> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">dgcAlg^{\geq 0}_{k}</annotation></semantics></math> <a class="maruku-eqref" href="#eq:CategoryOfdgcAlgebras">(1)</a>:</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> is given by the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+algebras">tensor product of algebras</a>;</p> <p>(see at <em><a href="category+of+monoids#PushoutOfCommutativeMonoids">pushouts of commutative monoids</a></em>)</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/product">product</a> is given by <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> on <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a></p> <p>(since the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> is a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>).</p> </li> </ol> </div> <div class="num_defn" id="dgcCochainAlgebraInNonNegDegreeOfFiniteType"> <h6 id="definition_3">Definition</h6> <p><strong>(finite type)</strong></p> <p>Say that a <a class="existingWikiWord" href="/nlab/show/dgc-algebra">dgc-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A \in dgcAlg^{\geq 0}_k</annotation></semantics></math> (def. <a class="maruku-ref" href="#dgcCochainAlgebrasInNonNegativeDegrees"></a>) is of <em><a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a></em> if its <a class="existingWikiWord" href="/nlab/show/forgetful+functor">underlying</a> <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> is in each degree of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> as a <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>-<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>.</p> </div> <div class="num_defn" id="ProjectiveModelStructureOnCdgAlg"> <h6 id="definition_4">Definition</h6> <p><strong>(projective model structure on rational connective dgc-algebras)</strong></p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> for the catgory of <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a> from def. <a class="maruku-ref" href="#dgcCochainAlgebrasInNonNegativeDegrees"></a> equipped with the following <a class="existingWikiWord" href="/nlab/show/classes">classes</a> of morphisms:</p> <ul> <li> <p><em><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a></em> are the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>;</p> </li> <li> <p><em><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a></em> are the degreewise <a class="existingWikiWord" href="/nlab/show/surjections">surjections</a>;</p> </li> </ul> </div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, Def. 4.2</a>, <a href="#GelfandManin96">Gelfand-Manin 96, Def. V.3.3</a>)</p> <div class="num_prop" id="IndeedProjectiveModelStructureOnCdgAlg"> <h6 id="proposition">Proposition</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from def. <a class="maruku-ref" href="#ProjectiveModelStructureOnCdgAlg"></a> is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a>, to be called the <em>projective model structure</em>.</p> </div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, Theorem 4.3</a>, <a href="#GelfandManin96">Gelfand-Manin 96, Theorem V.3.4</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>)</strong></p> <p>Evidently every <a class="existingWikiWord" href="/nlab/show/object">object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> (Def. <a class="maruku-ref" href="#ProjectiveModelStructureOnCdgAlg"></a>, prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a>) is a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a>. Therefore these model categories structures are in particular also structures of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>.</p> </div> <p>The nature of the cofibrations is discussed <a href="#SullivanAlgebras">below</a>.</p> <h3 id="properties">Properties</h3> <h4 id="SullivanAlgebras">Cofibrations and Sullivan algebras</h4> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p><strong>(sphere and disk algebras)</strong></p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[n]</annotation></semantics></math> for the graded vector space which is the ground field <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> in degree <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> and 0 in all other degrees. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>, consider the <a class="existingWikiWord" href="/nlab/show/semifree+dgc-algebras">semifree dgc-algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≔</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> S(n) \coloneqq (\wedge^\bullet k[n], 0) </annotation></semantics></math></div> <p>and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/semifree+dgc-algebras">semifree dgc-algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">(</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>⊕</mo><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mtd> <mtd><mo stretchy="false">(</mo><mi>n</mi><mo>></mo><mn>0</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding="application/x-tex"> D(n) \coloneqq \left\lbrace \array{ 0 & (n = 0) \\ (\wedge^\bullet (k[n] \oplus k[n-1]), 0) & (n \gt 0) } \right. </annotation></semantics></math></div> <p>for which the differential sends the generator of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[n-1]</annotation></semantics></math> to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[n]</annotation></semantics></math></p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> i_n \colon S(n) \to D(n) </annotation></semantics></math></div> <p>for the obvious morphism that takes the generator in degree <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> to the generator in degree <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> (and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n = 0</annotation></semantics></math> it is the unique morphism from the <a class="existingWikiWord" href="/nlab/show/initial+object">initial object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0,0)</annotation></semantics></math>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n \gt 0</annotation></semantics></math> write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>n</mi></msub><mo lspace="verythinmathspace">:</mo><mi>k</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>D</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> j_n \colon k[0] \to D(n) \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p><strong>(generating cofibrations)</strong></p> <p>The sets</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>i</mi> <mi>n</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><mo>∪</mo><mo stretchy="false">{</mo><mi>k</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>S</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>,</mo><mi>S</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mi>k</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> I = \{i_n \}_{n \geq 1} \cup \{k[0] \to S(0), S(0) \to k[0]\} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mo stretchy="false">{</mo><msub><mi>j</mi> <mi>n</mi></msub><msub><mo stretchy="false">}</mo> <mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex"> J = \{j_n \}_{n \gt 1} </annotation></semantics></math></div> <p>are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting the model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a> as a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>.</p> </div> <p>review includes (<a href="#Hess06">Hess 06, p. 6</a>)</p> <p>In this section we describe the cofibrations in the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcalg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcalg^{\geq 0}_k)_{proj}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ProjectiveModelStructureOnCdgAlg"></a>, prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a>). Notice that it is these that are in the image of the dual <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a>.</p> <p>Before we characterize the cofibrations, first some notation.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi></mrow><annotation encoding="application/x-tex">\wedge^\bullet V</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> over it. Equipped with the trivial differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d = 0</annotation></semantics></math> this is a <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>=</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V, d=0)</annotation></semantics></math>.</p> <p>With <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> our ground field we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0)</annotation></semantics></math> for the corresponding dg-algebra, the tensor unit for the standard <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgAlg</mi></mrow><annotation encoding="application/x-tex">dgAlg</annotation></semantics></math>. This is the <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> on the 0-vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) = (\wedge^\bullet 0, 0)</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p><strong>(Sullivan algebras)</strong></p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/relative+Sullivan+algebra">relative Sullivan algebra</a></strong> is a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> of dg-algebras that is an inclusion</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>A</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (A,d) \to (A \otimes_k \wedge^\bullet V, d') </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,d)</annotation></semantics></math> some dg-algebra and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> some graded vector space, such that</p> <ul> <li> <p>there is a <a class="existingWikiWord" href="/nlab/show/well+ordered+set">well ordered set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi></mrow><annotation encoding="application/x-tex">J</annotation></semantics></math></p> </li> <li> <p>indexing a basis <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>v</mi> <mi>α</mi></msub><mo>∈</mo><mi>V</mi><mo stretchy="false">|</mo><mi>α</mi><mo>∈</mo><mi>J</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{v_\alpha \in V| \alpha \in J\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>;</p> </li> <li> <p>such that with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mo>=</mo><mi>span</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mi>α</mi></msub><mo stretchy="false">|</mo><mi>α</mi><mo><</mo><mi>β</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_{\lt \beta} = span(v_\alpha | \alpha \lt \beta)</annotation></semantics></math> for all basis elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">v_\beta</annotation></semantics></math> we have that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>′</mo><msub><mi>v</mi> <mi>β</mi></msub><mo>∈</mo><mi>A</mi><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,. </annotation></semantics></math></div></li> </ul> <p>This is called a <strong>minimal</strong> <a class="existingWikiWord" href="/nlab/show/relative+Sullivan+algebra">relative Sullivan algebra</a> if in addition the condition</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>α</mi><mo><</mo><mi>β</mi><mo stretchy="false">)</mo><mo>⇒</mo><mo stretchy="false">(</mo><mi>deg</mi><msub><mi>v</mi> <mi>α</mi></msub><mo>≤</mo><mi>deg</mi><msub><mi>v</mi> <mi>β</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta) </annotation></semantics></math></div> <p>holds. For a Sullivan algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> relative to the tensor unit we call the <a class="existingWikiWord" href="/nlab/show/semifree+dga">semifree dga</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V,d)</annotation></semantics></math> simply a <strong>Sullivan algebra</strong>. And a <strong>minimal Sullivan algebra</strong> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> is a minimal relative Sullivan algebra.</p> </div> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Sullivan algebras were introduced by <a class="existingWikiWord" href="/nlab/show/Dennis+Sullivan">Dennis Sullivan</a> in his development of <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>. This is one of the key application areas of the model structure on dg-algebras.</p> </div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p><strong>(<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras)</strong></p> <p>Because they are <a class="existingWikiWord" href="/nlab/show/semifree+dgas">semifree dgas</a>, Sullivan dg-algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V,d)</annotation></semantics></math> are (at least for degreewise finite dimensional <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math>) <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a>s of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a>s.</p> <p>The co-commutative differential co-algebra encoding the corresponding <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a> is the free cocommutative algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\vee^\bullet V^*</annotation></semantics></math> on the degreewise dual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> with differential <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo>=</mo><msup><mi>d</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">D = d^*</annotation></semantics></math>, i.e. the one given by the formula</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∨</mo><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">(</mo><mi>d</mi><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\omega \in V</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">v_i \in V^*</annotation></semantics></math>.</p> </div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p><strong>(cofibrations are relative Sullivan algebras)</strong></p> <p>The cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_{k})_{proj}</annotation></semantics></math> are precisely the <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> of <a class="existingWikiWord" href="/nlab/show/relative+Sullivan+algebras">relative Sullivan algebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,d) \to (A\otimes_k \wedge^\bullet V, d')</annotation></semantics></math>.</p> <p>Accordingly, the cofibrant objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_{k})_{proj}</annotation></semantics></math> are precisely the Sullivan algebras <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\wedge^\bullet V, d)</annotation></semantics></math></p> </div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, Prop. 7.11</a><a href="#GelfandManin96">Gelfand-Manin 96., Prop. V.5.4</a>)</p> <h4 id="HomComplexes">Simplicial hom-complexes</h4> <p>We discuss <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial</a> <a class="existingWikiWord" href="/nlab/show/mapping+spaces">mapping spaces</a> between <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a>. These <em>almost</em> make the projective model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a> into a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>, except that the <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a>/<a class="existingWikiWord" href="/nlab/show/powering">powering</a> <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> holds only for <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> or else on <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a> of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a>. Still, this has useful implications, for instance it implies that the <a class="existingWikiWord" href="/nlab/show/reduced+suspension">reduced suspension</a> and <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> on [augmented algebras|augmented]] <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>.</p> <div class="num_defn" id="MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"> <h6 id="definition_7">Definition</h6> <p><strong>(simplicial mapping spaces)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A,B \in dgcAlg^{\geq 0}_k</annotation></semantics></math> (def. <a class="maruku-ref" href="#dgcCochainAlgebrasInNonNegativeDegrees"></a>), let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> Maps(A,B) \in sSet </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> whose <a class="existingWikiWord" href="/nlab/show/n-simplices">n-simplices</a> are the dg-algebra <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> into the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <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> with the de Rham complex of <a class="existingWikiWord" href="/nlab/show/polynomial+differential+forms+on+the+n-simplex">polynomial differential forms on the n-simplex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega_{poly}^\bullet(\Delta^n)</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>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">)</mo> <mi>n</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> Maps(A,B)_n \;\coloneqq\; Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) </annotation></semantics></math></div> <p>and whose face and degeneracy maps are the obvious ones induced from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo lspace="verythinmathspace">:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\Omega_{poly}^\bullet \colon \Delta^{op} \to dgcAlg^{\geq 0}_k</annotation></semantics></math> is canonically a <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a> in dgc-algebras.</p> <p>We also call this the <em>simplicial <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a></em> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to <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>. This construction naturally extends to a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>×</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><mo>⟶</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Maps(-,-) \;\colon\; (dgcAlg^{\geq 0}_k)^{op} \times dgcAlg^{\geq 0}_k \longrightarrow dgcAlg^{\geq 0}_k </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a> with the category itself.</p> <p>Observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mspace width="thickmathspace"></mspace><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mrow></mrow> <mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow></msub><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>A</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \;\simeq\; {}_{\Omega^\bullet_{poly}}Hom_{dgcAlg^{\geq 0}_k} \left( \Omega^\bullet_{poly}(\Delta^n) \otimes_k A \,,\, \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \,, </annotation></semantics></math></div> <p>where on the right we have those dg-algebra homomorphism which in addition preserves the left <a class="existingWikiWord" href="/nlab/show/dg-module">dg-module</a> structure over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly}(\Delta^n)</annotation></semantics></math>. This induces for any three <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A,B,C \in dgcAlg^{\geq 0}_k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/composition">composition</a> homomorphism of <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> out of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of mapping spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>∘</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow> <mi>sSet</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \circ^{sSet}_{A,B,C} \;\colon\; Maps(A,B) \times Maps(B,C) \longrightarrow Maps(A,C) \,. </annotation></semantics></math></div></div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, 5.1</a>)</p> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>The set of 0-simplices of of the <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Maps(A,B)</annotation></semantics></math> in def. <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a> is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the ordinary <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of dg-algebras:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Maps(A,B)_0 \simeq Hom_{dgcAlg^{\geq 0}_k}(A,B) </annotation></semantics></math></div> <p>and under this identification the two notions of <a class="existingWikiWord" href="/nlab/show/composition">composition</a> agree.</p> </div> <p>Definition <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a> makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">dgcAlg^{\geq 0}_k</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> (“<a class="existingWikiWord" href="/nlab/show/simplicial+category">simplicial category</a>”). The follows says that it is also <a class="existingWikiWord" href="/nlab/show/powering">powered</a>, not over all of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math>, but over finite simplicial sets:</p> <div class="num_prop" id="PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets"> <h6 id="proposition_4">Proposition</h6> <p><strong>(powering over finite simplicial sets)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">A, B \in dgcAlg^{\geq 0}_k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">S \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>, there is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mo>≥</mo></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{dgcAlg^{\geq}_k}(A, \Omega^\bullet_{poly}(S) \otimes_k B) \longrightarrow Hom_{sSet}( S, Maps(A,B) ) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of <a class="existingWikiWord" href="/nlab/show/dgc-algebras">dgc-algebras</a> into the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> with the <a class="existingWikiWord" href="/nlab/show/polynomial+differential+forms+on+n-simplices">polynomial differential forms on n-simplices</a> from def. <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a> to the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> in <a class="existingWikiWord" href="/nlab/show/simplicial+sets">simplicial sets</a> into the simplicial <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> from def. <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a>.</p> <p>Moreover, this morphism is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> if one of the following conditions holds:</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finite+set">finite</a> <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>;</p> </li> <li> <p><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 of <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> (def. <a class="maruku-ref" href="#dgcCochainAlgebraInNonNegDegreeOfFiniteType"></a>).</p> </li> </ul> </div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, lemma 5.2</a>)</p> <div class="num_prop"> <h6 id="proposition_5">Proposition</h6> <p><strong>(pullback powering axiom)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">i \colon V \to W</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p \colon X \to Y</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">dgcAlg^{\geq 0}_k</annotation></semantics></math>. Then their <a class="existingWikiWord" href="/nlab/show/pullback+power">pullback power</a> with respect to the simplicial <a class="existingWikiWord" href="/nlab/show/mapping+space">mapping space</a> functor (def. <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>p</mi> <mi>i</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Maps</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Maps</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Maps</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></munder><mi>Maps</mi><mo stretchy="false">(</mo><mi>W</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> p^i \;\colon\; Maps(W,X) \longrightarrow Maps(V,X) \underset{Maps(V,Y)}{\times} Maps(W,Y) </annotation></semantics></math></div> <p>is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is a cofibration and <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> a fibration in the projective <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure from prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a>;</p> </li> <li> <p>in addition a <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a> (i.e. a weak equivalence in the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a>) if at least one of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> or <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 weak equivalence in the projective model structure from prop. <a class="maruku-ref" href="#IndeedProjectiveModelStructureOnCdgAlg"></a>.</p> </li> </ol> </div> <p>(<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, prop. 5.3</a>)</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>Prop. <a class="maruku-ref" href="#PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets"></a> <em>would</em> say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> with respect to the simplicial enrichment from def. <a class="maruku-ref" href="#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees"></a> were it not for the fact that prop. <a class="maruku-ref" href="#PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets"></a> gives the <a class="existingWikiWord" href="/nlab/show/powering">powering</a> only over finite simplicial sets.</p> </div> <h4 id="relation_to_simplicial_sets">Relation to simplicial sets</h4> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Quillen+adjunction+between+simplicial+sets+and+connective+dgc-algebras">Quillen adjunction between simplicial sets and connective dgc-algebras</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/PL+de+Rham+complex">PL de Rham complex</a>-construction is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> in a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/opposite+model+structure">opposite</a> of the <a class="existingWikiWord" href="/nlab/show/projective+model+structure+on+connective+dgc-algebras">projective model structure on connective dgc-algebras</a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure on simplicial sets</a></p> </li> </ul> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>DiffGradedCommAlgebras</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msubsup><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi> <mi>op</mi></msubsup><munderover><mrow><msub><mo>⊥</mo> <mpadded width="0"><mi>Qu</mi></mpadded></msub></mrow><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>exp</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder><mover><mo>⟵</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></munderover><msub><mi>SimplicialSets</mi> <mi>Qu</mi></msub></mrow><annotation encoding="application/x-tex"> \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} SimplicialSets_{Qu} </annotation></semantics></math></div></div> <h4 id="relation_to_cosimplicial_commutative_algbras">Relation to cosimplicial commutative algbras</h4> <p>The <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> gives a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+algebras">projective model structure on cosimplicial commutative algebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>cAlg</mi> <mi>k</mi> <mi>Δ</mi></msubsup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">(cAlg_k^{\Delta})_{proj}</annotation></semantics></math>.</p> <h4 id="PreservationOfWeakEquivalencesUnderPushout">Preservation of weak equivalences under pushout</h4> <div class="num_prop" id="PushoutAlongRelativeSullivanModelsPreservesQuasiIsos"> <h6 id="proposition_7">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> along <a class="existingWikiWord" href="/nlab/show/relative+Sullivan+models">relative Sullivan models</a> preserves <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a>)</strong></p> <p>In the projective model structure on connective dgc-algebras (Def. <a class="maruku-ref" href="#ProjectiveModelStructureOnCdgAlg"></a>), the operation of <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> along a <a class="existingWikiWord" href="/nlab/show/relative+Sullivan+model">relative Sullivan model</a> preserves weak equivalences (quasi-isomorphisms).</p> </div> <p>This is <a href="#FelixHalperinThomas00">Felix-Halperin-Thomas 00, Lemma 14.2, using Prop. 6.7 (ii)</a>.</p> <p>The same statement for augmented dgc-algebras is in <a href="#Baues88">Baues 88, Section I.8, Lemma 8.16</a>.</p> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>Lemma 14.2 in <a href="#FelixHalperinThomas00">Felix-Halperin-Thomas 00</a> is stated under the additional assumption that the dgc-algebras being pushed put have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>0</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">H^0(-) = k</annotation></semantics></math>. But this is only used to find the stronger statement that the pushout is itself a Sullivan model. The argument via Prop. 6.7 that the pushout is a quasi-iso does not use this assumption.</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Prop. <a class="maruku-ref" href="#PushoutAlongRelativeSullivanModelsPreservesQuasiIsos"></a> comes close to saying that the projective model structure on connective dgc-algebras is a <a class="existingWikiWord" href="/nlab/show/left+proper+model+category">left proper model category</a>, but not quite: The class of all <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> is larger than that of <a class="existingWikiWord" href="/nlab/show/relative+Sullivan+algebras">relative Sullivan algebras</a>, it includes also their <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a> (see e.g <a href="#Hess06">Hess 06</a>).</p> </div> <h4 id="change_of_scalars">Change of scalars</h4> <p> <div class='num_prop' id='ExtensionOfScalarsQuillenAdjunction'> <h6>Proposition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℚ</mi><mover><mo>↪</mo><mi>i</mi></mover><mi>k</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q} \xhookrightarrow{i} k</annotation></semantics></math> the corresponding inclusion of the <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>. Then the iduced <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace><mo>⊣</mo><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;\dashv\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</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 maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub><munderover><mrow><msub><mo>⊥</mo> <mpadded width="0"><mi>Qu</mi></mpadded></msub></mrow><munder><mo>⟶</mo><mrow><msub><mi>res</mi> <mi>ℚ</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℚ</mi></msub><mi>k</mi></mrow></mover></munderover><mo maxsize="1.2em" minsize="1.2em">(</mo><msubsup><mi>dgcAlg</mi> <mi>ℚ</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo maxsize="1.2em" minsize="1.2em">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex"> \big( dgcAlg^{\geq 0}_k \big)_{proj} \underoverset {\underset{res_{\mathbb{Q}}}{\longrightarrow}} {\overset{ (-) \otimes_{\mathbb{Q}} k }{\longleftarrow}} {\bot_{\mathrlap{Qu}}} \big( dgcAlg^{\geq 0}_{\mathbb{Q}} \big)_{proj} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between the respective projective model categories (Def. <a class="maruku-ref" href="#ProjectiveModelStructureOnCdgAlg"></a>).</p> <p></p> </div> (<a href="#BousfieldGugenheim76">Bousfield & Gugenheim 1976, Lemma 11.6</a>) <div class='proof'> <h6>Proof</h6> <p>It is immediate that <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a> is a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a>:</p> <ol> <li> <p>It preserves <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>, since these are the <a class="existingWikiWord" href="/nlab/show/surjections">surjections</a> of <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/sets">sets</a>, and restriction of scalars does not change the underlying sets.</p> </li> <li> <p>It preserves <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>, since these are the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> on <a class="existingWikiWord" href="/nlab/show/cochain+cohomology">cochain cohomology</a> <a class="existingWikiWord" href="/nlab/show/cohomology+groups">groups</a> of <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a>, and, again, restriction of scalars does not change the underlying sets of the underlying cochain complexes.</p> </li> </ol> <p></p> </div> </p> <h4 id="CommVsNoncomm">Commutative vs. non-commutative dg-algebras</h4> <blockquote> <p>this needs harmonization</p> </blockquote> <div class="num_prop"> <h6 id="proposition_8">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><msub><mi>dgcAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> F dgcAlg_k \to dgAlg_k </annotation></semantics></math></div> <p>from (graded-)commutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s to dg-algebras is the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> part of a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ab</mi><mo lspace="verythinmathspace">:</mo><mi>dgAlg</mi><mover><mo>→</mo><mo>←</mo></mover><mi>CdgAlg</mi><mo>:</mo><mi>F</mi></mrow><annotation encoding="application/x-tex"> Ab \colon dgAlg \stackrel{\leftarrow}{\to} CdgAlg : F </annotation></semantics></math></div></div> <blockquote> <p>boundedness?</p> </blockquote> <div class="proof"> <h6 id="proof">Proof</h6> <p>The forgetful functor clearly preserves fibrations and cofibrations. It has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, the free abelianization functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Ab</annotation></semantics></math>, which sends a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> to its quotient <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A/[A,A]</annotation></semantics></math>.</p> </div> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>Let the ground <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <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> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>. Then every <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> which has the structure of an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over</a> the <a class="existingWikiWord" href="/nlab/show/E-k-operad">E-∞ operad</a> has a <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">A \to A_c</annotation></semantics></math> to a commutative dg-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">A_c</annotation></semantics></math> which is</p> <ul> <li> <p>a morphism of <a class="existingWikiWord" href="/nlab/show/E-k-operad">E-∞ algebras</a> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding="application/x-tex">A_c</annotation></semantics></math> has the obvious <a class="existingWikiWord" href="/nlab/show/E-k-operad">E-∞ algebras</a> structure)</p> </li> <li> <p>a weak weak equivalence in the model structure on dg-algebras (i.e. a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> of the underlying cochain complexes).</p> </li> </ul> </div> <p>This is in (<a href="#KrizMay95">Kriz-May 95, II.1.5</a>).</p> <p>So this says that the weak equivalence classes of the commutative dg-algebras in the model category of all dg-algebras already exhaust the most general non-commutative but homotopy-commutative dg-algebras.</p> <div class="num_remark" id="HomotopyFaithfulnessOfforgettingCommutativity"> <h6 id="remark_8">Remark</h6> <p>Discussion of a restricted kind of homotopy-faithfulness of the forgetful functor from the homotopy theory of commutative to not-necessarily commutative dg-algebras is in (<a href="#Amrani14">Amrani 14</a>).</p> </div> <h2 id="Unbounded">Unbounded dg-algebras</h2> <p>We discuss now the case of unbounded dg-algebras. For these there is no longer the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> available. Instead, these can be understood as arising naturally as function <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 in the <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived</a> <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a> over formal duals of bounded dg-algebras, see <a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a>.</p> <h3 id="GradingsAndConventions">Gradings and conventions</h3> <p>In <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> two categorical gradings interact: a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> <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>-groupoid <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> has a space of <a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">X_k</annotation></semantics></math> for all non-negative <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 each such has itself a <em><a class="existingWikiWord" href="/nlab/show/simplicial+T-algebra">simplicial T-algebra</a></em> of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from <a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a>.</p> <p>Functions on a bare <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>-groupoid <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>, modeled as a <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>, form a <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(K)</annotation></semantics></math>, which under the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> identifies with a cochain <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> (meaning: with positively graded differential) in non-negative degree</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>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mo>∂</mo> <mn>2</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></msup><msup><mo stretchy="false">↓</mo> <mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>↦</mo><mi>𝒪</mi></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>K</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msubsup><mo>∂</mo> <mn>0</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy="false">↑</mo> <mrow><msubsup><mo>∂</mo> <mn>1</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy="false">↑</mo> <mrow><msubsup><mo>∂</mo> <mn>2</mn> <mo>*</mo></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>K</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mrow><msubsup><mo>∂</mo> <mn>0</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy="false">↑</mo> <mrow><msubsup><mo>∂</mo> <mn>1</mn> <mo>*</mo></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>K</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>↔</mo><mo>∼</mo></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mi>i</mi></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left( \array{ \vdots \\ \downarrow \downarrow \downarrow \downarrow \\ K_2 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \downarrow^{\partial_2} \\ K_1 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \\ K_0 } \right) \;\;\;\;\; \stackrel{\mathcal{O}}{\mapsto} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \uparrow \uparrow \uparrow \\ \mathcal{O}(K_2) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \uparrow^{\partial_2^*} \\ \mathcal{O}(K_1) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \\ \mathcal{O}(K_0) } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \cdots \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_2 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_1 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_0 \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \vdots } \right) \,. </annotation></semantics></math></div> <p>On the other hand, a representable <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> has itself a <em><a class="existingWikiWord" href="/nlab/show/simplicial+T-algebra">simplicial T-algebra</a></em> of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mover><mo>↔</mo><mo>∼</mo></mover><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(X) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \mathcal{O}(X)_0 \\ \uparrow \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \uparrow \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \uparrow \uparrow \uparrow \\ \vdots } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \mathcal{O}(X)_0 \\ \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \\ \vdots } \right) \,. </annotation></semantics></math></div> <p>Taking this together, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">X_\bullet</annotation></semantics></math> a general <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>, its function algebra is generally an <em>unbounded</em> cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>k</mi><mo>−</mo><mi>p</mi><mo>=</mo><mi>q</mi></mrow></munder><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>k</mi></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>p</mi></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>3</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mn>0</mn></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>3</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(X_\bullet) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ \bigoplus_{k-p = q} \mathcal{O}(X_k)_{-p} \\ \uparrow \\ \vdots \\ \uparrow^d \\ \mathcal{O}(X_1)_0 \oplus \mathcal{O}(X_2)_{-1} \oplus \mathcal{O}(X_3)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_0 \oplus \mathcal{O}(X_1)_{-1} \oplus \mathcal{O}(X_2)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_{-1} \oplus \mathcal{O}(X_1)_{-2} \oplus \mathcal{O}(X_2)_{-3}\oplus \cdots \\ \uparrow^{d} \\ \vdots } \right) \,. </annotation></semantics></math></div> <h3 id="ForUnboundedDGAlgebrasDefinition">Definition</h3> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>For <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0 let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>cdgAlg</mi><mo>=</mo><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> cdgAlg = CMon(Ch_\bullet(k)) </annotation></semantics></math></div> <p>be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> this is a</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> <p>which is</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a>.</p> </li> </ul> </div> <p>The existence of the model structure follows from the general discussion at <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a>.</p> <p>Properness and combinatoriality is discussed in (<a href="#ToenVezzosi">ToënVezzosi</a>):</p> <ul> <li> <p>in lemma 2.3.1.1 they state that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mo>+</mo></msub></mrow><annotation encoding="application/x-tex">cdgAlg_+</annotation></semantics></math> constitutes the first two items in a triple which they call an <em>HA context</em> .</p> </li> <li> <p>this implies their assumption 1.1.0.4 which asserts properness and combinatoriality</p> </li> </ul> <p>Discussion of cofibrations in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgAlg</mi> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">dgAlg_{proj}</annotation></semantics></math> is in (<a href="#Keller">Keller</a>).</p> <h3 id="properties_2">Properties</h3> <h4 id="Properness">Properness</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAg_k</annotation></semantics></math> be the projective model structure on commutative unbounded dg-algebras from above.</p> <p>This is a <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>. See MO discussion <a href="http://mathoverflow.net/q/204414/381">here</a>.</p> <h4 id="derived_tensor_product">Derived tensor product</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAg_k</annotation></semantics></math> be the projective model structure on commutative unbounded dg-algebras from above</p> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>For cofibrant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A \in cdgAlg_k</annotation></semantics></math>, the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><mi>k</mi><mi>Mod</mi><mo>→</mo><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> A\otimes_k (-) : k Mod \to A Mod </annotation></semantics></math></div> <p>preserves <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A,B \in cdgAlg_k</annotation></semantics></math>, their <a class="existingWikiWord" href="/nlab/show/derived+functor">derived</a> <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">k Mod</annotation></semantics></math> coincides in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> with the derived <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">k Mod</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><mi>A</mi><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mi>k</mi> <mi>L</mi></munderover><mi>B</mi><mover><mo>→</mo><mrow></mrow></mover><mi>A</mi><msubsup><mo>⊗</mo> <mi>k</mi> <mi>L</mi></msubsup><mi>B</mi></mrow><annotation encoding="application/x-tex"> A \coprod_k^{L} B \stackrel{}{\to} A \otimes_k^L B </annotation></semantics></math></div> <p>is an isomorphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(k Mod)</annotation></semantics></math>.</p> </div> <p>This follows by the above with (<a href="#ToenVezzosi">ToënVezzosi, assumption 1.1.0.4, and page 8</a>).</p> <h4 id="SimplicialHomObjects">Derived hom-functor</h4> <p>The model structure on unbounded dg-algebras is <em>almost</em> a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>. See the section <em><a href="model+structure+on+dg-algebras+over+an+operad#SimplicialEnrichment">simplicial enrichment</a></em> at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a></em> for details.</p> <div class="num_defn"> <h6 id="definition_9">Definition</h6> <p>Let <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> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy="false">(</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly} : sSet \to (cdgAlg_k)^{op}</annotation></semantics></math> be the functor that assigns polynomial <a class="existingWikiWord" href="/nlab/show/differential+forms+on+simplices">differential forms on simplices</a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>dgcAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A,B \in dgcAlg_k</annotation></semantics></math> define the <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">(</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><msub><mi>Hom</mi> <mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> cdgAlg_k(A,B) : ([n] \mapsto Hom_{cdgAlg_k}(A, B \otimes_k \Omega^\bullet_{poly}(\Delta[n])) \,. </annotation></semantics></math></div> <p>This extends to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo>×</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo>→</mo><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> cdgAlg_k(-,-) : cdgAlg_k^{op} \times cdgAlg_k \to sSet \,. </annotation></semantics></math></div></div> <div class="num_prop"> <h6 id="proposition_10">Proposition</h6> <p>The functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cdgAlg_k(-,-)</annotation></semantics></math> satisfies the dual of the <a class="existingWikiWord" href="/nlab/show/pushout-product+axiom">pushout-product axiom</a>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">i : A \to B</annotation></semantics></math> any cofibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p : X \to Y</annotation></semantics></math> any fibration, the canonical morphism</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>i</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy="false">)</mo><mo>:</mo><msub><mi>cdgalg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><msub><mo>×</mo> <mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msub><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (i^*, p_*) : cdgalg_k(A,B) \to cdgAlg_k(A,X) \times_{cdgAlg_k(A,Y)} cdgAlg_k(B,Y) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Kan+fibration">Kan fibration</a>, which is acyclic if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> or <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.</p> </div> <p>This implies in particular that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> cofibrant, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cdgAlg_k(A,B)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> <p>The proof works along the lines of (<a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76, prop. 5.3</a>). See also the discussion at <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a>.</p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We give the proof for a special case. The general case is analogous.</p> <p>We show that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> cofibrant, and for any <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> (automatically fibrant), <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cdgAlg_k(A,B)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>.</p> <p>By a standard fact in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a> (due to <a href="#BousfieldGugenheim76">Bousfield-Gugenheim 76</a>, discussed at <a class="existingWikiWord" href="/nlab/show/differential+forms+on+simplices">differential forms on simplices</a>) we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy="false">(</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mo>+</mo></msubsup><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly} : sSet \to (cdgAlg^+_k)^{op}</annotation></semantics></math> is a left <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen functor</a>, hence in particular sends acyclic cofibrations to acyclic cofibrations, hence acyclic <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s of <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a>s to acyclic fibrations of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s.</p> <p>Specifically for each <a class="existingWikiWord" href="/nlab/show/horn">horn</a> inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\Lambda[n]_k \hookrightarrow \Delta[n]</annotation></semantics></math> we have that the restriction map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>→</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly}(\Delta[n]) \to \Omega^\bullet_{poly}(\Lambda[n]_k)</annotation></semantics></math> is an acyclic fibration in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mo>*</mo></msubsup></mrow><annotation encoding="application/x-tex">cdgAlg_k^*</annotation></semantics></math>, hence in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math>.</p> <p>A <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>-horn in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cdgAlg_k(A,B)</annotation></semantics></math> is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \to B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)</annotation></semantics></math>. A filler for this horn is a lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Λ</mi><mo stretchy="false">[</mo><mi>n</mi><msub><mo stretchy="false">]</mo> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && B \otimes \Omega^\bullet_{poly}(\Delta[n]) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ A &\to& B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k) } \,. </annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is cofibrant, then such a lift does always exist.</p> </div> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>cdgAlg</mi></mrow><annotation encoding="application/x-tex">A \in cdgAlg</annotation></semantics></math> cofibrant, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">cdgAlg_k(A,B)</annotation></semantics></math> is the correct <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℝ</mi><mi>Hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> cdgAlg_k(A,B) \simeq \mathbb{R}Hom(A,B) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>By the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is cofibrant and according to the facts discussed at <a class="existingWikiWord" href="/nlab/show/derived+hom-space">derived hom-space</a>, we need to show that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>B</mi><mo>:</mo><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>↦</mo><mi>B</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> s B : [n] \mapsto B\otimes_k \Omega^\bullet_{poly}(\Delta[n]) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/simplicial+resolution">resolution</a>, or <em>simplicial frame</em> for <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>. (Notice that every object is fibrant in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math>).</p> <p>Since polynomial differential forms are acyclic on simplices (discussed <a href="http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices">here</a>) it follows that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mi>B</mi><mo>→</mo><mi>s</mi><mi>B</mi></mrow><annotation encoding="application/x-tex"> const B \to s B </annotation></semantics></math></div> <p>is degreewise a weak equivalence. It remains to show that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">s A</annotation></semantics></math> is fibrant 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><mo stretchy="false">[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msub><mo stretchy="false">]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding="application/x-tex">[\Delta^{op}, cdgAlg_k]_{Reedy}</annotation></semantics></math>.</p> <p>One finds that the matching object is given by</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>match</mi><mi>s</mi><mi>B</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo>=</mo><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (match s B)_k = B \otimes \Omega^\bullet_{poly}(\partial \Delta[k]) \,. </annotation></semantics></math></div> <p>Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>B</mi></mrow><annotation encoding="application/x-tex">s B</annotation></semantics></math> is Reedy fibrant if in each degree the morphism</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>s</mi><msub><mi>B</mi> <mi>k</mi></msub><mo>→</mo><mo stretchy="false">(</mo><mi>match</mi><mi>s</mi><mi>B</mi><msub><mo stretchy="false">)</mo> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (s B_k \to (match s B)_k ) = (\Omega^\bullet_{poly}(\partial \Delta[k] \hookrightarrow \Delta[k])) </annotation></semantics></math></div> <p>is a fibration. But this follows from the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}</annotation></semantics></math> is a left <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen functor</a> (as discussed at <a class="existingWikiWord" href="/nlab/show/differential+forms+on+simplices">differential forms on simplices</a>).</p> </div> <h4 id="DerivedCopowering">Derived copowering over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math></h4> <p>We discuss a concrete model for the <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>-copowering of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">(cdgAlg_k)^\circ</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> in terms of an operation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>First notice a basic fact about ordinary commutative algebras.</p> <div class="num_prop"> <h6 id="proposition_12">Proposition</h6> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">CAlg_k</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> is given by the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> over <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>:</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>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mi>A</mi></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>e</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>Id</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left( \array{ A &\stackrel{i_A}{\to}& A \coprod B &\stackrel{i_B}{\leftarrow}& B } \right) \simeq \left( \array{ A &\stackrel{Id_A \otimes_k e_B}{\to}& A \otimes_k B & \stackrel{e_A \otimes Id_B}{\leftarrow}& B } \right) </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>We check the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the coproduct: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">C \in CAlg_k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">f,g : A,B \to C</annotation></semantics></math> two morphisms, we need to show that there is a unique morphism <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>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">(f,g) : A \otimes_k B \to C</annotation></semantics></math> such that the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>e</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>Id</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{Id_A \otimes e_B}{\to}& A \otimes_k B &\stackrel{e_A \otimes Id_B}{\leftarrow}& B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}} \\ && C } </annotation></semantics></math></div> <p>commutes. For the left triangle to commute we need that <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>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> sends elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a,e_B)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f(a)</annotation></semantics></math>. For the right triangle to commute we need that <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>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> sends elements of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(e_A, b)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g(b)</annotation></semantics></math>. Since every element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mrow><annotation encoding="application/x-tex">A \otimes_k B</annotation></semantics></math> is a product of two elements of this form</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>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (a,b) = (a,e_B) \cdot (e_A, b) </annotation></semantics></math></div> <p>this already uniquely determines <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>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> to be given on elements by the map</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>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>f</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>g</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (a,b) \mapsto f(a) \cdot g(b) \,. </annotation></semantics></math></div> <p>That this is indeed an <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>-algebra homomorphism follows from the fact that <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi></mrow><annotation encoding="application/x-tex">g</annotation></semantics></math> are</p> </div> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>For these derivations it is crucial that we are working with commutative algebras.</p> </div> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>We have that the <a class="existingWikiWord" href="/nlab/show/copower">copower</a>ing of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with the map of sets from two points to the single point</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><mo>*</mo><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><mo>*</mo><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>A</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>A</mi><mover><mo>→</mo><mi>μ</mi></mover><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (* \coprod * \to *) \cdot A \simeq ( A \otimes_k A \stackrel{\mu}{\to} A ) </annotation></semantics></math></div> <p>is the product morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. And that the tensoring with the map from the empty set to the point</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>∅</mi><mo>→</mo><mo>*</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>A</mi><mo>≃</mo><mo stretchy="false">(</mo><mi>k</mi><mover><mo>→</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></mover><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\emptyset \to *)\cdot A \simeq (k \stackrel{e_A}{\to} A) </annotation></semantics></math></div> <p>is the unit morphism on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">f : S \to T</annotation></semantics></math> any map of sets we have that the tensoring</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>S</mi><mover><mo>→</mo><mi>f</mi></mover><mi>T</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>A</mi><mo>=</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">|</mo><mi>S</mi><mo stretchy="false">|</mo></mrow></msup><mo>→</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">|</mo><mi>T</mi><mo stretchy="false">|</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> (S \stackrel{f}{\to} T) \cdot A = A^{\otimes_k |S|} \to A^{\otimes_k |T|} </annotation></semantics></math></div> <p>is the morphism between <a class="existingWikiWord" href="/nlab/show/tensor+power">tensor power</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> of the cardinalities of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>, respectively, whose component over a copy of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> on the right corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">t \in T</annotation></semantics></math> is the iterated product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">|</mo><msup><mi>f</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">{</mo><mi>t</mi><mo stretchy="false">}</mo><mo stretchy="false">|</mo></mrow></msup><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A^{\otimes_k |f^{-1}\{t\}|} \to A</annotation></semantics></math> on as many tensor powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as there are elements in the preimage of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>.</p> </div> <p>The analogous statements hold true with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">CAlg_k</annotation></semantics></math> replaced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math>: for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S \in sSet</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> A \in cdgAlg_k</annotation></semantics></math> we obtain a simplicial cdg-algebra</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⋅</mo><mi>A</mi><mo>∈</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup></mrow><annotation encoding="application/x-tex"> S \cdot A \in cdgAlg_k^{\Delta^{op}} </annotation></semantics></math></div> <p>by the ordinary degreewise <a class="existingWikiWord" href="/nlab/show/copower">copower</a>ing over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math> has coproducts (equal to the tensor product over <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>).</p> <p>This is equivalently a commutative monoid in simplicial unbounded chain complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup><mo>≃</mo><mi>CMon</mi><mo stretchy="false">(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><msup><mo stretchy="false">)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> cdgAlg_k^{\Delta^{op}} \simeq CMon(Ch^\bullet(k)^{\Delta^{op}}) \,. </annotation></semantics></math></div> <p>By the logic of the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> the symmetric <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal</a> <a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a> functor (via the <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a>) sends this to a commutative <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> in non-positively graded cochain complexes in unbounded cochain complexes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><msubsup><mi>Ch</mi> <mo>−</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\bullet(S \cdot A) \in CMon(Ch^\bullet_-(Ch^\bullet(k))) \,. </annotation></semantics></math></div> <p>Since the <a class="existingWikiWord" href="/nlab/show/total+complex">total complex</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Tot</mi><mo>:</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)</annotation></semantics></math> is itself symmetric lax monoidal (…), this finally yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Tot</mi><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> Tot C^\bullet(S \cdot A) \in CMon(Ch^\bullet(k)) \simeq cdgAlg_k </annotation></semantics></math></div> <div class="num_defn"> <h6 id="definition_10">Definition</h6> <p>Define the functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CC</mi><mo>:</mo><mi>sSet</mi><mo>×</mo><mi>cdgAlg</mi><mo>→</mo><mi>cdgAlg</mi></mrow><annotation encoding="application/x-tex"> CC : sSet \times cdgAlg \to cdgAlg </annotation></semantics></math></div> <p>by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CC</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>Tot</mi><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy="false">(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CC(S,A) := Tot C^\bullet(S \cdot A) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_10">Remark</h6> <p>We have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CC</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">⨁</mo> <mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><mo stretchy="false">(</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">|</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy="false">|</mo></mrow></msup><msub><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> CC(Y,A)^n := \bigoplus_{k \geq 0} (A^{\otimes_k |Y_k| })_{n+k} </annotation></semantics></math></div></div> <p>This appears essentially (…) as (<a href="#GinotTradlerZeinalian">GinotTradlerZeinalian, def 3.1.1</a>).</p> <div class="num_prop"> <h6 id="proposition_13">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limit">(∞,1)-copowering</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>dgcAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">(dgcAlg_k)^\circ</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> is modeled by the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CC</mi></mrow><annotation encoding="application/x-tex">CC</annotation></semantics></math>.</p> </div> <p>This follows from (<a href="#GinotTradlerZeinalian">GinotTradlerZeinalian, theorem 4.2.7</a>), which asserts that the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> of this tensoring is the unique <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>, up to equivalence, satisfying the axioms 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>-copowering.</p> <div class="num_prop"> <h6 id="proposition_14">Proposition</h6> <p>The functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CC</mi><mo>:</mo><mi>sSet</mi><mo>×</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> CC : sSet \times cdgAlg_k \to cdgAlg_k </annotation></semantics></math></div> <p>preserves weak equivalences in both arguments.</p> </div> <p>This is essentially due to (<a href="#Pirashvili">Pirashvili</a>). The full statement is (<a href="#GinotTradlerZeinalian">GinotTradlerZeinalian, prop. 4.2.1</a>).</p> <div class="num_remark"> <h6 id="remark_11">Remark</h6> <p>This means that the assumption for the copowering models of higher order <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> are satsified in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math> which are described in the section <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#PirashviliHigherOrder">Pirashvili's higher Hochschild homology</a> is satisfied:</p> <p>this means that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>cdgAlg</mi></mrow><annotation encoding="application/x-tex">A \in cdgAlg</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">S \in sSet</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CC</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CC(S,A)</annotation></semantics></math> is a model for the function <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>-algebra on the <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mi>A</mi></mrow><annotation encoding="application/x-tex">Spec A</annotation></semantics></math>. See the section <a href="http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#OvercdgAlgs">Higher order Hochschild homology modeled on cdg-algebras</a> for more details.</p> </div> <h4 id="DerivedPowering">Derived powering over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math></h4> <div class="num_prop"> <h6 id="claim">Claim</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">S \in \infty Grpd</annotation></semantics></math> be presented by a degreewise finite <a class="existingWikiWord" href="/nlab/show/simplicial+set">simplicial set</a> (which we denote by the same symbol).</p> <p>Then the <a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">cdgAlg_k</annotation></semantics></math> over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>-shaped diagram constant on <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> is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly}(S)</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><mi>const</mi><mi>k</mi><mo>≃</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}{\lim_{\leftarrow}}_S const k \simeq \Omega^\bullet_{poly}(S) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>We show dually that for degreewise finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> the assignment <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>Spec</mi><mi>A</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>Spec</mi><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(S, Spec A) \mapsto Spec (\Omega^\bullet_{poly}(S) \otimes A)</annotation></semantics></math> models the <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>-copowering in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">cdgAlg_k^{op}</annotation></semantics></math>.</p> <p>By the discussion at <a href="http://nlab.mathforge.org/nlab/show/limit+in+a+quasi-category#Tensoring">(∞,1)-copowering</a> it is sufficient to to establish an equivalence</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>dgcAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mo stretchy="false">(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (dgcAlg_{k}^{op})^\circ(Spec (\Omega^\bullet_{poly}(S) \otimes A), Spec B) \simeq \infty Grpd(S, (dgcAlg_{k}^{op})^\circ(Spec A, Spec B)) </annotation></semantics></math></div> <p>natural in <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>. Consider a cofibrant model of <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>, which we denote by the same symbol. The we compute with 1-categorical <a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a> calculus</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>sSet</mi><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>,</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo stretchy="false">(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy="false">]</mo><mo>,</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy="false">(</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mo stretchy="false">(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} sSet(S, cdgAlg_k^{op}(Spec A,Spec B)) & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{sSet}(S \times \Delta[r], cdgAlg_k^{op}(Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{Set}(S_k \times \Delta[k,r], Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{cdgAlg_k^{op}}((S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(\int^{[k] \in \Delta} (S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S \times \Delta[r]) \times Spec A, Spec B)) \end{aligned} \,, </annotation></semantics></math></div> <p>where all steps are <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>s and the dot denotes the ordinary 1-categorical <a class="existingWikiWord" href="/nlab/show/copower">copower</a>ing of the 1-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>cdgAlg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">cdgAlg^{op}</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. In the last step we are using that the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> commutes with finite limits of dg-algebras. (This is where the finiteness assumption is needed).</p> <p>Now we use that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow><annotation encoding="application/x-tex">\Omega^\bullet_{poly}</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/product">product</a>s up to <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> (as discussed <a href="http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices#Properties">here</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega^\bullet_{poly}(S \times \Delta[r]) \simeq \Omega^\bullet_{poly}(S) \otimes \Omega_{poly}^\bullet(\Delta[r]) \,. </annotation></semantics></math></div> <p>This being a weak equivalence between fibrant objects and since <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 assumed cofibrant, we have by the <a href="#SimplicialHomObjects">above discussion</a> of the derived hom-functor (and using the <a class="existingWikiWord" href="/nlab/show/factorization+lemma">factorization lemma</a>) a weak equivalence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy="false">(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mi>Δ</mi><mo stretchy="false">[</mo><mi>r</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdots \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S) \times Spec \Omega^\bullet_{poly}\Delta[r]) \times Spec A, Spec B)) \,. </annotation></semantics></math></div> <p>Since all this is <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> in <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>, this proves the claim.</p> </div> <h4 id="PathObjectsForUnboundedCommutative">Path objects</h4> <div class="num_prop"> <h6 id="proposition_15">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">A \in cdgAlg_k</annotation></semantics></math>, a <a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>P</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mi>fib</mi></mover><mi>A</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is given by</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 stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> P(A) := A \otimes_k \Omega^\bullet_{poly}(\Delta[1]) </annotation></semantics></math></div></div> <p>This follows along the above lines. The statement appears for instance as (<a href="#Behrend">Behrend, lemma 1.19</a>).</p> <h4 id="RelationToAInfinityAlgebras">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math>-algebra spectra</h4> <p>For every <a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</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> there is the notion of <a class="existingWikiWord" href="/nlab/show/algebra+spectra">algebra spectra</a> over <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>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>:</mo><mo>=</mo><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R := H \mathbb{Z}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> for the <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s. Then unbounded dg-algebras (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>) are one model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math>-algebra spectra.</p> <div class="num_prop"> <h6 id="proposition_16">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the standard <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">H \mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebra+spectra">algebra spectra</a> and the model structure on unbounded differential graded algebras.</p> </div> <p>See <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a> for details.</p> <h4 id="RelationToEInfinityAlgebras">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝔼</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{E}_\infty</annotation></semantics></math>-algebras</h4> <p>Commutative dg-algebras over a field <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> of characteristic 0 constitute a presentation of <a class="existingWikiWord" href="/nlab/show/E-infinity+algebras">E-infinity algebras</a> over <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> ([Lurie, prop. A.7.1.4.11]).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-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+chain+complexes">model structure on chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-operads">model structure on dg-operads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a></p> <ul> <li> <p><strong>model structure on dg-algebras</strong></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">model structure on dg-categories</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="on_connective_dgcalgebras_2">On connective dgc-algebras</h3> <p>The cofibrantly generated model structure on <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+algebras">differential graded-commutative algebras</a> is originally due to</p> <ul> <li id="BousfieldGugenheim76"><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Gugenheim">Victor Gugenheim</a>, <em><a class="existingWikiWord" href="/nlab/show/On+PL+deRham+theory+and+rational+homotopy+type">On PL deRham theory and rational homotopy type</a></em>, Memoirs of the AMS 179 (1976) (<a href="https://bookstore.ams.org/memo-8-179">ams:memo-8-179</a>)</li> </ul> <p>Textbook account:</p> <ul> <li id="GelfandManin96"><a class="existingWikiWord" href="/nlab/show/Sergei+Gelfand">Sergei Gelfand</a>, <a class="existingWikiWord" href="/nlab/show/Yuri+Manin">Yuri Manin</a>, Chapter V of: <em><a class="existingWikiWord" href="/nlab/show/Methods+of+homological+algebra">Methods of homological algebra</a></em>, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (<a href="https://doi.org/10.1007/978-3-662-12492-5">doi:10.1007/978-3-662-12492-5</a>)</li> </ul> <p>Review:</p> <ul> <li id="Hess06"> <p><a class="existingWikiWord" href="/nlab/show/Kathryn+Hess">Kathryn Hess</a>, p. 6 of: <em>Rational homotopy theory: a brief introduction</em>, contribution to <em><a href="https://jdc.math.uwo.ca/summerschool/">Summer School on Interactions between Homotopy Theory and Algebra</a></em>, University of Chicago, July 26-August 6, 2004, Chicago (<a href="http://arxiv.org/abs/math.AT/0604626">arXiv:math.AT/0604626</a>), chapter in Luchezar Lavramov, <a class="existingWikiWord" href="/nlab/show/Dan+Christensen">Dan Christensen</a>, <a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a> (eds.), <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007 (<a href="http://dx.doi.org/10.1090/conm/436">doi:10.1090/conm/436</a>)</p> </li> <li id="Moerman15"> <p><a class="existingWikiWord" href="/nlab/show/Joshua+Moerman">Joshua Moerman</a>, Section II of: <em>Rational Homotopy Theory</em>, 2015 (<a href="https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/MoermanRationalHomotopyTheory.pdf" title="pdf">pdf</a>)</p> </li> </ul> <p>The approach in <a href="#Hess06">Hess 06</a> makes use of the general discussion in section 3 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Kirsten+Schemmerhorn">Kirsten Schemmerhorn</a>, <em>Model categories and simplicial methods</em>, Notes from lectures given at the University of Chicago, August 2004, in: <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007(<a href="http://arxiv.org/abs/math.AT/0609537">arXiv:math.AT/0609537</a>, <a href="http://dx.doi.org/10.1090/conm/436">doi:10.1090/conm/436</a>)</li> </ul> <p>that obtains the model structure from the <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a>.</p> <p>See also</p> <ul> <li id="Baues88"> <p>Baues, Chapter I.8 of: <em>Algebraic Homotopy</em> (<a href="https://www.maths.ed.ac.uk/~v1ranick/papers/baues4.pdf">pdf</a></p> </li> <li id="FelixHalperinThomas00"> <p><a class="existingWikiWord" href="/nlab/show/Yves+F%C3%A9lix">Yves Félix</a>, <a class="existingWikiWord" href="/nlab/show/Stephen+Halperin">Stephen Halperin</a>, <a class="existingWikiWord" href="/nlab/show/Jean-Claude+Thomas">Jean-Claude Thomas</a>, <em>Rational Homotopy Theory</em>, Graduate Texts in Mathematics, 205, Springer-Verlag, 2000 (<a href="https://link.springer.com/book/10.1007/978-1-4613-0105-9">doi:10.1007/978-1-4613-0105-9</a>)</p> </li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/equivariant+rational+homotopy+theory">equivariant rational homotopy theory</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Laura+Scull">Laura Scull</a>, <em>A model category structure for equivariant algebraic models</em>, Transactions of the American Mathematical Society 360 (5), 2505-2525, 2008 (<a href="https://doi.org/10.1090/S0002-9947-07-04421-2">doi:10.1090/S0002-9947-07-04421-2</a>)</li> </ul> <h3 id="on_noncommutative_dgalgebras">On non-commutative dg-algebras</h3> <p>For general <strong>non-commutative</strong> (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in</p> <ul> <li id="Jardine97"><a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, <em><a class="existingWikiWord" href="/nlab/files/JardineModelDG.pdf" title="A Closed Model Structure for Differential Graded Algebras">A Closed Model Structure for Differential Graded Algebras</a></em>, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.</li> </ul> <p>This is also the structure used in</p> <ul> <li>J. L Castiglioni, G. Cortiñas, <em>Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence</em>, J. Pure Applied Algebra <strong>191</strong> (2004) 119-142 [<a href="https://arxiv.org/abs/math/0306289">arXiv:math/0306289</a>, <a href="https://doi.org/10.1016/j.jpaa.2003.11.009">doi:10.1016/j.jpaa.2003.11.009</a>]</li> </ul> <p>where aspects of its relation to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">model structure on cosimplicial rings</a> is discussed. (See <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> for more on this).</p> <h3 id="on_unbounded_dgalgebras">On unbounded dg-algebras</h3> <p>Discussion of the model structure on unbounded dg-algebras over a field of characteristic 0 is in</p> <ul> <li id="ToenVezzosi"><a class="existingWikiWord" href="/nlab/show/Bertrand+To%C3%ABn">Bertrand Toën</a>, <a class="existingWikiWord" href="/nlab/show/Gabriele+Vezzosi">Gabriele Vezzosi</a>, <em>HAG II, geometric stacks and applicatons</em> (<a href="http://arxiv.org/abs/math/0404373v4">arXiv:math/0404373v4</a>)</li> </ul> <p>A general discussion of algebras over an operad in unbounded chain complexes is in</p> <ul> <li id="Hinich"><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>Homological algebra of homotopy algebras</em>, Communications in Algebra, 25(10). 3291-3323 (1997)(<a href="http://arxiv.org/abs/q-alg/9702015">arXiv:q-alg/9702015</a>, <em>Erratum</em> (<a href="http://arxiv.org/abs/math/0309453">arXiv:math/0309453</a>))</li> </ul> <p>A survey of some useful facts with an eye towards <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a> is in</p> <ul> <li id="Behrend"><a class="existingWikiWord" href="/nlab/show/Kai+Behrend">Kai Behrend</a>, <em>Differential graded schemes I: prefect resolving algebras</em> (<a href="http://arxiv.org/abs/math/0212225">arXiv:0212225</a>)</li> </ul> <p>Discussion of cofibrations in unbounded dg-algebras are in</p> <ul> <li id="Keller"><a class="existingWikiWord" href="/nlab/show/Bernhard+Keller">Bernhard Keller</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-algebras, modules and functor categories</em> (<a href="http://www.math.jussieu.fr/~keller/publ/ainffun.pdf">pdf</a>)</li> </ul> <h3 id="more">More</h3> <p>The derived copowering of unbounded commutative dg-algebras over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sSet</mi></mrow><annotation encoding="application/x-tex">sSet</annotation></semantics></math> is discussed (somewhat implicitly) in</p> <ul> <li id="GinotTradlerZeinalian"><a class="existingWikiWord" href="/nlab/show/Gr%C3%A9gory+Ginot">Grégory Ginot</a>, Thomas Tradler, Mahmoud Zeinalian, <em>Derived higher Hochschild homology, topological chiral homology and factorization algebras</em>, (<a href="http://arxiv.org/abs/1011.6483">arxiv/1011.6483</a>)</li> </ul> <p>The <em>commutative</em> product on the dg-algebra of the higher order Hochschild complex is discussed in</p> <ul id="GinotTradlerZeinalianChenModel"> <li><a class="existingWikiWord" href="/nlab/show/Gr%C3%A9gory+Ginot">Grégory Ginot</a>, Thomas Tradler, Mahmoud Zeinalian, <em>A Chen model for mapping spaces and the surface product</em> (<a href="http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2231v1.pdf">pdf</a>)</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/E-infinity+algebras">E-infinity algebras</a> is discussed in</p> <ul> <li id="KrizMay95"> <p><a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a> and <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Operads, algebras, modules and motives</em> , Astérisque No 233 (1995)</p> </li> <li id="Lurie"> <p><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, section 7.1 of <em>Higher algebra</em> (<a href="http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf">pdf</a>)</p> </li> </ul> <p>The relation between commutative and non-commutative dgas is further discussed in</p> <ul> <li id="Amrani14"> <p><a class="existingWikiWord" href="/nlab/show/Ilias+Amrani">Ilias Amrani</a>, <em>Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view</em> (<a href="http://arxiv.org/abs/1401.7285">arXiv:1401.7285</a>)</p> </li> <li id="Amrani14b"> <p><a class="existingWikiWord" href="/nlab/show/Ilias+Amrani">Ilias Amrani</a>, <em>Rational homotopy theory of function spaces and Hochschild cohomology</em> (<a href="http://arxiv.org/abs/1406.6269">arXiv:1406.6269</a>)</p> </li> </ul> <p>For more see also at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a></em>.</p> <p>Discussion of <a class="existingWikiWord" href="/nlab/show/homotopy+limits">homotopy limits</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+colimits">homotopy colimits</a> of dg-algebras is in</p> <ul> <li id="Walter06"><a class="existingWikiWord" href="/nlab/show/Ben+Walter">Ben Walter</a>, <em>Rational Homotopy Calculus of Functors</em> (<a href="http://arxiv.org/abs/math/0603336">arXiv:math/0603336</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 25, 2023 at 19:21:01. 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