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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 chain complexes </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/6275/#Item_35" 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="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</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> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#in_nonnegative_degree'>In non-negative degree</a></li> <li><a href='#for_unbounded_chain_complexes'>For unbounded chain complexes</a></li> </ul> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#CochainNonNeg'>In non-negative degree</a></li> <ul> <li><a href='#StandardQuillenOnBounded'>The standard (Quillen) model structures</a></li> <ul> <li><a href='#ProjectiveStructureOnChainComplexes'>Projective structure on chain complexes</a></li> <li><a href='#InjectiveStructureOnCochainComplexes'>Injective structure on cochain complexes</a></li> </ul> <li><a href='#GeneralResults'>Resolution model structures</a></li> <li><a href='#CochainNonNegProj'>Projective model structure on connective cochain complexes</a></li> </ul> <li><a href='#OnUnbounded'>In unbounded degree</a></li> <ul> <li><a href='#ProjectiveModelStructureOnUnboundedChainComplexes'>Standard projective model structure on unbounded chain complexes</a></li> <li><a href='#InjectiveModelStructureOnUnboundedChainComplexes'>Standard injective model structure on unbounded chain complexes</a></li> <li><a href='#InUnboundedDegreeGeneralResults'>Christensen-Hovey model structures using projective classes</a></li> <li><a href='#ExamplesOfStructuresOnUnboundedComplexes'>Examples</a></li> <ul> <li><a href='#CategoricalProjectiveClass'>Categorical projective class structure</a></li> <li><a href='#PureProjectiveClass'>Pure projective class structure</a></li> </ul> <li><a href='#gillespies_approach_using_cotorsion_pairs'>Gillespie’s approach using cotorsion pairs</a></li> <li><a href='#cisinskideglise_approach_using_descent_structures'>Cisinski-Deglise approach using descent structures</a></li> </ul> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#Properness'>Properness</a></li> <li><a href='#ExactFuncsAndQuillenFuncs'>Left/right exact functors and Quillen adjunctions</a></li> <li><a href='#cofibrant_generation'>Cofibrant generation</a></li> <li><a href='#inclusion_into_simplicial_objects'>Inclusion into simplicial objects</a></li> <li><a href='#cofibrations'>Cofibrations</a></li> <li><a href='#relation_to_module_spectra'>Relation to module spectra</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#References'>References</a></li> <ul> <li><a href='#for_bounded_chain_complexes'>For bounded chain complexes</a></li> <li><a href='#ForUnboundedChainComplexes'>For unbounded chain complexes</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>Model structures on chain complexes are <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures on <a class="existingWikiWord" href="/nlab/show/categories+of+chain+complexes">categories of chain complexes</a> whose weak equivalences are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a>. (There is also a <a class="existingWikiWord" href="/nlab/show/Hurewicz+model+structure+on+chain+complexes">Hurewicz model structure on chain complexes</a> whose weak equivalences are <a class="existingWikiWord" href="/nlab/show/chain+homotopy+equivalences">chain homotopy equivalences</a>.)</p> <p>Via these model structures, all of the standard techniques in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a>, such as <a class="existingWikiWord" href="/nlab/show/injective+resolutions">injective resolutions</a> and <a class="existingWikiWord" href="/nlab/show/projective+resolutions">projective resolutions</a>, are special cases of constructions in <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, such as <a class="existingWikiWord" href="/nlab/show/cofibrant+resolutions">cofibrant resolutions</a> and <a class="existingWikiWord" href="/nlab/show/fibrant+resolutions">fibrant resolutions</a>.</p> <p>The existence of these model structures depends subtly on whether the chain complexes in question are bounded or not.</p> <h3 id="in_nonnegative_degree">In non-negative degree</h3> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">Chain complexes</a> in non-negative degree in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</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> are special in that they may be identified via the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> as <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>A</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ch_{\bullet \geq 0}(A) \simeq A^{\Delta^{op}} \,. </annotation></semantics></math></div> <p>Similarly, <a class="existingWikiWord" href="/nlab/show/cochain+complex">cochain complexes</a> are identified with <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>A</mi> <mi>Δ</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Ch^{\bullet \geq 0}(A) \simeq A^{\Delta} \,. </annotation></semantics></math></div> <p>At least 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 the category of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">A^{\Delta^{op}}</annotation></semantics></math> is the category of abelian <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>s it inherits naturally a model category structure from the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>, which <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presents</a> the <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-groupoid">∞-groupoid</a>s.</p> <p>The model structure on chain complexes transports this presentation of the special <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 given by abelian simplicial groups along the Dold-Kan correspondence to chain complexes.</p> <p>Analogous statements apply to the category of unbounded chain complexes and the canonical <a class="existingWikiWord" href="/nlab/show/stable+%28infinity%2C1%29-category">stable (infinity,1)-category</a> <a class="existingWikiWord" href="/nlab/show/Spec">Spec</a> of <a class="existingWikiWord" href="/nlab/show/spectrum">spectra</a>.</p> <p>This we discuss below in</p> <ul> <li><em><a href="#CochainNonNeg">In non-negative degree</a></em></li> </ul> <h3 id="for_unbounded_chain_complexes">For unbounded chain complexes</h3> <p>Model structures on unbounded (co)chain complexes can be understood as presentations of <a class="existingWikiWord" href="/nlab/show/spectrum+objects">spectrum objects</a> in model structures of bounded (co)chain complexes.</p> <p>See</p> <ul> <li><em><a href="#ForUnboundedChainComplexes">For unbounded chain complexes</a></em></li> </ul> <h2 id="definitions">Definitions</h2> <h3 id="CochainNonNeg">In non-negative degree</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <p>Recall that by the dual <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">C^\Delta</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalent to the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^\bullet_+(C)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a> in non-negative degree. This means that we can transfer results discussed at <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+objects">model structure on cosimplicial objects</a> to cochain complexes (see <a href="http://arxiv.org/PS_cache/math/pdf/0312/0312531v1.pdf">Bousfield2003, section 4.4</a> for more).</p> <h4 id="StandardQuillenOnBounded">The standard (Quillen) model structures</h4> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≔</mo><mi>R</mi></mrow><annotation encoding="application/x-tex"> \mathcal{A} \coloneqq R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> for its <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/modules">modules</a>.</p> <p>We discuss the</p> <ul> <li> <p><a href="#ProjectiveStructureOnChainComplexes">Projective structure on chain complexes</a></p> </li> <li> <p><a href="#InjectiveStructureOnCochainComplexes">Injective structure on cochain complexes</a></p> </li> </ul> <h5 id="ProjectiveStructureOnChainComplexes">Projective structure on chain complexes</h5> <div class="num_theorem" id="ProjectiveModelStructure"> <h6 id="theorem">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_{\bullet \geq 0 }(\mathcal{A})</annotation></semantics></math> (in non-negative degree) whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> are the morphisms that are <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> in each <em><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a></em> degree;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are degreewise <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> with degreewise <a class="existingWikiWord" href="/nlab/show/projective+object">projective</a> <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>;</p> </li> </ul> <p>called the <strong>projective model structure</strong>.</p> </div> <p>The projective model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msub></mrow><annotation encoding="application/x-tex">Ch_{\bullet \geq 0}</annotation></semantics></math> is originally due to (<a href="#Quillen67">Quillen 67, II.4, pages II.4.11, II.4.12</a>). See also (<a href="#GoerssSchemmerhorn06">Goerss-Schemmerhorn 06, Theorem 1.5</a>, <a href="#Dungan10">Dungan 10, 2.4.2, proof in section 2.5</a>).</p> <p> <div class='num_prop' id='ProjectiveModelStructureOnConnectiveChainComplexesIsProper'> <h6>Proposition</h6> <p>The projective model structure on connective chain complexes is a <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>.</p> </div> (e.g. <a href="#Jardine03">Jardine 03, Prop. 1.5</a>)</p> <p> <div class='num_prop' id='ProjectiveModelStructureOnConnectiveChainComplexesIsMonoidal'> <h6>Proposition</h6> <p>With respect to the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> this is a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>.</p> </div> (e.g. <a href="#Jardine03">Jardine 03, Prop. 1.5</a>, <a href="#SchwedeShipley03">Schwede & Shipley 2003, p. 312 (26 of 48)</a>).</p> <h5 id="InjectiveStructureOnCochainComplexes">Injective structure on cochain complexes</h5> <p>Dually</p> <div class="num_theorem" id="InjectiveModelStructure"> <h6 id="theorem_2">Theorem</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on non-negatively graded <a class="existingWikiWord" href="/nlab/show/cochain+complexes">cochain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\bullet \geq 0 }(\mathcal{A})</annotation></semantics></math> whose</p> <ul> <li> <p>weak equivalences are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a>;</p> </li> <li> <p>cofibrations are the morphisms that are <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math> in each <em><a class="existingWikiWord" href="/nlab/show/positive+number">positive</a></em> degree;</p> </li> <li> <p>fibrations are degreewise <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> with <a class="existingWikiWord" href="/nlab/show/injective+object">injective</a> <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>,</p> </li> </ul> <p>called the <strong>injective model structure</strong>.</p> </div> <p>(<a href="#Dungan10">Dungan 10, Theorem 2.4.5</a>)</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>This means that a chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mo>•</mo></msub><mo>∈</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C_\bullet \in Ch_{\bullet}(\mathcal{A})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> in the projective model structure, theorem <a class="maruku-ref" href="#ProjectiveModelStructure"></a>, precisely if it consists of <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a>. Accordingly, a <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> in the projective model structure is precisely what in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> is called a <em><a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></em>. Dually for <a class="existingWikiWord" href="/nlab/show/fibrant+resolutions">fibrant resolutions</a> in the injective model structure, theorem <a class="maruku-ref" href="#InjectiveModelStructure"></a>, and <a class="existingWikiWord" href="/nlab/show/injective+resolutions">injective resolutions</a> in homological algebra.</p> <p>This way the traditional definition of <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a> relates to the general construction of <a class="existingWikiWord" href="/nlab/show/derived+functors">derived functors</a> in <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> theory. See there for more details. Similar comments apply to the various other model structures below.</p> </div> <h4 id="GeneralResults">Resolution model structures</h4> <p>There are <em>resolution model structures</em> on cosimplicial objects in a model category, due to (<a href="#DwyerKanStover">DwyerKanStover</a>), reviewed in (<a href="#Bousfield">Bousfield</a>)</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>(…)</p> </div> <div class="num_theorem"> <h6 id="theorem_3">Theorem</h6> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>∈</mo><mi>Obj</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} \in Obj(A)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/class">class</a> of objects, such 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> has enough <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">\mathcal{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>s.</p> <p>Then there is a model category structure on non-negatively graded cochain complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>𝒢</mi></msub></mrow><annotation encoding="application/x-tex">Ch^{\bullet \geq 0}(A)_{\mathcal{G}}</annotation></semantics></math> whose</p> <ul> <li> <p>weak equivalences are maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f : X \to Y</annotation></semantics></math> such that for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">K \in A</annotation></semantics></math> the induced map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo><mo>→</mo><mi>A</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>K</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A(Y,K) \to A(X,K)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> of chain complexes of abelian groups;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a cofibration if it is <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">\mathcal{G}</annotation></semantics></math>-monic in positive degree;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a fibration if it is degreewise a <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a> with degreewise <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">\mathcal{G}</annotation></semantics></math>-injective kernel.</p> </li> </ul> </div> <p>See <a href="http://arxiv.org/PS_cache/math/pdf/0312/0312531v1.pdf">Bousfield2003, section 4.4</a>.</p> <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> has enough <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>s 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">\mathcal{G}</annotation></semantics></math> is the class of all of them, this reproduces the <a href="#StandardQuillenOnBounded">standard Quillen model structure</a> discussed above:</p> <div class="num_cor"> <h6 id="corollary">Corollary</h6> <p>Let <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> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with enough <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>s. Then there is a model category structure on non-negatively graded cochain complexes <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>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\geq 0}(A)</annotation></semantics></math> whose</p> <ul> <li> <p>weak equivalences are the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s;</p> </li> <li> <p>fibrations are the morphisms that are <a class="existingWikiWord" href="/nlab/show/epimorphism">epimorphism</a>s with injective <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> in each degree;</p> </li> <li> <p>cofibrations are the morphisms which are <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s in <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> in each <em>positive</em> degree.</p> </li> </ul> </div> <p>If we take <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">\mathcal{G}</annotation></semantics></math> to be the class of all objects 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> this gives the following structure.</p> <div class="num_cor"> <h6 id="corollary_2">Corollary</h6> <p>There is a model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>A</mi><msub><mo stretchy="false">)</mo> <mi>tot</mi></msub></mrow><annotation encoding="application/x-tex">Ch^{\bullet\geq 0}(A)_{tot}</annotation></semantics></math> whose</p> <ul> <li> <p>weak equivalences are cochain <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a>s;</p> </li> <li> <p>fibrations are the morphisms that are degreewise <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a>s and whose <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>s are <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>s;</p> </li> <li> <p>cofibration are the morphisms that are in positive degree <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s.</p> </li> </ul> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> is a category of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over some <a class="existingWikiWord" href="/nlab/show/field">field</a>, we have that every epi/mono splits and that every <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> is a homotopy equivalence. Moreover, in this case every chain complex is quasi-isomorphic to its <a class="existingWikiWord" href="/nlab/show/homology">homology</a> (regarded as a chain complex with zero differentials).</p> <p>This is the model structure which induces the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a> over a <a class="existingWikiWord" href="/nlab/show/field">field</a> that is used in <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>.</p> </div> <h4 id="CochainNonNegProj">Projective model structure on connective cochain complexes</h4> <p>We discuss a model structure on connective cochain complexes of abelian groups in which the fibrations are the degreewise epis. This follows an analogous proof in (<a href="model+structure+on+dg-algebras#Jardine97">Jardine 97</a>).</p> <div class="num_theorem" id="ProjectiveModelStructureOnConnectiveCochainComplexes"> <h6 id="theorem_4">Theorem</h6> <p><strong>(projective model structure on connective cochain complexs )</strong></p> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\bullet \geq 0}(Ab)</annotation></semantics></math> of non-negatively graded cochain complexes of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> becomes a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> with</p> <ul> <li> <p>fibrations the degreewise surjections;</p> </li> <li> <p>weak equivalences the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s.</p> </li> </ul> <p>Moreover this is a <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a>-structure with respect to the canonical structure of an <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> induced from the dual <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan equivalence</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ch^\bullet_+(Ab) \simeq Ab^\Delta</annotation></semantics></math> by the fact that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ab</mi> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">Ab^\Delta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/category+of+cosimplicial+objects">category of cosimplicial objects</a> (see there) in a category with all limits and colimits.</p> </div> <p>The first part of this theorem is claimed, without proof, in <a href="#CastiglioniCortinas03">Castiglioni-Cortinas 03, Def. 4.7</a>.</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>We spell out a proof of the model structure below in a sequence of lemmas. The proof that this is a simplicial model category is at <a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a>.</p> </div> <p>We record a detailed proof of the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>•</mo><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\bullet \geq 0}(Ab)</annotation></semantics></math> with fibrations the degreewise surjections, following the appendix of (<a href="#Stel">Stel 10</a>).</p> <p>As usual, 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> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n]</annotation></semantics></math> for the complex concentrated on the additive group of <a class="existingWikiWord" href="/nlab/show/integer">integer</a>s 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>1</mn></mrow><annotation encoding="application/x-tex">n \geq 1</annotation></semantics></math> write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n-1,n]</annotation></semantics></math> for the cochain complex <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><mi>⋯</mi><mn>0</mn><mo>→</mo><mi>ℤ</mi><mover><mo>→</mo><mi>Id</mi></mover><mi>ℤ</mi><mo>→</mo><mn>0</mn><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(0 \to \cdots 0 \to \mathbb{Z} \stackrel{Id}{\to} \mathbb{Z} \to 0 \cdots)</annotation></semantics></math> with the two copies of <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> in degree <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-1</annotation></semantics></math> and <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>.</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 = 0</annotation></semantics></math> let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}[-1,0] = 0</annotation></semantics></math>, for convenience.</p> <div class="num_lemma" id="ProjStructGenCofibs"> <h6 id="lemma">Lemma</h6> <p>For all <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> the canonical maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \to \mathbb{Z}[n]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]</annotation></semantics></math> are cofibrations, in that they have the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> against acyclic fibrations.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>:</mo><mi>A</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">p : A \stackrel{\simeq}{\to} B</annotation></semantics></math> be degreewise surjective and an isomorphism on cohomology.</p> <p>First consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo>,</mo><mn>0</mn><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}[0]\to \mathbb{Z}[-1,0] = 0</annotation></semantics></math>. We need to construct lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mn>0</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z}[0] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ 0 &\stackrel{}{\to}& B } \,. </annotation></semantics></math></div> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">p(f_0(1)) = 0</annotation></semantics></math> we have by using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a quasi-iso that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mspace width="thickmathspace"></mspace><mi>mod</mi><mspace width="thickmathspace"></mspace><mi>im</mi><msub><mi>d</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">f_0(1) = 0 \; mod\; im d_A</annotation></semantics></math>. But in degree 0 this means that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">f_0(1) = 0</annotation></semantics></math>. And so the unique possible lift in the above diagram does exist.</p> <p>Consider now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]</annotation></semantics></math> 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>. We need to construct a lift in all diagrams of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z}[n] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B } \,. </annotation></semantics></math></div> <p>Such a lift is equivalently an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma \in A_{n-1}</annotation></semantics></math> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mi>σ</mi><mo>=</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A \sigma = f_n(1)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_{n-1}(\sigma) = g_{n-1}(1)</annotation></semantics></math>.</p> </li> </ul> <p>Since <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 quasi-isomorphism, and since it takes the closed element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">f_n(1) \in A_n</annotation></semantics></math> to the exact element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mi>B</mi></msub><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p_n(f_n(1)) = d_B g_{n-1}(1)</annotation></semantics></math> it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_n(1)</annotation></semantics></math> itself must be exact in that there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>z</mi><mo>∈</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">z \in A_{n-1}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mi>z</mi><mo>=</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A z = f_n(1)</annotation></semantics></math>. Pick such.</p> <p>So then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_B ( p(z) - g_{n-1}(1) ) = 0</annotation></semantics></math> and again using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is a quasi-isomorphism this means that there must be a closed <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> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a \in A_{n-1}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>d</mi> <mi>B</mi></msub><mi>b</mi></mrow><annotation encoding="application/x-tex">p(a) = p(z)- g_{n-1}(1) + d_B b</annotation></semantics></math> for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow></msub></mrow><annotation encoding="application/x-tex">b \in B_{n-2}</annotation></semantics></math>. Choose such <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math>.</p> <p>Since <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 degreewise onto, there is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">a'</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mo>=</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">p(a') = b</annotation></semantics></math>. Choosing this the above becomes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(a) = p(z) - g_{n-1}(1) + p(d_A a')</annotation></semantics></math>.</p> <p>Set then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mo>=</mo><mi>z</mi><mo>−</mo><mi>a</mi><mo>+</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>′</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma := z - a + d_A a' \,. </annotation></semantics></math></div> <p>It follows with the above that this satisfies the two conditions on <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>:</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><msub><mi>d</mi> <mi>A</mi></msub><mi>σ</mi></mtd> <mtd><mo>=</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>z</mi><mo>−</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>+</mo><msub><mi>d</mi> <mi>A</mi></msub><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>′</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>z</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} d_A \sigma &= d_A z - d_A a + d_A d_A a' \\ & = d_A z \\ & = f_n(1) \end{aligned} </annotation></semantics></math></div><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>p</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo>−</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>+</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><msub><mi>g</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} p( \sigma ) &= p(z) - p(a) + p(d_A a') \\ & = g_{(n-1)}(1) \end{aligned} \,. </annotation></semantics></math></div> <p>Finally consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \to \mathbb{Z}[n]</annotation></semantics></math> for all <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>. We need to produce lifts 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><mn>0</mn></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mi>p</mi></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 0 &\stackrel{}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{p} \\ \mathbb{Z}[n] &\stackrel{g}{\to}& B } \,. </annotation></semantics></math></div> <p>Such a lift is a choice of element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in A_n</annotation></semantics></math> such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mi>σ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_A \sigma = 0</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(\sigma) = g_n(1)</annotation></semantics></math>;</p> </li> </ul> <p>Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_n(1)</annotation></semantics></math> is closed 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 surjective quasi-isomorphism, we may find a closed <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>n</mi></msub></mrow><annotation encoding="application/x-tex">a \in A_n</annotation></semantics></math> and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>′</mo><mo>∈</mo><msub><mi>A</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">a' \in A_{n-1}</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>+</mo><msub><mi>d</mi> <mi>B</mi></msub><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">(</mo><mi>a</mi><mo>′</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p (a) = g_{n}(1) + d_B(p(a'))</annotation></semantics></math>. Set then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>:</mo><mo>=</mo><mi>a</mi><mo>−</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>a</mi><mo>′</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \sigma := a - d_A a' \,. </annotation></semantics></math></div></div> <div class="num_lemma"> <h6 id="lemma_2">Lemma</h6> <p>For all <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>, the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \to \mathbb{Z}[n-1,n]</annotation></semantics></math> are acyclic cofibrations, in that they have the <a class="existingWikiWord" href="/nlab/show/left+lifting+property">left lifting property</a> again all degreewise surjections.</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <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 = 0</annotation></semantics></math> this is trivial. 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> a 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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A \\ \downarrow && \downarrow^{\mathrlap{p}} \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B } </annotation></semantics></math></div> <p>is equivalently just any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">g_{n-1}(1) \in B</annotation></semantics></math> and 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> accordingly just any element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\sigma \in A</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">p(\sigma) = g_{n-1}(1)</annotation></semantics></math>. Such exists because <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 degreewise surjctive by assumption.</p> </div> <div class="num_lemma" id="ProjStructCharAcyclicFibrations"> <h6 id="lemma_3">Lemma</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> is an acyclic fibration precisely if it has the <a class="existingWikiWord" href="/nlab/show/right+lifting+property">right lifting property</a> against <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">0 \to \mathbb{Z}[n]</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n] \to \mathbb{Z}[n-1,n]</annotation></semantics></math> for all <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>.</p> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>By the above lemmas, it remains to show only one direction: if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the RLP, then it is an acyclic fibration.</p> <p>So assume <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> has the RLP. Then from the existence of the lifts</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><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A \\ \downarrow && \downarrow \\ \mathbb{Z}[n] &\stackrel{g}{\to}& B } </annotation></semantics></math></div> <p>one deduces 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> is degreewise surjective on closed elements. In particular this means it is surjective in cohomology.</p> <p>With that, it follows from the existence of all the lifts</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></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>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mi>g</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathbb{Z}[n] &\stackrel{f}{\to}& A \\ \downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow \\ \mathbb{Z}[n-1,n] &\stackrel{g}{\to}& B } </annotation></semantics></math></div> <p>for <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> a lift of the closed element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">g_n(1)</annotation></semantics></math> 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> is degreewise surjective on all elements.</p> <p>Moreover, these lifts say that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_n(1)</annotation></semantics></math> is any closed element such that under <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> it becomes exact (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>B</mi></msub><msub><mi>g</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mo stretchy="false">(</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_B g_{n-1}(1) = p(f_n(1))</annotation></semantics></math>), then it must already be exact itself (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><msub><mi>σ</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><msub><mi>f</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A \sigma_{n-1}(1) = f_n(1)</annotation></semantics></math>). Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is also injective on cohomology and hence by the above is an isomorphism on cohomology.</p> </div> <div class="num_lemma" id="ProjStrucFactAxiomI"> <h6 id="lemma_4">Lemma</h6> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> can be factored as a morphism with left lifting property against all fibrations followed by a fibration.</p> </div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>Apply the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a>-reasoning to the maps in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>J</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex"> J = \{0 \to \mathbb{Z}[n-1,n]\}</annotation></semantics></math>.</p> <p>Since 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> a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}[n,n+1]\to B</annotation></semantics></math> corresponds to an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">b \in B_n</annotation></semantics></math>. From the commuting diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>𝕟</mi></mrow></mrow><mrow><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow></mrow></mfrac></munder><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\to& A \\ \downarrow && \downarrow^{\mathrlap{f}} \\ \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1] &\stackrel{}{\to}& B } </annotation></semantics></math></div> <p>one obtains a factorization through its <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>j</mi></mpadded></msup><mo>↙</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>𝕟</mi></mrow></mrow><mrow><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow></mrow></mfrac></munder><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mo>↘</mo> <mi>p</mi></msub></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && A \\ &{}^{\mathllap{j}}\swarrow& \downarrow \\ A \coprod \coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1] && \downarrow^{\mathrlap{f}} \\ &\searrow_{p}& \downarrow \\ && B } \,. </annotation></semantics></math></div> <p>Since <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> is the pushout of an acyclic cofibration, it is itself an acyclic cofibration. Moreover, since the cohomology of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo lspace="thinmathspace" rspace="thinmathspace">∐</mo> <mfrac linethickness="0"><mrow><mrow><mi>n</mi><mo>∈</mo><mi>𝕟</mi></mrow></mrow><mrow><mrow><mi>b</mi><mo>∈</mo><msub><mi>B</mi> <mi>n</mi></msub></mrow></mrow></mfrac></msub><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>,</mo><mi>n</mi><mo>+</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\coprod_{{n \in \mathbb{n}} \atop {b \in B_n}} \mathbb{Z}[n,n+1]</annotation></semantics></math> clearly vanishes, it is a quasi-isomorphism.</p> <p>The map <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 manifestly degreewise onto and hence a fibration.</p> </div> <div class="num_lemma"> <h6 id="lemma_5">Lemma</h6> <p>Every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> may be factored as a cofibration followed by an acyclic fibration.</p> </div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>By a <a href="#ProjStructCharAcyclicFibrations">lemma above</a> acyclic fibrations are precisely the maps with the right lifting property against morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy="false">{</mo><mn>0</mn><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">I = \{0 \to \mathbb{Z}[n], \mathbb{Z}[n]\to \mathbb{Z}[n-1,n]\}</annotation></semantics></math>, which by the <a href="#ProjStructGenCofibs">first lemma above</a> are cofibrations.</p> <p>The claim then follows again from the <a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a> apllied to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>.</p> </div> <div class="num_lemma"> <h6 id="lemma_6">Lemma</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> that is both a cofibration (:= LLP against acyclic fibrations ) and a weak equivalence has the left lifting property against all fibrations.</p> </div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>By a standard argument, this follows from the factorization lemma proven <a href="#ProjStrucFactAxiomI">above</a>, which says that we may find a factorization</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>j</mi></mover></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></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"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{j}{\to}& \hat B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{p}} \\ && B } </annotation></semantics></math></div> <p>with <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> having LLP against all fibrations and being a weak equivalence, 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. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is assumed to be a weak equivalence, it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> is an acyclic fibration. By definition of cofibrations as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>LLP</mi><mo stretchy="false">(</mo><mi>Fib</mi><mo>∩</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">LLP(Fib \cap W)</annotation></semantics></math> this implies that we have the lift 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><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>j</mi></mover></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>Id</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{j}{\to}& \hat B \\ {}^{\mathllap{f}}\downarrow &{}^{\mathllap{\sigma}}\nearrow& \downarrow^{\mathrlap{p}} \\ B &\stackrel{Id}{\to}& B } \,. </annotation></semantics></math></div> <p>Equivalently redrawing this as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>Id</mi></mover></mtd> <mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mi>Id</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>p</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>i</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd><mover><mi>B</mi><mo stretchy="false">^</mo></mover></mtd> <mtd><mover><mo>→</mo><mi>p</mi></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\stackrel{Id}{\to}& A &\stackrel{Id}{\to}& A \\ {}^{\mathllap{f}}\downarrow && {}^{\mathllap{p}} \downarrow && {}^{\mathllap{i}}\downarrow \\ B &\stackrel{\sigma}{\to}& \hat B & \stackrel{p}{\to} & B } </annotation></semantics></math></div> <p>makes manifest that this exhibts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> as a retract of <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> and as such inherits its left lefting properties.</p> </div> <p>This series of lemmas establishes the claimed model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>+</mo> <mo>•</mo></msubsup><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^\bullet_+(Ab)</annotation></semantics></math>.</p> <h3 id="OnUnbounded">In unbounded degree</h3> <p>There are several approaches to defining model structures on the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> -</p> <h4 id="ProjectiveModelStructureOnUnboundedChainComplexes">Standard projective model structure on unbounded chain complexes</h4> <p> <div class='num_prop' id='StandardModelStructureOnUnboundedComplexes'> <h6>Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(R Mod)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> admits the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> of a</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/cofibrantly+generated+model+category">cofibrantly generated</a></p> <p id="GeneratingCofibrationsOfUnboundedProjectiveStructure"> with generating (acyclic) cofibrations being, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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style="fill:rgb(0%,0%,0%);fill-opacity:1;"> <use xlink:href="#Ccnm0syzGp0_vodvXIR_aykN3gI=-glyph8-1" x="153.315864" y="41.76835"></use> <use xlink:href="#Ccnm0syzGp0_vodvXIR_aykN3gI=-glyph8-2" x="155.666285" y="41.76835"></use> </g> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 55.50726 -12.128635 L 64.595304 -12.128635 " transform="matrix(0.999338,0,0,-0.999338,125.838054,32.457514)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.486247 2.870241 C -2.032822 1.146444 -1.020433 0.333406 -0.00022732 0.00115533 C -1.020433 -0.335005 -2.032822 -1.148042 -2.486247 -2.86793 " transform="matrix(0.999338,0,0,-0.999338,190.629133,44.57928)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:butt;stroke-linejoin:miter;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M 81.618284 -12.128635 L 90.706327 -12.128635 " transform="matrix(0.999338,0,0,-0.999338,125.838054,32.457514)"></path> <path style="fill:none;stroke-width:0.47818;stroke-linecap:round;stroke-linejoin:round;stroke:rgb(0%,0%,0%);stroke-opacity:1;stroke-miterlimit:10;" d="M -2.485923 2.870241 C -2.032499 1.146444 -1.02011 0.333406 0.0000961261 0.00115533 C -1.02011 -0.335005 -2.032499 -1.148042 -2.485923 -2.86793 " transform="matrix(0.999338,0,0,-0.999338,216.72256,44.57928)"></path> </g> </svg> <ul> <li><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a></li> </ul> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a> with</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> the (degreewise) <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> the degreewise <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split injections</a> with cofibrant <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>.</p> </li> </ul> <p></p> </div> (For partial characterization of the <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant objects</a> see further <a href="#BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant">below</a>.) <div class='proof'> <h6>Proof</h6> <p>Properness and cofibrant generation are discussed in <a href="#HoveyPalmieriStrickland97">Hovey, Palmieri & Strickland (1997), remark after theorem 9.3.1</a> and <a href="#SchwedeShipley98">Schwede & Shipley (1998), p. 7</a>, see also <a href="#Fausk06">Fauk (2006), Thm. 3.2</a>. The characterization of the cofibrations is in <a href="#Hovey99">Hovey (1999), Lem. 2.3.6</a> and that of the generating cofibrations are made explicit in <a href="#Hovey99">Hovey (1999), Def. 2.3.3</a>. Cf. also <a href="#MuroRoitzheim19">Muro & Roitzheim (2019), pp. 3</a>.</p> <p id="LocalPresentability"> It remains to see that the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> is <a class="existingWikiWord" href="/nlab/show/locally+presentable+category">locally presentable</a>, so that the model structure is <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a>. (This minor but important point must be clear to the above authors, but seems not to be made explicit in any of the references.) This follows because:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+abelian+category">Grothendieck abelian category</a> (by <a href="Grothendieck+category#CateoriesOfModules">this example</a>);</p> </li> <li> <p>when <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">\mathcal{A}</annotation></semantics></math> is a Grothendieck abelian category then so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(\mathcal{A})</annotation></semantics></math> (by <a href="Grothendieck+category#ChainComplexesInGrothAbCat">this example</a>);</p> </li> <li> <p>all Grothendick abelian categories are locally presentable (by <a href="locally+presentable+category#GrothAbCatsAreLocPresntbl">this example</a>).</p> </li> </ol> <p></p> </div> </p> <p> <div class='num_remark' id='NotAllUnboundedComplexesAreProjectivelyCofibrant'> <h6>Remark</h6> <p>It is clear that every chain complex in the model structure of Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a> is <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant</a>. However, over general rings (not though over <a class="existingWikiWord" href="/nlab/show/fields">fields</a>, see Prop. <a class="maruku-ref" href="#UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant"></a> below) <em>not</em> every chain complex is <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a>, not even those consisting of <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> – a <a class="existingWikiWord" href="/nlab/show/counterexample">counterexample</a> is given in <a href="#Hovey99">Hovey (1999), Rem. 2.3.7</a>:</p> <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">\mathbb{K}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/field">field</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>≔</mo><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mo>⋅</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">R \coloneqq \mathbb{K} \oplus \mathbb{K} \cdot x</annotation></semantics></math> be its <a class="existingWikiWord" href="/nlab/show/ring+of+dual+numbers">ring of dual numbers</a>, i.e. with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>x</mi> <mn>2</mn></msup><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">x^2 = 0</annotation></semantics></math>.</p> <p>Denote its <a class="existingWikiWord" href="/nlab/show/augmented+algebra">augmentation</a> by</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><mpadded width="0" lspace="-100%width"><mrow><mi>ϵ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace></mrow></mpadded><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mo>⋅</mo><mi>x</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>𝕂</mi></mtd></mtr> <mtr><mtd><mi>a</mi><mo>+</mo><mi>b</mi><mi>x</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>a</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \mathllap{ \epsilon \;\colon\; } \mathbb{K} \oplus \mathbb{K} \cdot x &\longrightarrow& \mathbb{K} \\ a + b x &\mapsto& a } </annotation></semantics></math></div> <p>Via this <a class="existingWikiWord" href="/nlab/show/ring+homomorphism">ring homomorphism</a> we regard <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">\mathbb{K}</annotation></semantics></math> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>.</p> <p>Now in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(R Mod)</annotation></semantics></math>, consider the following unbounded chain complex:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>⋯</mi><mo>→</mo><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⋅</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⋅</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi><mo>→</mo><mi>⋯</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \;\coloneqq\; \left( \cdots \to \mathbb{K} \oplus \mathbb{K}x \xrightarrow{ \;\; \cdot x \;\; } \mathbb{K} \oplus \mathbb{K}x \xrightarrow{ \;\; \cdot x \;\; } \mathbb{K} \oplus \mathbb{K}x \to \cdots \right) \,. </annotation></semantics></math></div> <p>Since its <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a> clearly vanishes in every degree, the morphism it receives out of the <a class="existingWikiWord" href="/nlab/show/zero+object">zero object</a> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> and hence a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><munder><mo>⟶</mo><mrow><mo>∈</mo><mi mathvariant="normal">W</mi></mrow></munder><mi>𝒜</mi></mrow><annotation encoding="application/x-tex"> 0 \underset{\in \mathrm{W}}{\longrightarrow} \mathcal{A} </annotation></semantics></math></div> <p>and hence would be an <a class="existingWikiWord" href="/nlab/show/acyclic+cofibration">acyclic cofibration</a> 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">\mathcal{A}</annotation></semantics></math> were cofibrant.</p> <p>But consider then the following <a class="existingWikiWord" href="/nlab/show/lifting+problem">lifting problem</a> with this morphism</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><mo>⟶</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>R</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="3em" minsize="3em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="3em" minsize="3em">↓</mo><msup><mrow></mrow> <mi>ϵ</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒜</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ϵ</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>⋯</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>𝕂</mi><mo>→</mo><mn>0</mn><mo>→</mo><mn>0</mn><mo>→</mo><mi>⋯</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ &\longrightarrow& \big( \cdots \to 0 \to 0 \to R \to 0 \to 0 \to \cdots \big) \\ \Bigg\downarrow && \Bigg\downarrow {}^{ \epsilon } \\ \mathcal{A} & \overset{ \;\; \epsilon \;\;\; }{ \longrightarrow } & \big( \cdots \to 0 \to 0 \to \mathbb{K} \to 0 \to 0 \to \cdots \big) \mathrlap{\,,.} } </annotation></semantics></math></div> <p>Since the morphism on the right is clearly degreewise surjective and hence a <a class="existingWikiWord" href="/nlab/show/fibration">fibration</a> in the model structure, cofibrancy 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">\mathcal{A}</annotation></semantics></math> would imply that a <a class="existingWikiWord" href="/nlab/show/lift">lift</a> in this diagram exists. But to be even a lift of the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/graded+modules">graded modules</a> this lift would have to be the <a class="existingWikiWord" href="/nlab/show/identity+morphism">identity morphism</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> in degree 0, in order to make (in degree 0), this <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>R</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mi>id</mi></msup></mrow></mpadded><mo>↗</mo></mtd> <mtd><mo stretchy="false">↓</mo><mpadded width="0"><mrow><msup><mo></mo><mi>ϵ</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>R</mi></mtd> <mtd><munder><mo>⟶</mo><mi>ϵ</mi></munder></mtd> <mtd><mi>𝕂</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && R \\ & \mathllap{^{id}}\nearrow & \downarrow \mathrlap{^\epsilon} \\ R &\underset{\epsilon}{\longrightarrow}& \mathbb{K} } </annotation></semantics></math></div> <p>But that underlying lift fails to be a <a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a> in degrees (-1,0), where the following <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> does <em>not</em> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi></mtd></mtr> <mtr><mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mo></mo><mi>id</mi></msup></mrow></mpadded><mo maxsize="1.8em" minsize="1.8em">↑</mo></mtd> <mtd><mstyle mathcolor="red"><mo>×</mo></mstyle></mtd> <mtd><mo maxsize="1.8em" minsize="1.8em">↑</mo><mpadded width="0"><mrow><msup><mo></mo><mi>id</mi></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mo>⋅</mo><mi>x</mi></mrow></munder></mtd> <mtd><mi>𝕂</mi><mo>⊕</mo><mi>𝕂</mi><mi>x</mi><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 0 &\longrightarrow& \mathbb{K} \oplus \mathbb{K}x \\ \mathllap{^{id}} \Big\uparrow &\color{red}\times& \Big\uparrow \mathrlap{^{id}} \\ \mathbb{K} \oplus \mathbb{K}x &\underset{\cdot x}{\longrightarrow}& \mathbb{K} \oplus \mathbb{K}x \mathrlap{\,.} } </annotation></semantics></math></div> <p>It follows that the lift does not exist, hence that we have found an object <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">\mathcal{A}</annotation></semantics></math> in the model structure from Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a> which is not cofibrant.</p> </div> </p> <p>On the other hand:</p> <p> <div class='num_prop' id='BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/bounded-below+chain+complexes">bounded-below chain complexes</a> of <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> are projectively <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a>)</strong> <br /> Every <a class="existingWikiWord" href="/nlab/show/bounded-below+chain+complex">bounded-below chain complex</a> of <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> is <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a> in the model structure of Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a>.</p> </div> (<a href="#Hovey99">Hovey (1999), Lem. 2.3.6</a>)</p> <p>This implies:</p> <p> <div class='num_prop' id='UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant'> <h6>Proposition</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>:</p> <ol> <li> <p>every <a class="existingWikiWord" href="/nlab/show/object">object</a> in the projective model structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(k Mod)</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a>) is <a class="existingWikiWord" href="/nlab/show/cofibrant+objects">cofibrant</a>.</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are exactly the <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a>.</p> </li> </ol> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>By the assumption that <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 a field, every <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>-module (i.e. every <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>) is <a class="existingWikiWord" href="/nlab/show/projective+module">projective</a> (<a href="projective+module#ModuleOverAFieldIsProjective">this Prop.</a>). Therefore Prop. <a class="maruku-ref" href="#BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant"></a> says, in this situation, that every <a class="existingWikiWord" href="/nlab/show/bounded-below+chain+complex">bounded-below chain complex</a> is cofibrant Moreover, since every injection of vector spaces splits (<a href="Vect#ShortExactSequencesSplit">here</a>) the characterization of cofibrations in Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a> says that every <a class="existingWikiWord" href="/nlab/show/injection">injection</a> into a <a class="existingWikiWord" href="/nlab/show/bounded-below+chain+complex">bounded-below chain complex</a> of vector spaces is a cofibration (since its <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> is clearly itself bounded-below and hence cofibrant by the previous statement) .</p> <p>Now every chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a> of its stages of lower <a class="existingWikiWord" href="/nlab/show/connective+covers">connective covers</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>cn</mi> <mn>0</mn></msub><msub><mi>V</mi> <mo>•</mo></msub><mo>↪</mo><msub><mi>cn</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msub><msub><mi>V</mi> <mo>•</mo></msub><mo>↪</mo><msub><mi>cn</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>2</mn></mrow></msub><msub><mi>V</mi> <mo>•</mo></msub><mo>↪</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> cn_0 V_\bullet \hookrightarrow cn_{-1} V_\bullet \hookrightarrow cn_{-2} V_\bullet \hookrightarrow \cdots \,. </annotation></semantics></math></div> <p>By the previous paragraph, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cn</mi> <mn>0</mn></msub><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">cn_0 V_\bullet</annotation></semantics></math> is cofibrant and each morphism in this <a class="existingWikiWord" href="/nlab/show/cotower">cotower</a> is a cofibration. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>cn</mi> <mn>0</mn></msub><msub><mi>V</mi> <mo>•</mo></msub><mo>↪</mo><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">cn_0 V_\bullet \hookrightarrow V_\bullet</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/transfinite+composition">transfinite composition</a> of cofibrations, hence a cofibration, and therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math> is cofibrant.</p> <p>This proves the first statement. From this the second follows by the characterization of the cofibrations in Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a> and using again that all injections here are split.</p> </div> (Alternatively one may argue via the generating cofibration, cf. <a href="https://math.stackexchange.com/a/2457259/58526">MO:a/2457259</a>.)</p> <p> <div class='num_prop' id='MonoidalProjectiveModelStructureOnUnboundedChainComplexes'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> makes the projective model structure on unbounded chain complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(R Mod)</annotation></semantics></math> (Prop. <a class="maruku-ref" href="#StandardModelStructureOnUnboundedComplexes"></a>) a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>.</p> </div> For the special case that all <a class="existingWikiWord" href="/nlab/show/submodules">submodules</a> of <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> are again free (such as over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">R = \mathbb{Z}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/integers">integers</a>, by <a href="free+abelian+group#SubgroupsOfFreeAbelianGroupsAreFree">this Prop.</a>, or for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">R = k</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/field">field</a> by <a href="free+module#EveryModuleOverAFieldIsFree">this Prop</a>, and in general for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/principal+ideal+domain">principal ideal domain</a>, by <a href="free+module#submod">this Prop.</a>) a short proof is given in <a href="#Strickland20">Strickland (2020), Prop. 25</a>. The general statement is also a special case of <a href="#Hovey01">Hovey (2001), Cor. 3.7</a> <a href="#Fausk06">Fausk (2006), Thm. 6.1</a> (who state an even more general result about <em><a class="existingWikiWord" href="/nlab/show/sheaves">sheaves</a></em> of chain complexes).</p> <p> <div class='num_prop' id='LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes'> <h6>Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+objects">category of simplicial objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sCh</mi><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><msub><mo stretchy="false">)</mo> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">sCh(R Mod)_\bullet</annotation></semantics></math> in the projective model structure on unbounded chain complexes (from Prop. <a class="maruku-ref" href="#BoundedBelowComplexesOfProjectivesAreProjectivelyCofibrant"></a>) carries the structure of a <a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> (obtained as a <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> of the <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a>), whose weak equivalences are the maps that are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> under the <a class="existingWikiWord" href="/nlab/show/total+chain+complex">total chain complex</a> functor, and such that the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> via :</p> <div class="maruku-equation" id="eq:QuillenEquivalenceBetweenProjectiveUnboundedChainAndSimpEnhancement"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>const</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo>⇄</mo><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><msub><mi>ev</mi> <mn>0</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> const \,\colon\, Ch_\bullet(R Mod) \rightleftarrows sCh_\bullet(R Mod) \,\colon\, ev_0 \,. </annotation></semantics></math></div> <p></p> </div> <div class='proof'> <h6>Proof</h6> <p>The existence as a simplicial model category and Quillen equivalence of the underlying categories is due to <a href="#RezkSchwedeShipley01">Rezk, Schwede & Shipley (2001), cor. 4.6</a>, using methods like those discussed at <em><a href="simplicial+model+category#SimpEquivMods">simplicial model category – Simplicial Quillen equivalent models</a></em>.</p> <p>Moreover, general facts imply that</p> <ol> <li> <p>a Reedy model structure with coefficients in a combinatorial model category is itself combinatorial (see <a href="Reedy+model+structure#CombinatorialStructure">here</a>)</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/left+Bousfield+localization">left Bousfield localization</a> of a combinatorial model category is itself combinatorial (see <a href="Bousfield+localization+of+model+categories#Existence">here</a>).</p> </li> </ol> <p></p> </div> </p> <p>Below, this model structure is recovered as example <a class="maruku-ref" href="#CategoricalProjectiveClasses"></a> of the <a href="#InUnboundedDegreeGeneralResults">Christensen-Hovey projective class construction</a>.</p> <p> <div class='num_prop' id='OverGroundFieldAllObjectsInSimplicialStructureOnUnboundedComplexesAreCofibrant'> <h6>Proposition</h6> <p>Over a <a class="existingWikiWord" href="/nlab/show/field">field</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>, every object in the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh_\bullet(k Mod)</annotation></semantics></math> (from Prop. <a class="maruku-ref" href="#LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes"></a>) is cofibrant.</p> </div> <div class='proof'> <h6>Proof</h6> <p>Since <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> does not change the class of cofibrations, we need to show that every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>V</mi> <mo>•</mo> <mo>•</mo></msubsup><mo>∈</mo><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V_\bullet^\bullet \in sCh_\bullet(k Mod)</annotation></semantics></math> is <a href="Reedy+model+structure#ReedyModelStructure">Reedy cofibrant</a>, hence (cf. <a href="Reedy+model+structure#ReedyCofibrantObjects">this Remark</a>) that the comparison morphisms from the <a class="existingWikiWord" href="/nlab/show/latching+objects">latching objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>r</mi></msub><msubsup><mi>V</mi> <mo>•</mo> <mi>r</mi></msubsup><mo>→</mo><msubsup><mi>V</mi> <mo>•</mo> <mi>r</mi></msubsup></mrow><annotation encoding="application/x-tex">L_r V^r_\bullet \to V^r_\bullet</annotation></semantics></math> are monomorphisms for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">r \in \mathbb{R}</annotation></semantics></math>. But since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(k Mod)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> (cf. <a href="category+of+chain+complexes#AbelianCategoryStructure">here</a>), <a href="Reedy+model+structure#LatchingInAbelianCategoryIsDegeneracySubobject">this Prop.</a> at <em><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></em> says that these are <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> and hence the claim follows by Prop. <a class="maruku-ref" href="#UnboundedChainComplexOfVectorSpacesProjectivelyCofibrant"></a>.</p> </div> </p> <p> <div class='num_prop' id='MonoidalSimplicialEnhancementOfUnboundedChainModelStructure'> <h6>Proposition</h6> <p>At least over a <a class="existingWikiWord" href="/nlab/show/field">field</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>, the local model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh_\bullet(k)</annotation></semantics></math> from Prop. <a class="maruku-ref" href="#LocalizedReedyModelStructureOnSimplicialUnboundedChainComplexes"></a> becomes a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> via the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math>-object-wise <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a>, and the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> <a class="maruku-eqref" href="#eq:QuillenEquivalenceBetweenProjectiveUnboundedChainAndSimpEnhancement">(1)</a> is a compatibly <a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a> with respect to the corresponding monoidal model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(k)</annotation></semantics></math> from Prop. <a class="maruku-ref" href="#MonoidalProjectiveModelStructureOnUnboundedChainComplexes"></a>, Prop. <a class="maruku-ref" href="#MonoidalProjectiveModelStructureOnUnboundedChainComplexes"></a>.</p> </div> This ought to be compatible with the simplicial structure such as to give a <a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a>. <div class='proof'> <h6>Proof</h6> <p>First, the plain <a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh_\bullet(k)</annotation></semantics></math> becomes a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a> under the objectwise <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a>, by <a href="monoidal+model+category#Barwick10">Barwick (2010), Thm. 3.51</a> (beware that the notation “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>M</mi></mstyle><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{M}(A)</annotation></semantics></math>” there does refer to the Reedy model structure on <em>presheaves</em>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mi>op</mi></msup><mo>,</mo><mstyle mathvariant="bold"><mi>M</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(A^{op}, \mathbf{M})</annotation></semantics></math> (cf. p. 265), which means that the condition that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>←</mo></msup></mrow><annotation encoding="application/x-tex">A^{\leftarrow}</annotation></semantics></math> consists of epimorphisms <em>is</em> satisfied for our case where, under this notational convention, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">A = \Delta</annotation></semantics></math>).</p> <p id="SimplicialMonoidalModelStructure"> Next, to see that this monoidal model structure passes to the <a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">left Bousfield localization</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sCh</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh_\bullet(k)</annotation></semantics></math> at the realization equivalences:</p> <p>Observing that every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh(k)</annotation></semantics></math> is a simplicial homotopy colimit of simplicially constant objects (by an argument as in <a href="homotopy+limit#SimplicialSetIsHomotopyColimitOverItself">this Prop.</a>) and recalling that the local objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh(k)</annotation></semantics></math> are the homotopically constant simplicial objects, it is sufficient to check — by <a href="Bousfield+localization+of+model+categories#Barwick10">Barwick (2010), Prop. 4.47</a> — that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V \,\in\, Ch(k)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">𝒲</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathscr{W} \,\in\, sCh(k)</annotation></semantics></math> homotopically constant, also their <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>const</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">𝒲</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[const(V),\,\mathscr{W}] \,\in\, sCh(k)</annotation></semantics></math> is homotopically constant.</p> <p>Now, on general grounds, the internal hom in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh(k)</annotation></semantics></math> is given by an <a class="existingWikiWord" href="/nlab/show/end">end</a> over the internal hom in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(k)</annotation></semantics></math> (to be denoted by the same angular bracket notation), as follows:</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 class="mathscript">𝒱</mi><mo>,</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">𝒲</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><msub><mo>∫</mo> <mrow><mo stretchy="false">[</mo><mi>s</mi><mo>′</mo><mo stretchy="false">]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><mo maxsize="1.2em" minsize="1.2em">[</mo><mo stretchy="false">(</mo><mi>Δ</mi><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo><mo>⋅</mo><mi class="mathscript">𝒱</mi><msub><mo stretchy="false">)</mo> <mrow><mi>s</mi><mo>′</mo></mrow></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi class="mathscript">𝒲</mi> <mrow><mi>s</mi><mo>′</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> [\mathscr{V},\,\mathscr{W}] \;\colon\; [s] \;\mapsto\; \int_{[s'] \in \Delta} \big[ (\Delta[s] \cdot \mathscr{V})_{s'} ,\, \mathscr{W}_{s'} \big] \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-)\cdot(-)</annotation></semantics></math> denotes the canonical <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sCh</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sCh(k)</annotation></semantics></math> over <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>.</p> <p>So in the case at hand, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi class="mathscript">𝒱</mi><mspace width="thinmathspace"></mspace><mo>=</mo><mspace width="thinmathspace"></mspace><mi>const</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathscr{V} \,=\, const(V)</annotation></semantics></math>, we find this to be:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mtable displaystyle="false" rowspacing="0.5ex" columnalign="left"><mtr><mtd><mo stretchy="false">[</mo><mi>const</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mspace width="thinmathspace"></mspace><mi class="mathscript">𝒲</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace></mtd></mtr> <mtr><mtd><msub><mo>∫</mo> <mrow><mi>s</mi><mo>′</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">[</mo><mi>Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>′</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>V</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi class="mathscript">𝒲</mi> <mrow><mi>s</mi><mo>′</mo></mrow></msub><mo maxsize="1.2em" minsize="1.2em">]</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">[</mo><mi>V</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mo>∫</mo> <mrow><mi>s</mi><mo>′</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi class="mathscript">𝒲</mi> <mrow><mi>s</mi><mo>′</mo></mrow></msub><msup><mo stretchy="false">)</mo> <mrow><mi>Δ</mi><mo stretchy="false">(</mo><mi>s</mi><mo>′</mo><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow></msup><mo maxsize="1.8em" minsize="1.8em">]</mo></mtd></mtr> <mtr><mtd><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mo maxsize="1.8em" minsize="1.8em">[</mo><mi>V</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi class="mathscript">𝒲</mi> <mi>s</mi></msub><mo maxsize="1.8em" minsize="1.8em">]</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mtd></mtr></mtable></mrow><annotation encoding="application/x-tex"> \begin{array}{l} [const(V),\, \mathscr{W}] \,\colon\, [s] \;\mapsto\; \\ \int_{s'} \big[ \Delta(s',s) \cdot V ,\, \mathscr{W}_{s'} \big] \\ \;\simeq\; \Big[ V ,\, \int_{s'} (\mathscr{W}_{s'})^{\Delta(s',s)} \Big] \\ \;\simeq\; \Big[ V ,\, \mathscr{W}_s \Big] \,, \end{array} </annotation></semantics></math></div> <p>where we passed from <a class="existingWikiWord" href="/nlab/show/tensoring">tensoring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>⋅</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S\cdot (-)</annotation></semantics></math> to its <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <a class="existingWikiWord" href="/nlab/show/powering">powering</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mi>S</mi></msup></mrow><annotation encoding="application/x-tex">(-)^S</annotation></semantics></math>, used that an <a href="hom-functor+preserves+limits#InternalHomFunctor">internal hom preserves limits</a> and then the <a class="existingWikiWord" href="/nlab/show/enriched+Yoneda+lemma">enriched Yoneda lemma</a> in its <a class="existingWikiWord" href="/nlab/show/end">end</a>-form (discussed at <em><a class="existingWikiWord" href="/nlab/show/co-Yoneda+lemma">co-Yoneda lemma</a></em>).</p> <p>But since the objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi class="mathscript">𝒲</mi> <mrow><mi>k</mi><mo>′</mo></mrow></msub><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">V,\,\mathscr{W}_{k'} \,\in\,Ch(k)</annotation></semantics></math> are cofibrant and fibrant (as all objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(k)</annotation></semantics></math>, by the above discussion), the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>V</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Ch</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[V,-] \,\colon\, Ch(k) \to Ch(k) </annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/right+Quillen+functor">right Quillen functor</a> and as such preserves weak equivalences between the fibrant objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi class="mathscript">𝒲</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathscr{W}_{\bullet}</annotation></semantics></math> (by <a href="Introduction+to+Homotopy}Theory#KenBrownLemma">Ken Brown’s lemma</a>). This means that if the simplicial chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi class="mathscript">𝒲</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\mathscr{W}_\bullet</annotation></semantics></math> is homotopically constant then so is the simplicial chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>const</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>,</mo><mi class="mathscript">𝒲</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mi>s</mi><mo stretchy="false">]</mo><mspace width="thinmathspace"></mspace><mo>↦</mo><mspace width="thinmathspace"></mspace><mo stretchy="false">[</mo><mi>V</mi><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi class="mathscript">𝒲</mi> <mi>s</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> [const(V),\mathscr{W}] \,\colon\, [s] \,\mapsto\, [V,\, \mathscr{W}_s]</annotation></semantics></math>, which was to be shown.</p> <p>It remains to observe that the Quillen equivalence to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(k)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a>, but this is immediate since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>const</mi></mrow><annotation encoding="application/x-tex">const</annotation></semantics></math> is by aconstruction a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> and the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> is <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant</a> (as all objects).</p> </div> </p> <p><br /></p> <h4 id="InjectiveModelStructureOnUnboundedChainComplexes">Standard injective model structure on unbounded chain complexes</h4> <p> <div class='num_prop' id='StandardInjectiveModelStructureOnUnboundedComplexes'> <h6>Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> the <a class="existingWikiWord" href="/nlab/show/category+of+unbounded+chain+complexes">category of unbounded chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(R Mod)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> carries the structure of a</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a> with</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> the <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> the (degreewise) <a class="existingWikiWord" href="/nlab/show/injections">injections</a>.</p> </li> </ul> <p></p> </div> This is <a href="#Hovey99">Hovey (1999), Thm. 2.3.13</a>.</p> <h4 id="InUnboundedDegreeGeneralResults">Christensen-Hovey model structures using projective classes</h4> <p>Let <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">\mathcal{A}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with all <a class="existingWikiWord" href="/nlab/show/limit">limit</a>s and <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s.</p> <p><a href="#ChristensenHovey">Christensen-Hovey</a> construct a family of model category structures on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> parameterized by a choice of <em>projective class</em> . The cofibrations, fibrations and weak equivalences all depend on the projective class.</p> <div class="num_defn" id="ProjectiveClass"> <h6 id="definition_2">Definition</h6> <p>A <strong>projective class</strong> 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">\mathcal{A}</annotation></semantics></math> is a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>⊂</mo><mi>ob</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} \subset ob \mathcal{A}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/object">object</a>s and a collection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi><mo>⊂</mo><mi>mor</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{E} \subset mor \mathcal{A}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s, such that</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> is precisely the collection of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">\mathcal{P}</annotation></semantics></math>-epic maps;</p> </li> <li> <p><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">\mathcal{P}</annotation></semantics></math> is precisely the collection of all objects <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> such that each map in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> is <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>-epic;</p> </li> <li> <p>for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, there is a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> with <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> in <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">\mathcal{P}</annotation></semantics></math>.</p> </li> </ul> </div> <div class="num_example" id="TrivialProjectiveClass"> <h6 id="example_2">Example</h6> <p>Taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒫</mi><mo>:</mo><mo>=</mo><mi>ob</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{P} := ob \mathcal{A}</annotation></semantics></math> to be the class of <em>all</em> objects yields a projective class – called the <em>trivial projective class</em> . The corresponding morphisms are the class <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{E}</annotation></semantics></math> of all <a class="existingWikiWord" href="/nlab/show/split+epimorphism">split epimorphism</a>s in <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">\mathcal{A}</annotation></semantics></math>.</p> </div> <div class="num_example" id="PullbackProjectiveClass"> <h6 id="example_3">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} = </annotation></semantics></math> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> be the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>s. Choosing <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">\mathcal{P}</annotation></semantics></math> to be the class of all summands of <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> of <a class="existingWikiWord" href="/nlab/show/finitely+presented">finitely presented</a> modules yields a projective class.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p>Given a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒜</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>ℬ</mi></mrow><annotation encoding="application/x-tex"> (F \dashv U) : \mathcal{A} \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} \mathcal{B} </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒫</mi><mo>,</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{P}, \mathcal{E})</annotation></semantics></math> a projective class 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">\mathcal{B}</annotation></semantics></math> then its <strong>pullback projective class</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>U</mi><mo>*</mo><mi>𝒫</mi><mo>,</mo><msup><mi>U</mi> <mo>*</mo></msup><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(U * \mathcal{P}, U^* \mathcal{E})</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> 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">\mathcal{A}</annotation></semantics></math> is defined by</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>U</mi> <mo>*</mo></msup><mi>𝒫</mi><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mi>retracts</mi><mspace width="thickmathspace"></mspace><mi>of</mi><mspace width="thickmathspace"></mspace><mi>F</mi><mi>P</mi><mo stretchy="false">|</mo><mi>P</mi><mo>∈</mo><mi>𝒫</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">U^* \mathcal{P} := \{retracts\;of\; F P | P \in \mathcal{P}\}</annotation></semantics></math></li> </ul> </div> <div class="num_theorem" id="ModelStructureOnUnboundedFromProjectiveClass"> <h6 id="definitiontheorem">Definition/Theorem</h6> <p>Given a projective class <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">\mathcal{P}</annotation></semantics></math> in <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">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ProjectiveClass"></a>), call a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in Ch(\mathcal{A})</annotation></semantics></math></p> <ul> <li> <p>a fibration if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(P,f)</annotation></semantics></math> is a surjection in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">P \in \mathcal{P}</annotation></semantics></math>;</p> </li> <li> <p>a weak equivalence if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A}(P,f)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(Ab)</annotation></semantics></math> for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>∈</mo><mi>𝒫</mi></mrow><annotation encoding="application/x-tex">P \in \mathcal{P}</annotation></semantics></math>.</p> </li> </ul> <p>Then this constitutes a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure precisely if cofibrant <a class="existingWikiWord" href="/nlab/show/resolution">resolution</a>s exist, which is the case in particular if</p> <ol> <li> <p><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">\mathcal{P}</annotation></semantics></math> is the pullback projective class (def. <a class="maruku-ref" href="#PullbackProjectiveClass"></a>) of a trivial projective class (def. <a class="maruku-ref" href="#TrivialProjectiveClass"></a>) along a functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> that preserves countable <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>;</p> </li> <li> <p>(…)</p> </li> </ol> <p>When the structure exists, it is a <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>.</p> </div> <p>This is theorem 2.2 in <a href="#ChristensenHovey">Christensen-Hovey</a>.</p> <p>We shall write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mi>𝒫</mi></msub></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})_{\mathcal{P}}</annotation></semantics></math> for this model category structure.</p> <h4 id="ExamplesOfStructuresOnUnboundedComplexes">Examples</h4> <p>We list some examples for the model structures on chain complexes in unbounded degree discussed <a href="#InUnboundedDegreeGeneralResults">above</a>.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> be an associative ring and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} = R</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>.</p> <ul> <li> <p><a href="#CategoricalProjectiveClass">Categorical projective class structure</a></p> </li> <li> <p><a href="#PureProjectiveClass">Pure projective class structure</a></p> </li> </ul> <h5 id="CategoricalProjectiveClass">Categorical projective class structure</h5> <div class="num_example" id="CategoricalProjectiveClasses"> <h6 id="example_5">Example</h6> <p>The <strong>categorical projective class</strong> 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">\mathcal{A}</annotation></semantics></math> is the projective class (def. <a class="maruku-ref" href="#ProjectiveClass"></a>) with <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">\mathcal{P}</annotation></semantics></math> the class of <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a>mands of free modules. The <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">\mathcal{P}</annotation></semantics></math>-model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> has</p> <ul> <li>as fibrations the degreewise surjections.</li> </ul> <p>So this reproduces the standard projective model structure from prop. <a class="maruku-ref" href="#ProjModelStructureOnUnbounded"></a>.</p> </div> <h5 id="PureProjectiveClass">Pure projective class structure</h5> <div class="num_example"> <h6 id="example_6">Example</h6> <p>The <strong>pure projective class</strong> 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">\mathcal{A}</annotation></semantics></math> has as <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">\mathcal{P}</annotation></semantics></math> summands of sums of finitely presented modules. Fibrations in the corresponding model structure are the maps that are degreewise those epimorphisms that appear in <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">\mathcal{P}</annotation></semantics></math>-exact sequences.</p> </div> <h4 id="gillespies_approach_using_cotorsion_pairs">Gillespie’s approach using cotorsion pairs</h4> <p><a href="#HoveyOverview">Hovey</a> has shown that, roughly speaking, <a class="existingWikiWord" href="/nlab/show/model+structures">model structures</a> on <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> correspond to cotorsion pairs. See <a class="existingWikiWord" href="/nlab/show/abelian+model+structure">abelian model structure</a>.</p> <p><a href="#HoveyOverview">Gillespie</a> shows that 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">\mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Grothendieck+abelian+category">Grothendieck abelian category</a>, then a <a class="existingWikiWord" href="/nlab/show/cotorsion+pair">cotorsion pair</a> induces an <a class="existingWikiWord" href="/nlab/show/abelian+model+structure">abelian model structure</a> on the category of (unbounded) complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math>, where the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a>.</p> <div class="num_theorem"> <h6 id="theorem_5">Theorem</h6> <p>Let <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">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Grothendieck+abelian+category">Grothendieck abelian category</a>. Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><mi>ℰ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D}, \mathcal{E})</annotation></semantics></math> is a hereditary <a class="existingWikiWord" href="/nlab/show/cotorsion+pair">cotorsion pair</a> that is cogenerated by a set, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> is a <span class="newWikiWord">Kaplansky class<a href="/nlab/new/Kaplansky+class">?</a></span> 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">\mathcal{A}</annotation></semantics></math> and <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">\mathcal{A}</annotation></semantics></math> has enough <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>-objects.</p> <p>Then there is an <a class="existingWikiWord" href="/nlab/show/abelian+model+structure">abelian model structure</a> on the category of complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> such that the trivial objects are the acyclic complexes.</p> </div> <p>Gillespie uses this result to get a <strong><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal</a></strong> <a class="existingWikiWord" href="/nlab/show/model+structure">model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>Qcoh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(Qcoh(X))</annotation></semantics></math>, the category of complexes of <a class="existingWikiWord" href="/nlab/show/quasi-coherent+sheaves">quasi-coherent sheaves</a> on a <a class="existingWikiWord" href="/nlab/show/quasi-compact+scheme">quasi-compact</a> <a class="existingWikiWord" href="/nlab/show/separated+scheme">separated</a> <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> <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>. This gives a better understanding of the <a class="existingWikiWord" href="/nlab/show/derived+category+of+quasi-coherent+sheaves">derived category of quasi-coherent sheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi><mo stretchy="false">(</mo><mi>Qcoh</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">D(Qcoh(X))</annotation></semantics></math>, and in particular gives immediately the <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><msup><mo>⊗</mo> <mstyle mathvariant="bold"><mi>L</mi></mstyle></msup><mo>⋅</mo></mrow><annotation encoding="application/x-tex">\cdot \otimes^{\mathbf{L}} \cdot</annotation></semantics></math> (which is usually a problem due to sheaves not having enough projectives).</p> <h4 id="cisinskideglise_approach_using_descent_structures">Cisinski-Deglise approach using descent structures</h4> <p>A third approach is due to <a href="#CisinskiDeglise">Cisinski-Deglise</a>.</p> <p>Let <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">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/Grothendieck+abelian+category">Grothendieck abelian category</a>. We will define a notion of <strong>descent structures</strong> 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">\mathcal{A}</annotation></semantics></math>.</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>For each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> and integer <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">n \in \mathbf{Z}</annotation></semantics></math>, we define the complexes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mi>n</mi></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">S^n E</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>D</mi> <mi>n</mi></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">D^n E</annotation></semantics></math> as follows: let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>S</mi> <mi>n</mi></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>=</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">(S^n E)^n = E</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 elsewhere; and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mo>=</mo><mo stretchy="false">(</mo><msup><mi>D</mi> <mi>n</mi></msup><mi>E</mi><msup><mo stretchy="false">)</mo> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mo>=</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">(D^n E)^n = (D^n E)^{n+1} = E</annotation></semantics></math> and 0 elsewhere. There are canonical morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>E</mi><mo>↪</mo><msup><mi>D</mi> <mi>n</mi></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">S^{n+1} E \hookrightarrow D^n E</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Let <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">\mathcal{G}</annotation></semantics></math> be an essentially small set of objects 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">\mathcal{A}</annotation></semantics></math>. A morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> is called a <strong><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">\mathcal{G}</annotation></semantics></math>-cofibration</strong> if it is contained in the smallest class of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> that is closed under <a class="existingWikiWord" href="/nlab/show/pushouts">pushouts</a>, <a class="existingWikiWord" href="/nlab/show/transfinite+compositions">transfinite compositions</a> and <a class="existingWikiWord" href="/nlab/show/retracts">retracts</a>, generated by the inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>S</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup><mi>E</mi><mo>→</mo><msup><mi>D</mi> <mi>n</mi></msup><mi>E</mi></mrow><annotation encoding="application/x-tex">S^{n+1} E \to D^n E</annotation></semantics></math>, for any integer <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 any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">E \in \mathcal{G}</annotation></semantics></math>. A complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> is called <strong><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">\mathcal{G}</annotation></semantics></math>-cofibrant</strong> if the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">0 \to C</annotation></semantics></math> is a <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">\mathcal{G}</annotation></semantics></math>-cofibration.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> is called <strong><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">\mathcal{G}</annotation></semantics></math>-local</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">E \in \mathcal{G}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">n \in \mathbf{Z}</annotation></semantics></math>, the canonical morphism</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><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mrow><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo>,</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathbf{K}(\mathcal{A})}(E[n], C) \to Hom_{\mathbf{D}(\mathcal{A})}(E[n], C) </annotation></semantics></math></div> <p>is an isomorphism. Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{K}(\mathcal{A})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>D</mi></mstyle><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{D}(\mathcal{A})</annotation></semantics></math> denote the <span class="newWikiWord">homotopy category of complexes<a href="/nlab/new/homotopy+category+of+complexes">?</a></span> and the <a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a> 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">\mathcal{A}</annotation></semantics></math>, respectively.</p> </div> <div class="num_defn"> <h6 id="definition_6">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> be a small family of complexes in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math>. An complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> is called <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>-flasque</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">n \in \mathbf{Z}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mo>∈</mo><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">H \in \mathcal{H}</annotation></semantics></math>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><mstyle mathvariant="bold"><mi>K</mi></mstyle><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><mi>C</mi><mo stretchy="false">[</mo><mi>n</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathbf{K}(\mathcal{A})}(H, C[n]) = 0. </annotation></semantics></math></div></div> <p>Finally we define:</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>A <strong>descent structure</strong> 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">\mathcal{A}</annotation></semantics></math> is a pair <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒢</mi><mo>,</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{G},\mathcal{H})</annotation></semantics></math>, where <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">\mathcal{G}</annotation></semantics></math> is an essentially small set of generators 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">\mathcal{A}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math> is an essentially small set of <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">\mathcal{G}</annotation></semantics></math>-cofibrant acyclic complexes such that any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>-flasque complex is <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">\mathcal{G}</annotation></semantics></math>-local.</p> </div> <p>Now one defines a <a class="existingWikiWord" href="/nlab/show/model+structure">model structure</a> associated to any such descent structure.</p> <div class="num_theorem"> <h6 id="theorem_6">Theorem</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒢</mi><mo>,</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{G},\mathcal{H})</annotation></semantics></math> be a descent structure on the <a class="existingWikiWord" href="/nlab/show/Grothendieck+abelian+category">Grothendieck abelian category</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">\mathcal{A}</annotation></semantics></math>. There is a <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a> <a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular</a> <a class="existingWikiWord" href="/nlab/show/model+structure">model structure</a> on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math>, where the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> of <a class="existingWikiWord" href="/nlab/show/complexes">complexes</a>, and <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are <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">\mathcal{G}</annotation></semantics></math>-cofibrations.</p> <p>Also, a complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/fibrant">fibrant</a> if and only if it is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℋ</mi></mrow><annotation encoding="application/x-tex">\mathcal{H}</annotation></semantics></math>-flasque or equivalently <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">\mathcal{G}</annotation></semantics></math>-local.</p> </div> <p>We call this the <strong><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">\mathcal{G}</annotation></semantics></math>-model structure</strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})</annotation></semantics></math>. As in Gillespie’s approach we can sometimes get a <a class="existingWikiWord" href="/nlab/show/monoidal+model+structure">monoidal model structure</a>. We refer to <a href="#CisinskiDeglise">Cisinski-Deglise</a> for the notion of a <strong>weakly flat</strong> descent structure.</p> <div class="num_theorem"> <h6 id="theorem_7">Theorem</h6> <p>Suppose <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒢</mi><mo>,</mo><mi>ℋ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{G}, \mathcal{H})</annotation></semantics></math> is a <em>weakly flat</em> descent structure 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">\mathcal{A}</annotation></semantics></math>. Then the <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">\mathcal{G}</annotation></semantics></math>-model structure is further <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal</a>.</p> </div> <h2 id="properties">Properties</h2> <blockquote> <p>Beware that this entry has evolved in a way that deserves re-organization now: What follows are mainly properties of or arguments for the model structures on not-unbounded chain complexes.</p> </blockquote> <h3 id="Properness">Properness</h3> <p>Model categories of chain complexes tend to be <a class="existingWikiWord" href="/nlab/show/proper+model+categories">proper model categories</a>.</p> <p> <div class='num_prop' id='PushoutAlongDegreewiseInjectionsPreservesQuasiIosmorphism'> <h6>Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> along degreewise <a class="existingWikiWord" href="/nlab/show/injections">injections</a> presrves <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>)</strong> <br /> Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mi>j</mi><mspace width="thickmathspace"></mspace></mrow></munder></mtd> <mtd><mi>D</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{\; i \;}{\longrightarrow}& B \\ {}^{\mathllap{f}} \big\downarrow &{}^{{}_{(po)}}& \big\downarrow {}^{\mathrlap{g}} \\ C &\underset{\; j \;}{\longrightarrow}& D } </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> <a class="existingWikiWord" href="/nlab/show/commuting+diagram">square</a> of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a> between (unbounded) <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>, such that</p> <ul> <li> <p><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 degreewise <a class="existingWikiWord" href="/nlab/show/injection">injection</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> </li> </ul> <p>Then also <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> is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>.</p> <p><a class="existingWikiWord" href="/nlab/show/formal+duality">Dually</a>, the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of a quasi-isomorphism along a degreewise <a class="existingWikiWord" href="/nlab/show/surjection">surjection</a> is again a quasi-isomorphism.</p> <p></p> </div> (e.g. <a href="#Strickland20">Strickland 2020, Prop. 24</a>) <div class='proof'> <h6>Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> of chain complexes is degreewise a pushout in the underlying <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. Since pushout in abelian categories preserves monomorphisms (<a href="abelian+category#PullbackPreservesEpimorphisms">this Prop.</a>) it follows that also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>j</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">j_n</annotation></semantics></math> is a monomorphism. Finally, the <a class="existingWikiWord" href="/nlab/show/pasting+law">pasting law</a> implies that the induced morphism of <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> and <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> is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. In summary, this means that we have a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of chain complexes as follows:</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><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>i</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><mi>B</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>cok</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><msup><mrow></mrow> <mrow><msub><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>po</mi><mo stretchy="false">)</mo></mrow></msub></mrow></msup></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mi>g</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>C</mi></mtd> <mtd><munder><mo>↪</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>j</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></munder></mtd> <mtd><mi>D</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>cok</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{\;\;\; i \;\;\;}{\hookrightarrow}& B &\longrightarrow& cok(i) \\ {}^{\mathllap{f}} \big\downarrow &{}^{{}_{(po)}}& \big\downarrow {}^{\mathrlap{g}} && \big\downarrow {}^{\mathrlap{\simeq}} \\ C &\underset{\;\;\; j \;\;\;}{\hookrightarrow}& D &\longrightarrow& cok(j) \,. } </annotation></semantics></math></div> <p>This implies a morphism of the corresponding <a class="existingWikiWord" href="/nlab/show/long+exact+sequences+in+chain+homology">long exact sequences in chain homology</a> of the form:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>cok</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>δ</mi><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><msub><mi>i</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>cok</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>δ</mi><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>g</mi> <mo>*</mo></msub></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>f</mi> <mo>*</mo></msub></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo><msup><mrow></mrow> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>⋯</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>H</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>cok</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>δ</mi><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><msub><mi>j</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mi>n</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>cok</mi><mo stretchy="false">(</mo><mi>j</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mspace width="thickmathspace"></mspace><mi>δ</mi><mspace width="thickmathspace"></mspace></mrow></mover></mtd> <mtd><msub><mi>H</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>⋯</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \cdots &\to& H_{n+1} \big( cok(i) \big) & \xrightarrow{\; \delta \;} & H_n(A) & \xrightarrow{\; i_\ast\;} & H_n(B) & \xrightarrow{\;\;\;} & H_n\big(cok(i)\big) & \xrightarrow{\;\delta\;} & H_{n-1}(A) & \to & \cdots \\ && {}^{\mathllap{}} \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{f_\ast}} \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{g_\ast}} \big\downarrow && \big\downarrow {}^{\mathrlap{\simeq}} && {}^{\mathllap{f_\ast}} \big\downarrow {}^{\mathrlap{\simeq}} \\ \cdots &\to& H_{n+1} \big( cok(j) \big) & \xrightarrow{\; \delta \;} & H_n(C) & \xrightarrow{\; j_\ast\;} & H_n(D) & \xrightarrow{\;\;\;} & H_n\big(cok(j)\big) & \xrightarrow{\;\delta\;} & H_{n-1}(C) & \to & \cdots } </annotation></semantics></math></div> <p>From this the <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a> implies that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mo>*</mo></msub></mrow><annotation encoding="application/x-tex">g_\ast</annotation></semantics></math> is an isomorphism on <a class="existingWikiWord" href="/nlab/show/chain+homology">chain homology</a>, hence that <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> is a quasi-isomorphism.</p> </div> </p> <h3 id="ExactFuncsAndQuillenFuncs">Left/right exact functors and Quillen adjunctions</h3> <p>Let <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">\mathcal{A}</annotation></semantics></math> and <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">\mathcal{B}</annotation></semantics></math> be <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a>. Let the <a class="existingWikiWord" href="/nlab/show/categories+of+chain+complexes">categories of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet^+(\mathcal{A})</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet^+(\mathcal{B})</annotation></semantics></math> be equipped with the model structure described <a href="#GeneralResults">above</a> where fibrations are the degreewise <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a>s with <a class="existingWikiWord" href="/nlab/show/injective+object">injective</a> <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>s.</p> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>If</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>L</mi><mo>⊣</mo><mi>R</mi><mo stretchy="false">)</mo><mo>:</mo><mi>𝒜</mi><mover><munder><mo>→</mo><mi>R</mi></munder><mover><mo>←</mo><mi>L</mi></mover></mover><mi>ℬ</mi></mrow><annotation encoding="application/x-tex"> (L \dashv R) : \mathcal{A} \stackrel{\overset{L}{\leftarrow}}{\underset{R}{\to}} \mathcal{B} </annotation></semantics></math></div> <p>is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a>s where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a>s, then</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>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo><mo>⊣</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow></munder><mover><mo>←</mo><mrow><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow></mover></mover><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Ch_\bullet^+(L) \dashv Ch_\bullet(R) : Ch_\bullet^+(\mathcal{A}) \stackrel{\overset{Ch_\bullet^+(L)}{\leftarrow}}{\underset{Ch_\bullet(R)}{\to}} Ch_\bullet^+(\mathcal{B}) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a>.</p> </div> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>Every functor preserves split epimorphism. Being a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/left+exact+functor">left exact functor</a> and hence preserves kernels. Using the characterization of <a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>s as those <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> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Hom</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom(-,I)</annotation></semantics></math> sends monomorphisms to epimorphisms, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> preserves injectives because <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves monomorphisms, by the adjunction isomorphism.</p> <p>Hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi></mrow><annotation encoding="application/x-tex">L</annotation></semantics></math> preserves all cofibrations and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> all fibrations.</p> </div> <h3 id="cofibrant_generation">Cofibrant generation</h3> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>The injective model structure on <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>R</mi><mi>Mod</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch^{\geq 0}(R Mod)</annotation></semantics></math> (from theorem <a class="maruku-ref" href="#InjectiveModelStructure"></a>) is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a>.</p> </div> <p>This appears for instance as <a href="#Hovey">Hovey, theorem 2.3.13</a>, where it is stated for unbounded (in both directions) chain complexes.</p> <p>For results on model structures on chain complexes that are provably not cofibrantly generated see section 5.4 of <a href="#ChristensenHovey">Christensen, Hovey</a>.</p> <h3 id="inclusion_into_simplicial_objects">Inclusion into simplicial objects</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> be the category of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s. The <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> provides a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>⊣</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>Ch</mi> <mo>•</mo> <mo>+</mo></msubsup><mover><munder><mo>→</mo><mi>Γ</mi></munder><mover><mo>←</mo><mi>N</mi></mover></mover><mi>sAb</mi></mrow><annotation encoding="application/x-tex"> (N \dashv \Gamma) : Ch_\bullet^+ \stackrel{\overset{N}{\leftarrow}}{\underset{\Gamma}{\to}} sAb </annotation></semantics></math></div> <p>between the projective model structure on connective chain complexes and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial abelian groups</a>. This in turns sits as a <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> along the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> over the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊣</mo><mi>U</mi><mo stretchy="false">)</mo><mo>:</mo><mi>sAb</mi><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><mi>sSet</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (F \dashv U) : sAb \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} sSet \,. </annotation></semantics></math></div> <p>The combined Quillen adjunction</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mi>F</mi><mo>⊣</mo><mi>U</mi><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mover><mo>→</mo><mo>←</mo></mover><mi>sSet</mi></mrow><annotation encoding="application/x-tex"> (N F \dashv U \Gamma) : Ch_\bullet \stackrel{\leftarrow}{\to} sSet </annotation></semantics></math></div> <p>prolongs to a Quillen adjunction on the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> on any <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> , which we denote by the same symbols</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>N</mi><mi>F</mi><mo>⊣</mo><mi>U</mi><mi>Γ</mi><mo stretchy="false">)</mo><mo>:</mo><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><msub><mi>Ch</mi> <mo>•</mo></msub><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mover><mo>→</mo><mo>←</mo></mover><mo stretchy="false">[</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mi>proj</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (N F \dashv U \Gamma) : [C,Ch_\bullet]_{proj} \stackrel{\leftarrow}{\to} [C, sSet]_{proj} \,. </annotation></semantics></math></div> <p>With due care this descends to the <a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure on simplicial presheaves</a> which <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presents</a> the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaf+%28%E2%88%9E%2C1%29-topos">(∞,1)-sheaf (∞,1)-topos</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. Then the above Quillen adjunction serves to embed <a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> into the larger context of <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. See <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a> for more on this.</p> <h3 id="cofibrations">Cofibrations</h3> <p>We discuss cofibrations in the <a href="#OnUnbounded">model structures on unbounded complexes</a>.</p> <p>Let <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">\mathcal{P}</annotation></semantics></math> be a given projective class on an abelian category <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">\mathcal{A}</annotation></semantics></math>, def. <a class="maruku-ref" href="#ProjectiveClass"></a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mi>𝒫</mi></msub></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})_{\mathcal{P}}</annotation></semantics></math> for the corresponding model structure on unbounded chain complexes, theorem <a class="maruku-ref" href="#ModelStructureOnUnboundedFromProjectiveClass"></a>.</p> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mi>𝒫</mi></msub></mrow><annotation encoding="application/x-tex">C \in Ch(\mathcal{A})_{\mathcal{P}}</annotation></semantics></math> is cofibrant precisely if</p> <ol> <li> <p>in each degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{Z}</annotation></semantics></math> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">C_n</annotation></semantics></math> is relatively projective in <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">\mathcal{A}</annotation></semantics></math>;</p> </li> <li> <p>every morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> into a weakly contractible complex in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mi>𝒫</mi></msub></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})_{\mathcal{P}}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopic</a> to the <a class="existingWikiWord" href="/nlab/show/zero+morphism">zero morphism</a>.</p> </li> </ol> </div> <p>This appears as (<a href="#ChristensenHovey">ChristensenHovey, lemma 2.4</a>).</p> <div class="num_prop"> <h6 id="proposition_4">Proposition</h6> <p>A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f : A \to B</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msub><mo stretchy="false">)</mo> <mi>𝒫</mi></msub></mrow><annotation encoding="application/x-tex">Ch(\mathcal{A})_{\mathcal{P}}</annotation></semantics></math> is a cofibration precisely if it is degreewise</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/split+monomorphism">split monomorphism</a>;</p> </li> <li> <p>with cofibrant <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a>.</p> </li> </ol> </div> <p>This appears as (<a href="#ChristensenHovey">ChristensenHovey, prop. 2.5</a>).</p> <h3 id="relation_to_module_spectra">Relation to module spectra</h3> <p>For <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> any ring, there is the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+spectrum">Eilenberg-MacLane spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">H R</annotation></semantics></math>. This is an <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a>, hence there is a notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi><mi>R</mi></mrow><annotation encoding="application/x-tex">H R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module+spectra">module spectra</a>. These are <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to chain complexes of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-modules. See <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a> for details.</p> <h2 id="related_concepts">Related concepts</h2> <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+chain+complexes+of+super+vector+spaces">model structure on chain complexes of super vector spaces</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-algebras">model structure on dg-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> </ul> <h2 id="References">References</h2> <h3 id="for_bounded_chain_complexes">For bounded chain complexes</h3> <p>The projective model structure on connective chain complexes (Theorem <a class="maruku-ref" href="#ProjectiveModelStructure"></a>) is due to</p> <ul> <li id="Quillen67"><a class="existingWikiWord" href="/nlab/show/Daniel+Quillen">Daniel Quillen</a>, Section II.4 item 5 in: <em><a class="existingWikiWord" href="/nlab/show/Homotopical+Algebra">Homotopical Algebra</a></em>, Lecture Notes in Mathematics 43, Springer 1967(<a href="https://doi.org/10.1007/BFb0097438">doi:10.1007/BFb0097438</a>)</li> </ul> <p>see also:</p> <ul> <li id="DwyerSpalinski95"> <p><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Jan+Spalinski">Jan Spalinski</a>, Section 7 of: <em><a class="existingWikiWord" href="/nlab/show/Homotopy+theories+and+model+categories">Homotopy theories and model categories</a></em> (<a class="existingWikiWord" href="/nlab/files/DwyerSpalinski_HomotopyTheories.pdf" title="pdf">pdf</a>)</p> <p>in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">I. M. James</a>, <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em>, North Holland 1995 (<a href="https://www.elsevier.com/books/handbook-of-algebraic-topology/james/978-0-444-81779-2">ISBN:9780080532981</a>, <a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>)</p> </li> <li id="GoerssSchemmerhorn06"> <p><a class="existingWikiWord" href="/nlab/show/Paul+Goerss">Paul Goerss</a>, <a class="existingWikiWord" href="/nlab/show/Kirsten+Schemmerhorn">Kirsten Schemmerhorn</a>, Theorem 1.5 in: <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>)</p> </li> </ul> <p>The projective model structure on connective <em>co</em>chain complexes (Theorem <a class="maruku-ref" href="#ProjectiveModelStructureOnConnectiveCochainComplexes"></a>) is claimed, without proof, in:</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="CastiglioniCortinas03"> <p>J. L. Castiglioni, G. Cortiñas, Def. 4.7 of: <em>Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence</em>, J. Pure Appl. Algebra 191 (2004), no. 1-2, 119–142, (<a href="http://arxiv.org/abs/math/0306289">arXiv:math.KT/0306289</a>, <a href="https://doi.org/10.1016/j.jpaa.2003.11.009">doi:10.1016/j.jpaa.2003.11.009</a>)</p> </li> </ul> <p>Of course the description of model categories of chain complexes as (<a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> of) special cases of (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 is exactly opposite to the historical development of these ideas.</p> <p>While the homotopical treatment of weak equivalences of chain complexes (<a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a>s) in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> is at the beginning of all studies of higher categories and a “folk theorem” ever since</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, Letter to Alexander Grothendieck. April 11, 1984</li> </ul> <p>it seems that the injective model structure on chain complexes has been made fully explicit in print only in proposition 3.13 (according to remark 3.14 there) of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Tibor+Beke">Tibor Beke</a><em>Sheafifiable homotopy model categories</em>, Math. Proc. Cambridge Philos. Soc. <strong>129</strong> 3 (2000) 447-475 [<a href="http://arxiv.org/abs/math/0102087">arXiv:math/0102087</a>, <a href="https://doi.org/10.1017/S0305004100004722">doi:10.1017/S0305004100004722</a>]</li> </ul> <p>The projective model structure is discussed after that (and shown to be a <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a>) in:</p> <ul> <li id="Hovey99KTh"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Model category structures on chain complexes of sheaves</em> (1999) [<a href="https://faculty.math.illinois.edu/K-theory/0366">K-theory:0366</a>, <a href="https://faculty.math.illinois.edu/K-theory/0366/sheaves.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Hovey-SheavesOfChainComplexesPreprint.pdf" title="pdf">pdf</a>]</p> </li> <li id="Hovey01"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Model category structures on chain complexes of sheaves</em>, Trans. Amer. Math. Soc. <strong>353</strong> 6 (2001) [<a href="https://www.ams.org/journals/tran/2001-353-06/S0002-9947-01-02721-0">ams:S0002-9947-01-02721-0</a>, <a href="https://www.jstor.org/stable/221954">jstor:221954</a>]</p> </li> </ul> <p>An explicit proof of the injective model structure with monos in positive degree is spelled out in</p> <ul> <li id="Dungan10"><a class="existingWikiWord" href="/nlab/show/Gregory+Dungan">Gregory Dungan</a>, <em>Review of model categories</em>, 2010 (<a href="http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.696.123&rep=rep1&type=pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DunganModelCategories.pdf" title="pdf">pdf</a>)</li> </ul> <p>An explicit proof of the model structure on cochain complexes of abelian groups with fibrations the degreewise surjections is recorded in the appendix of</p> <ul id="Stel"> <li><a class="existingWikiWord" href="/nlab/show/Herman+Stel">Herman Stel</a>, <em><a class="existingWikiWord" href="/schreiber/show/master+thesis+Stel">∞-Stacks and their Function Algebras – with Applications to ∞-Lie Theory</a></em> (2010)</li> </ul> <p>The resolution model structures on cofibrant objects go back to</p> <ul id="DwyerKanStover"> <li><a class="existingWikiWord" href="/nlab/show/William+Dwyer">William Dwyer</a>, <a class="existingWikiWord" href="/nlab/show/Dan+Kan">Dan Kan</a>, C. Stover, <em>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_2</annotation></semantics></math> model category structure for pointed simplicial spaces</em>, J. Pure and Applied Algebra 90 (1993) 137–152</li> </ul> <p>and are reviewed in</p> <ul> <li id="Bousfield"><a class="existingWikiWord" href="/nlab/show/Aldridge+Bousfield">Aldridge Bousfield</a>, <em>Cosimplicial resolutions and homotopy spectral sequences in model categories</em> Geometry and Topology, volume 7 (2003)</li> </ul> <h3 id="ForUnboundedChainComplexes">For unbounded chain complexes</h3> <p>Influential precusor discussion of homotopy theory of unbounded chain complexes (introducing notions like K-projective and K-injective complexes):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolas+Spaltenstein">Nicolas Spaltenstein</a>, <em>Resolutions of unbounded complexes</em>, Compositio Mathematica <strong>65</strong> 2 (1988) 121-154 [<a href="http://www.numdam.org/item?id=CM_1988__65_2_121_0">numdam:CM_1988__65_2_121_0</a>, <a href="https://eudml.org/doc/89885">eudml:89885</a>]</li> </ul> <p>The observation that from this one obtains a model category structure on unbounded chain complexes is due to:</p> <ul> <li id="Hinich97"><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>Homological algebra of homotopy algebras</em>, Communications in Algebra <strong>25</strong> 10 (1997) 3291-3323 [<a href="http://arxiv.org/abs/q-alg/9702015">arXiv:q-alg/9702015</a>, <a href="https://doi.org/10.1080/00927879708826055">doi:10.1080/00927879708826055</a>, Erratum: <a href="http://arxiv.org/abs/math/0309453">arXiv:math/0309453</a>]</li> </ul> <p>and maybe independently due to:</p> <ul> <li id="HoveyPalmieriStrickland97"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <a class="existingWikiWord" href="/nlab/show/John+Palmieri">John Palmieri</a>, <a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, after theorem 9.3.1 in: <em>Axiomatic stable homotopy theory</em>, Memoirs Amer. Math. Soc. <strong>610</strong> (1997) [<a href="https://bookstore.ams.org/memo-128-610">ISBN:978-1-4704-0195-5</a>, <a href="http://www.math.rochester.edu/people/faculty/doug/otherpapers/axiomatic.pdf">pdf</a>]</li> </ul> <p>(where the relevant insights are credited to <a href="https://ncatlab.org/nlab/show/homological+algebra#Weibel94">Weibel (1994)</a>)</p> <p>and shown to be <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> in:</p> <ul> <li id="SchwedeShipley98"> <p><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, p. 7 of: <em>Algebras and modules in monoidal model categories</em>, Proc. London Math. Soc. <strong>80</strong> 2 (2000) 491-511 [<a href="https://arxiv.org/abs/math/9801082">arXiv:math/9801082</a>, <a href="https://doi.org/10.1112/S002461150001220X">doi:10.1112/S002461150001220X</a>]</p> </li> <li id="Hovey99"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, Thm 2.3.1 in: <em><a class="existingWikiWord" href="/nlab/show/Model+Categories">Model Categories</a></em>, Mathematical Surveys and Monographs, <strong>63</strong> AMS (1999) [<a href="https://bookstore.ams.org/surv-63-s">ISBN:978-0-8218-4361-1</a>, <a href="https://doi.org/http://dx.doi.org/10.1090/surv/063">doi:10.1090/surv/063</a>, <a href="https://people.math.rochester.edu/faculty/doug/otherpapers/hovey-model-cats.pdf">pdf</a>, <a href="http://books.google.co.uk/books?id=Kfs4uuiTXN0C&printsec=frontcover">Google books</a>]</p> </li> </ul> <p>and shown to be (<a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> and in addition) <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper</a> and <a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal</a> in:</p> <ul> <li id="Strickland20"><a class="existingWikiWord" href="/nlab/show/Neil+Strickland">Neil Strickland</a>, <em>The model structure for chain complexes</em> [<a href="https://arxiv.org/abs/2001.08955">arXiv:2001.08955</a>]</li> </ul> <p>In the context of the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> with the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+abelian+groups">model structure on simplicial abelian groups</a>:</p> <ul> <li id="SchwedeShipley03"><a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <em>Equivalences of monoidal model categories</em>, Algebr. Geom. Topol. <strong>3</strong> (2003) 287-334 [<a href="http://arxiv.org/abs/math.AT/0209342">arXiv:math.AT/0209342</a>, <a href="https://projecteuclid.org/euclid.agt/1513882376">euclid:euclid.agt/1513882376</a>]</li> </ul> <p>That the corresponding <a class="existingWikiWord" href="/nlab/show/category+of+simplicial+objects">category of simplicial objects</a> in unbounded chain complexes is thus a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> <a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a> is</p> <ul> <li id="RezkSchwedeShipley01"><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <a class="existingWikiWord" href="/nlab/show/Stefan+Schwede">Stefan Schwede</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, Cor. 4.6 in: <em>Simplicial structures on model categories and functors</em>, American Journal of Mathematics <strong>123</strong> 3 (2001) 551-575 [<a href="http://arxiv.org/abs/math/0101162">arXiv:0101162</a>, <a href="https://www.jstor.org/stable/25099071">jstor:25099071</a>]</li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> of chain complexes:</p> <ul> <li id="Fausk06"><a class="existingWikiWord" href="/nlab/show/Halvard+Fausk">Halvard Fausk</a>, <em>T-model structures on chain complexes of presheaves</em> [<a href="https://arxiv.org/abs/math/0612414">arXiv:math/0612414</a>]</li> </ul> <p>The article</p> <ul> <li id="ChristensenHovey"><a class="existingWikiWord" href="/nlab/show/Dan+Christensen">Dan Christensen</a>, <a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Quillen model structures for relative homological algebra</em>, Math. Proc. Cambridge Philos. Soc. <strong>133</strong> 2 (2002) 261-293 [<a href="https://arxiv.org/abs/math/0011216">arXiv:math/0011216</a>, <a href="https://doi.org/10.1017/S0305004102006126">doi:10.1017/S0305004102006126</a>]</li> </ul> <p>discusses model structures on unbounded chain complexes with generalized notions of epimorphisms induced from “projective classes”. See also:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Dan+Christensen">Dan Christensen</a>, <em>Derived categories and projective classes</em> , (2005) (<a href="http://hopf.math.purdue.edu/cgi-bin/generate?/Christensen/derived">hopf archive</a>)</li> </ul> <p>Another approach is due to James Gillespie, using <a class="existingWikiWord" href="/nlab/show/cotorsion+pairs">cotorsion pairs</a>. An overview of this work is in</p> <ul> <li id="HoveyOverview"><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <em>Cotorsion pairs and model categories</em>, 2006 (<a href="http://homepages.math.uic.edu/~bshipley/hovey.pdf">pdf</a>)</li> </ul> <p>Some generalizations and simplifications of the original approach are discussed in</p> <ul> <li id="Gillespie">James Gillespie, <em>Kaplansky classes and derived categories</em>, 2007, <a href="http://phobos.ramapo.edu/~jgillesp/updated%20qcsheaf.pdf">pdf</a></li> </ul> <p>Generalization to <a class="existingWikiWord" href="/nlab/show/bicomplexes">bicomplexes</a>:</p> <ul> <li id="MuroRoitzheim19"><a class="existingWikiWord" href="/nlab/show/Fernando+Muro">Fernando Muro</a>, <a class="existingWikiWord" href="/nlab/show/Constanze+Roitzheim">Constanze Roitzheim</a>, <em>Homotopy Theory of Bicomplexes</em>, Journal of Pure and Applied Algebra <strong>223</strong> 5 (2019) 1913-1939 [<a href="https://arxiv.org/abs/1802.07610">arXiv:1802.07610</a>, <a href="https://doi.org/10.1016/j.jpaa.2018.08.007">doi:10.1016/j.jpaa.2018.08.007</a>]</li> </ul> <p>Finally a third approach to the unbounded case is discussed (in a context of <a class="existingWikiWord" href="/nlab/show/mixed+motives">mixed motives</a>) in:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+D%C3%A9glise">Frédéric Déglise</a>, <em>Local and stable homological algebra in Grothendieck abelian categories</em>, Homology, Homotopy and Applications <strong>11</strong> 1 (2009) 219–260 [<a href="http://arxiv.org/abs/0712.3296">arXiv:0712.3296</a>, <a href="https://projecteuclid.org/journals/homology-homotopy-and-applications/volume-11/issue-1/Local-and-stable-homological-algebra-in-Grothendieck-abelian-categories/hha/1251832567.full">hha:1251832567</a>]</p> </li> <li id="CisinskiDeglise09"> <p><a class="existingWikiWord" href="/nlab/show/Denis-Charles+Cisinski">Denis-Charles Cisinski</a>, <a class="existingWikiWord" href="/nlab/show/Fr%C3%A9d%C3%A9ric+D%C3%A9glise">Frédéric Déglise</a>, <em>Triangulated categories of mixed motives</em>, Springer Monographs in Mathematics, Springer (2019) [<a href="http://arxiv.org/abs/0912.2110">arXiv:0912.2110</a>, <a href="https://doi.org/10.1007/978-3-030-33242-6">doi:10.1007/978-3-030-33242-6</a>]</p> </li> </ul> <p>A discussion of the homotopy theory of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> of unbounded chain complex is in</p> <ul> <li id="Jardine03"><a class="existingWikiWord" href="/nlab/show/J.+F.+Jardine">J. F. Jardine</a>, <em>Presheaves of chain complexes</em>, K-theory 30.4 (2003): 365-420 (<a href="https://pdfs.semanticscholar.org/360c/fd4eb76da956e0c8dd897377f6d6f342e44e.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/JardinePresheavesOfChainComplexes.pdf" title="pdf">pdf</a>)</li> </ul> <p>A model structure on noncommutative <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s whose proof strategy is useful also for cochain complexes is in</p> <ul> <li><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 <strong>17</strong>, AMS (1997) 55-58.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on April 16, 2024 at 14:38:38. 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