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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 for L-infinity algebras </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/2746/#Item_21" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="model_category_theory">Model category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></strong>, <a class="existingWikiWord" href="/nlab/show/model+%28infinity%2C1%29-category">model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mn>∞</mn> </mrow> <annotation encoding="application/x-tex">\infty</annotation> </semantics> </math>-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/relative+category">relative category</a>, <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibration">fibration</a>, <a class="existingWikiWord" href="/nlab/show/cofibration">cofibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+factorization+system">weak factorization system</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/resolution">resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cell+complex">cell complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+%28as+an+operation%29">homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thickmathspace"></mspace></mrow><annotation encoding="application/x-tex">\;</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+a+model+category">of a model category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Quillen+bifunctor">Quillen bifunctor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+Kan+extension">homotopy Kan extension</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+weighted+colimit">homotopy weighted (co)limit</a></p> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coend">homotopy (co)end</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield-Kan+map">Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+model+category">monoidal model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/monoidal+Quillen+adjunction">monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+model+category">enriched model category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/enriched+Quillen+adjunction">enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+enriched+model+category">monoidal enriched model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+Quillen+adjunction">simplicial Quillen adjunction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+monoidal+model+category">simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/combinatorial+model+category">combinatorial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cellular+model+category">cellular model category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+model+category">algebraic model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compactly+generated+model+category">compactly generated model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+model+category">cartesian closed model category</a>, <a class="existingWikiWord" href="/nlab/show/locally+cartesian+closed+model+category">locally cartesian closed model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> </ul> <p><strong>Producing new model structures</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/global+model+structures+on+functor+categories">on functor categories (global)</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Reedy+model+structure">Reedy model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+an+overcategory">on slice categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bousfield+localization+of+model+categories">Bousfield localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebraic+fibrant+objects">on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction+for+model+categories">Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom-space">(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cisinski+model+structure">Cisinski model structure</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%E2%88%9E-groupoids">for ∞-groupoids</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+topological+spaces">on topological spaces</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+topological+spaces">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+compactly+generated+topological+spaces">on compactly generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+Delta-generated+topological+spaces">on Delta-generated spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+diffeological+spaces">on diffeological spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Str%C3%B8m+model+structure">Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+over+a+test+category">model structure on presheaves over a test category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">on simplicial sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+semi-simplicial+sets">on semi-simplicial sets</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/classical+model+structure+on+simplicial+sets">classical model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/constructive+model+structure+on+simplicial+sets">constructive model structure</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+right+fibrations">for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groupoids">model structure on simplicial groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cubical+sets">on cubical sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-groupoids">on strict ∞-groupoids</a>, <a class="existingWikiWord" href="/nlab/show/natural+model+structure+on+groupoids">on groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">on chain complexes</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+abelian+groups">model structure on cosimplicial abelian groups</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+simplicial+sets">model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fine+model+structure+on+topological+G-spaces">fine model structure on topological G-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coarse+model+structure+on+topological+G-spaces">coarse model structure on topological G-spaces</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dgc-algebras">model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+chain+complexes">model structure on equivariant chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+equivariant+dgc-algebras">model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-groupoids</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+for+n-groupoids">for n-types</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure+on+groupoids">for 1-groupoids</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+groups">model structure on simplicial groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+reduced+simplicial+sets">model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+monoids">on monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">on simplicial T-algebras</a>, on <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+a+monad">on algebas over a monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> </ul> <p><em>specific <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">on dg-algebras over an operad</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">on dg-algebras</a> and on <a class="existingWikiWord" href="/nlab/show/simplicial+ring">on simplicial rings</a>/<a class="existingWikiWord" href="/nlab/show/model+structure+on+cosimplicial+rings">on cosimplicial rings</a></p> <p>related by the <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-modules">model structure on dg-modules</a></p> </li> </ul> <p><em>for stable/spectrum objects</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+spectra">model structure on spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+ring+spectra">model structure on ring spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+parameterized+spectra">model structure on parameterized spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+spectra">model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+categories+with+weak+equivalences">on categories with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+quasi-categories">Joyal model for quasi-categories</a> (and its <a class="existingWikiWord" href="/nlab/show/model+structure+for+cubical+quasicategories">cubical version</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">on sSet-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+complete+Segal+spaces">for complete Segal spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</em></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-categories">on dg-categories</a></li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-operads</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">on operads</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+Segal+operads">for Segal operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">on algebras over an operad</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">on modules over an algebra over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dendroidal+sets">on dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+complete+Segal+spaces">for dendroidal complete Segal spaces</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+for+dendroidal+Cartesian+fibrations">for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>n</mi><mo>,</mo><mi>r</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(n,r)</annotation></semantics></math>-categories</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Theta+space">for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+weak+complicial+sets">for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+cellular+sets">on cellular sets</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/canonical+model+structure">on higher categories in general</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+strict+%E2%88%9E-categories">on strict ∞-categories</a></p> </li> </ul> <p><em>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves / <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+homotopical+presheaves">on homotopical presheaves</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">on simplicial presheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/global+model+structure+on+simplicial+presheaves">global model structure</a>/<a class="existingWikiWord" href="/nlab/show/Cech+model+structure+on+simplicial+presheaves">Cech model structure</a>/<a class="existingWikiWord" href="/nlab/show/local+model+structure+on+simplicial+presheaves">local model structure</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sheaves">on simplicial sheaves</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+presheaves+of+simplicial+groupoids">on presheaves of simplicial groupoids</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-enriched+presheaves">on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+%282%2C1%29-sheaves">model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div></div> <h4 id="lie_theory"><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>-Lie theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+theory">∞-Lie theory</a></strong> (<a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure 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> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><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>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><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>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><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>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><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>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition_as_algebras_over_an_operad'>Definition as algebras over an operad</a></li> <li><a href='#DefinitionsAndQuillenEquivalences'>Definitions as formal/infinitesimal <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</a></li> <ul> <li><a href='#DefinitionsAndQuillenEquivalencesSummary'>Summary</a></li> <li><a href='#OndgLieAlgebras'>On dg-Lie algebras</a></li> <li><a href='#OndgCoalgebras'>On dg-coalgebras</a></li> <li><a href='#SheavesOverCosimplicialInfinitesimallyThickenedPoints'>On simplicial presheaves over cosimplicial formal spaces</a></li> <li><a href='#OnDgInfinitesimallyThickenedPoints'>On dg formal spaces</a></li> <li><a href='#OnCosimplicialAlgebras'>On cosimplicial algebras (and dual Dold-Kan correspondence)</a></li> </ul> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#general'>General</a></li> <li><a href='#HomotopiesAndDerivedHomSpaces'>Homotopies and derived hom spaces</a></li> <li><a href='#HomotopyFiberProducts'>Homotopy fiber products</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>There exist various <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures which <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</a> the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a>.</p> <p>By definition <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> are the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebras">∞-algebras</a> in the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> over the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>. As such they carry a <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a>. There is a strictification which leads equivalently to a <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a>.</p> <p>A more geometric way is to think of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> as being the <a class="existingWikiWord" href="/nlab/show/tangent+spaces">tangent spaces</a> to connected <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>, hence to the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a>/<a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groups">smooth ∞-groups</a>, hence as the first order infinitesimal neighbourhood</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔤</mi><mo>↪</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}\mathfrak{g} \hookrightarrow \mathbf{B}G </annotation></semantics></math></div> <p>of the essentially unique point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">* \to \mathbf{B}G</annotation></semantics></math> (see at <em><a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></em>). Since this is equivalently the first order neighbourhood of the <em><a class="existingWikiWord" href="/nlab/show/formal+geometry">formal</a></em> neighbourhood (the <a class="existingWikiWord" href="/nlab/show/jets">jets</a>) and the formal neighbourhood can be described purely in terms of <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>/<a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> (see at <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a>) many models for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are formulated in terms of such data.</p> <p>In particular, one succinct way to present <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> (as discussed there) is as <a class="existingWikiWord" href="/nlab/show/dg-coalgebras">dg-coalgebras</a>:</p> <div class="num_prop" id="LInfinityAlgebraIsQuasiFreeDgCoalgebra"> <h6 id="propositiondefinition">Proposition/Definition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a> <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><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>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo>,</mo><mi>⋯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}, [-], [-,-], [-,-,-], \cdots)</annotation></semantics></math> structure on a <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</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">\mathfrak{g}</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> structure on the graded-commutative <a class="existingWikiWord" href="/nlab/show/cofree+coalgebra">cofree coalgebra</a> over <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">\mathfrak{g}</annotation></semantics></math>.</p> <p>Conversely, the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> (and general “weak” morphisms between them) is the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of that of counital cocommutative <a class="existingWikiWord" href="/nlab/show/dg-coalgebras">dg-coalgebras</a> on those whose underlying bare graded-commutative <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> (forgetting the codifferential) is free</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo>↪</mo><mi>dgCoCAlg</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_\infty Alg \hookrightarrow dgCoCAlg \,. </annotation></semantics></math></div></div> <p>Accordingly it is of interest to have also <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a> or dually (on <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> of) <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> which presents <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras. Prop. <a class="maruku-ref" href="#LInfinityAlgebraIsFibrantObjectIndgFormalSpace"></a> below identifies the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math> as the (<a class="existingWikiWord" href="/nlab/show/full+subcategory">full sub-</a>)<a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> inside such a model structure of “differential graded formal spaces”, which in turn is related by a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> to various other models</p> <div class="num_remark"> <h6 id="remarkwarning">Remark/Warning</h6> <p>So we write “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math>” here for the <em><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras and general morphisms between them, since this is an entry on model category presentations. If we want to refer to the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras we here write explicitly “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_W( L_\infty Alg)</annotation></semantics></math>”, referring to the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a> of this 1-category.</p> </div> <div class="num_remark"> <h6 id="remarkwarning_2">Remark/Warning</h6> <p>All gradings in the following are <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>-gradings, unless explicitly stated otherwise. In terms of the underlying <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> this means that we are dealing with <a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a> (see below the section <em><a href="##SheavesOverCosimplicialInfinitesimallyThickenedPoints">Simplicial sheaves over comsimplicial formal spaces</a></em> for details): the algebra elements in positive degree correspond to <a class="existingWikiWord" href="/nlab/show/category+theory">categorical</a>/<a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a>-degree, and those in negative degree to the <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> degree of the derived site of cosimplicial formal spaces.</p> <p>Technically this affects for instance the nature of <a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a>: for instance the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a> <a href="#OndgLieAlgebras">below</a> is <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> from a <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a>. For unbounded chain complexes this is the “<a href="model+structure+on+chain+complexes#CategoricalProjectiveClass">Categorical projective class structure</a>” whose fibrations are the <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</a> that are surjective in every degree. This appears for instance in prop. <a class="maruku-ref" href="#ModelStructureOnDgLieAlgebras"></a> and prop. <a class="maruku-ref" href="#LInfinityAlgebrasAsACategoryOfFibrantObjects"></a> below.</p> <p>On the other hand, if one considered chain complexes in non-negative degree (for tangent complexes in “higher but non-derived geometry”), then one would use the <a href="model+structure+on+chain+complexes#ProjectiveStructureOnChainComplexes">Projective structure on chain complexes in non-negative degree</a>. This has as fibrations precisely the chain maps that are surjective in every <em>positive</em> degree. This case is (currently) <em>not</em> discussed in the following.</p> </div> <div class="num_remark" id="NonWeakMaps"> <h6 id="remarkwarning_3">Remark/Warning</h6> <p>Some of the model structures below are on the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras with “strict” morphisms between them, namely for those morphisms which are morphisms of <a class="existingWikiWord" href="/nlab/show/algebras+over+an+operad">algebras over an operad</a> for an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra regarded as an algebra over a <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>. We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>str</mi></msub><mo>→</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex"> L_\infty Alg_{str} \to L_\infty Alg </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/wide+subcategory">wide subcategory</a> on the strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-morphisms.</p> </div> <h2 id="definition_as_algebras_over_an_operad">Definition as algebras over an operad</h2> <p>As their name indicates, <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a> are the <a class="existingWikiWord" href="/nlab/show/homotopy+algebras">homotopy algebras</a> <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">over</a> the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a> (in a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>). As such, the general theory of <a class="existingWikiWord" href="/nlab/show/model+structures+on+algebras+over+an+operad">model structures on algebras over an operad</a> provides a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on the category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>This we discuss here. But there is also a natural identification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras with <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a> <a class="existingWikiWord" href="/nlab/show/derived+stack">derived</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>. For expressing this a host of other, Quillen equivalent model structures are available. These we discuss below in <em><a href="#DefinitionsAndQuillenEquivalences">Definitions as formal/infinitesimal ∞-stacks</a></em>.</p> <p>By the general discussion at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a></em>, if <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 <a class="existingWikiWord" href="/nlab/show/field">field</a> which contains the field of <a class="existingWikiWord" href="/nlab/show/rational+numbers">rational numbers</a>, then for every <a class="existingWikiWord" href="/nlab/show/symmetric+operad">symmetric operad</a> (uncolored) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> (unbounded) <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>, the <a class="existingWikiWord" href="/nlab/show/free-forgetful+adjunction">free-forgetful adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><mo stretchy="false">(</mo><mi>𝒪</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Alg(\mathcal{O}) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Ch_\bullet(k) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/algebras+over+an+operad">algebras over an operad</a> and the underlying chain complexes induces a <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> from the projective unbounded <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a> where hence on both sides</p> <ul> <li> <p>the weak equivalences are the morphisms that are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> on the (underlying) chain complexes;</p> </li> <li> <p>the fibrations are the morphisms that are degreewise <a class="existingWikiWord" href="/nlab/show/surjections">surjections</a> on the (underlying) chain complexes.</p> </li> </ul> <p>Hence in particular <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo>⊢</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F \vdash U)</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a> between these model structures.</p> <p>(<a href="#Hinich97">Hinich97, theorem 4.1.1</a>)</p> <p>So this is in particular true for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>=</mo><mover><mi>Lie</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\mathcal{O} = \widehat Lie</annotation></semantics></math> the standard <a class="existingWikiWord" href="/nlab/show/cofibrant+resolution">cofibrant resolution</a> of the <a class="existingWikiWord" href="/nlab/show/Lie+operad">Lie operad</a>. In this case <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Alg</mi><mo stretchy="false">(</mo><mover><mi>Lie</mi><mo>^</mo></mover><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>str</mi></msub></mrow><annotation encoding="application/x-tex">Alg(\widehat Lie) \simeq L_\infty Alg_{str}</annotation></semantics></math> is the category of (unbounded) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras (with strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-maps between them as in remark <a class="maruku-ref" href="#NonWeakMaps"></a> above) and hence is equipped with a <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> this way</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>str</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mover><munder><mo>→</mo><mi>U</mi></munder><mover><mo>←</mo><mi>F</mi></mover></mover><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_\infty Alg_{str}(k) \stackrel{\overset{F}{\leftarrow}}{\underset{U}{\to}} Ch_\bullet(k) \,. </annotation></semantics></math></div> <p>Moreover, by the rectification result discussed at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras+over+an+operad">model structure on dg-algebras over an operad</a></em>, the resolution map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Lie</mi><mo>^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>Lie</mi></mrow><annotation encoding="application/x-tex">\widehat Lie \stackrel{\simeq}{\to} Lie</annotation></semantics></math> induces 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><msub><mi>L</mi> <mn>∞</mn></msub><msub><mi>Alg</mi> <mi>str</mi></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>dgLieAlg</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> L_\infty Alg_{str}(k) \stackrel{\simeq}{\to} dgLieAlg(k) </annotation></semantics></math></div> <p>with the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a>, similarly <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred</a> from the <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a>.</p> <h2 id="DefinitionsAndQuillenEquivalences">Definitions as formal/infinitesimal <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</h2> <p>We list here definitions of various further <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structures that all <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">present</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/L-%E2%88%9E+algebras">L-∞ algebras</a> and describe a web of <a class="existingWikiWord" href="/nlab/show/zig-zags">zig-zags</a> of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> between them. These Quillen equivalences may be thought of as presenting an equivalence between the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras and that of <a class="existingWikiWord" href="/nlab/show/infinitesimal+object">infinitesimal</a> <a class="existingWikiWord" href="/nlab/show/derived+stack">derived</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> (“<a class="existingWikiWord" href="/nlab/show/formal+moduli+problems">formal moduli problems</a>”).</p> <ul> <li> <p><a href="#DefinitionsAndQuillenEquivalencesSummary">Summary</a></p> </li> <li> <p><a href="#OndgLieAlgebras">On dg-Lie algebras</a></p> </li> <li> <p><a href="#OndgCoalgebras">On dg-Coalgebras</a></p> </li> <li> <p><a href="#SheavesOverCosimplicialInfinitesimallyThickenedPoints">On simplicial sheaves over cosimplicial formal spaces</a></p> </li> <li> <p><a href="#OnDgInfinitesimallyThickenedPoints">On dg-formal spaces</a></p> </li> </ul> <h3 id="DefinitionsAndQuillenEquivalencesSummary">Summary</h3> <p>The following tabulates the main categories considered below, the functors relating them and their homotopy theoretic nature. The last row points to the relevant definitions and propositions of the following text.</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a></th><th>form <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> in commutative <a class="existingWikiWord" href="/nlab/show/Artin+algebra">Artin</a> <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a></th><th>dualize</th><th>commutative <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a></th><th>form <a class="existingWikiWord" href="/nlab/show/tangent+space">tangents</a></th><th><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>↪</mo><mi>CE</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{CE}{\hookrightarrow}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><mi>dgArtinCAlg</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Pro(dgArtinCAlg)^{op} </annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munderover><mo>⟶</mo><mrow><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace></mrow><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>*</mo></msup></mrow></munderover></mrow><annotation encoding="application/x-tex">\underoverset{\;\simeq\;}{(-)^*}{\longrightarrow}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgCoCAlg</mi></mrow><annotation encoding="application/x-tex">dgCoCAlg</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mo>→</mo><mi>ℒ</mi></mover></mrow><annotation encoding="application/x-tex">\stackrel{\mathcal{L}}{\to}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgLieAlg</mi></mrow><annotation encoding="application/x-tex">dgLieAlg</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>=</mo><mo>:</mo><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">=: dgFormalSpace</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td><td style="text-align: left;"></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> under <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/opposite+model+structure">opposite model structure</a> of <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a></td></tr> <tr><td style="text-align: left;">prop. <a class="maruku-ref" href="#LInfinityAlgebraIsQuasiFreeDgCoalgebra"></a></td><td style="text-align: left;">def. <a class="maruku-ref" href="#ChevalleyEilenbergAlgebraConstruction"></a></td><td style="text-align: left;">def. <a class="maruku-ref" href="#dgFormalSpace"></a></td><td style="text-align: left;">prop. <a class="maruku-ref" href="#DualizingInclusionOfDGFormalSpaceIntoDgCoalgebras"></a></td><td style="text-align: left;">def. <a class="maruku-ref" href="#DGCoalgebrasCategory"></a></td><td style="text-align: left;">prop. <a class="maruku-ref" href="#LeftAdjointFromDgCoAlgToDgAlg"></a></td><td style="text-align: left;">def. <a class="maruku-ref" href="#dgLieAlgebraCategory"></a></td></tr> </tbody></table> <p>Here we are trying to use suggestive names of the categories involved. The notation used here corresponds to that in (<a href="#Pridham">Pridham</a>) by the following dictionary</p> <blockquote> <p>(handle with care, may still need attention)</p> </blockquote> <table><thead><tr><th>notation used here</th><th>notation in <a href="#Pridham">Pridham</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DerivedFormalSpace</mi></mrow><annotation encoding="application/x-tex">DerivedFormalSpace</annotation></semantics></math>, def. <a class="maruku-ref" href="#DerivedFormalSpace"></a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>scSp</mi></mrow><annotation encoding="application/x-tex">scSp</annotation></semantics></math>, def. 1.32</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math>, def. <a class="maruku-ref" href="#dgFormalSpace"></a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>DG</mi> <mi>ℤ</mi></msub><mi>Sp</mi></mrow><annotation encoding="application/x-tex">DG_\mathbb{Z}Sp</annotation></semantics></math>. def. 3.1</td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>FormalSpace</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">FormalSpace^{\Delta^{op}}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sDGSp</mi></mrow><annotation encoding="application/x-tex">sDGSp</annotation></semantics></math>, def. 4.6</td></tr> </tbody></table> <h3 id="OndgLieAlgebras">On dg-Lie algebras</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0.</p> <div class="num_defn" id="dgLieAlgebraCategory"> <h6 id="definition">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgLieAlg</mi> <mi>k</mi></msub><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">dgLieAlg_k \in Cat</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="num_prop" id="ModelStructureOnDgLieAlgebras"> <h6 id="proposition">Proposition</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgLieAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">dgLieAlg_k</annotation></semantics></math> carries a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure in which</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> are the degreewise <a class="existingWikiWord" href="/nlab/show/surjective+maps">surjective maps</a>;</p> </li> <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> </ul> <p>on the underlying <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>. This is the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> of the corresponding <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a> along the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> to the <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a>.</p> </div> <p>(<a href="#Pridham">Pridham, lemma 3.24</a>)</p> <p>We call this the <strong><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a></strong>.</p> <div class="num_defn" id="SendingDGLieAlgebraToDgCoalgebra"> <h6 id="definition_2">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>dgLieAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C} \;\colon\; dgLieAlg_k \to dgCoCAlg_k </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> which sends a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> <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>d</mi><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g},d,[-,-])</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> whose underlying <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> is free on the underlying <a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</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">\mathfrak{g}</annotation></semantics></math> and whose <a class="existingWikiWord" href="/nlab/show/coderivation">coderivation</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>↦</mo><mi>d</mi><msub><mi>v</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> \delta \colon v_1 \mapsto d v_1 </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>δ</mi><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">[</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \delta \colon (v_1, v_2) \mapsto [v_1, v_2] </annotation></semantics></math></div> <p>and then extended as a coderivation.</p> <p>(The <em>Chevalley-Eilenberg</em> dg-coalgebra.)</p> </div> <div class="num_prop" id="LeftAdjointFromDgCoAlgToDgAlg"> <h6 id="proposition_2">Proposition</h6> <p>The functor from def. <a class="maruku-ref" href="#SendingDGLieAlgebraToDgCoalgebra"></a> has a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgLieAlg</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{L} \;\colon\; dgCoCAlg_k \to dgLieAlg_k \,. </annotation></semantics></math></div></div> <p>(<a href="#Quillen69">Quillen 69, App. B6</a>, <a href="#Hinich98">Hinich98, 1.2.1, 2.2.5</a>, see also <a href="#Pridham">Pridham, def. 3.23</a>).</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>If one thinks of a <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> as presenting a a derived formal space, as discuss <a href="#SheavesOverCosimplicialInfinitesimallyThickenedPoints">below</a> then its image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math>, prop. <a class="maruku-ref" href="#LeftAdjointFromDgCoAlgToDgAlg"></a>, may be thought of as its <em>tangent dg-Lie algebra</em>. Therefore <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi></mrow><annotation encoding="application/x-tex">\mathcal{L}</annotation></semantics></math> is also called the <strong>tangent Lie algebra functor</strong>.</p> </div> <p>(<a href="#Hinich98">Hinich98, 1.2.1</a>)</p> <h3 id="OndgCoalgebras">On dg-coalgebras</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/field">field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0.</p> <div class="num_defn" id="DGCoalgebrasCategory"> <h6 id="definition_3">Definition</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex">dgCoCAlg_k \in Cat</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of co-commutative counital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+object">graded</a> <a class="existingWikiWord" href="/nlab/show/dg-coalgebras">dg-coalgebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="num_prop" id="ModelStructureOnCocommutativeDGCoalgebras"> <h6 id="proposition_3">Proposition</h6> <p>There exists a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">dgCoCAlg_k</annotation></semantics></math> for which</p> <ul> <li> <p>the <a class="existingWikiWord" href="/nlab/show/cofibrations">cofibrations</a> are the morphisms that are degreewise <a class="existingWikiWord" href="/nlab/show/injections">injections</a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are the morphisms <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> whose corresponding morphisms of <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℒ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{L}(f)</annotation></semantics></math>, prop. <a class="maruku-ref" href="#LeftAdjointFromDgCoAlgToDgAlg"></a>, is a <a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a> on the underlying <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>.</p> </li> </ul> </div> <p>(<a href="#Hinich98">Hinich98, theorem 3.1</a>) See also (<a href="#Pridham">Pridham, lemma 3.25</a>).</p> <p>We call this the <strong><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a></strong>.</p> <div class="num_remark" id="OndgCoAlgWEs"> <h6 id="remarkwarning_4">Remark/Warning</h6> <p>Beware that the class of weak equivalences in prop. <a class="maruku-ref" href="#ModelStructureOnCocommutativeDGCoalgebras"></a> is <em>not</em> that of quasi-isomorphisms on the chain complexes underlying the dg-coalgebras.</p> </div> <p>But they form a sub-class:</p> <div class="num_prop" id="dgLieQIsTodgCoAlgs"> <h6 id="proposition_4">Proposition</h6> <p>The Chevalley-Eilenberg functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>dgLieAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C} \;\colon\; dgLieAlg_k \to dgCoCAlg_k </annotation></semantics></math></div> <p>from def. <a class="maruku-ref" href="#SendingDGLieAlgebraToDgCoalgebra"></a> sends <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> on the <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a> underlying <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> to <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> on the chain complexes underlying their Chevalley-Eilenberg <a class="existingWikiWord" href="/nlab/show/dg-coalgebras">dg-coalgebras</a>.</p> </div> <p>(<a href="#Lurie">Lurie, prop. 2.2.6</a>)</p> <div class="num_prop" id="QuillenEquivalenceBetweendgLieAnddgCoCAlg"> <h6 id="proposition_5">Proposition</h6> <p>The 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>ℒ</mi><mo>⊣</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>dgLie</mi> <mi>k</mi></msub><mover><munder><mo>→</mo><mi>𝒞</mi></munder><mover><mo>←</mo><mi>ℒ</mi></mover></mover><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> (\mathcal{L} \dashv \mathcal{C}) \;\colon\; dgLie_k \stackrel{\overset{\mathcal{L}}{\leftarrow}}{\underset{\mathcal{C}}{\to}} dgCoCAlg_k </annotation></semantics></math></div> <p>from prop. <a class="maruku-ref" href="#LeftAdjointFromDgCoAlgToDgAlg"></a> constitutes a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a>, prop. <a class="maruku-ref" href="#ModelStructureOnDgLieAlgebras"></a>, and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a>, prop. <a class="maruku-ref" href="#ModelStructureOnCocommutativeDGCoalgebras"></a>.</p> </div> <p>(<a href="#Hinich98">Hinich 1998, theorem 3.2</a>)</p> <div class="num_remark"> <h6 id="remark_2">Remark</h6> <p>Since every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">dgCoCAlg_k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/cofibrant+object">cofibrant object</a> and every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>dgLie</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">dgLie_k</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/fibrant+object">fibrant object</a>, the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>ℒ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgCocAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \mathcal{C}\mathcal{L} \colon dgCoCAlg_k \to dgCocAlg_k </annotation></semantics></math></div> <p>is already its <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a> and the <a class="existingWikiWord" href="/nlab/show/unit+of+an+adjunction">unit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝔤</mi><mo>→</mo><mi>𝒞</mi><mi>ℒ</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex"> \mathfrak{g} \to \mathcal{C}\mathcal{L}\mathfrak{g} </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> that exhibits <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mi>ℒ</mi><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}\mathcal{L}\mathfrak{g}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/fibrant+resolution">fibrant resolution</a> and moreover, if <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">\mathfrak{g}</annotation></semantics></math> was already fibrant, hence by prop. <a class="maruku-ref" href="#LInfinityAlgebraIsFibrantObjectIndgFormalSpace"></a> below an <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>, as a <strong>strictification</strong> 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">\mathfrak{g}</annotation></semantics></math>: because a <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a> is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra in which the Lie bracket satisfies its <a class="existingWikiWord" href="/nlab/show/Jacobi+identity">Jacobi identity</a> strictly (not just up to a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> measured by the trinary bracket) and in which the “Jacobiator identity” holds strictly, etc.</p> </div> <h3 id="SheavesOverCosimplicialInfinitesimallyThickenedPoints">On simplicial presheaves over cosimplicial formal spaces</h3> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>InfThPoint</mi><mo>↪</mo><msup><mi>Alg</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> InfThPoint \hookrightarrow Alg^{op} </annotation></semantics></math></div> <p>for the category of <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a>, the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <a class="existingWikiWord" href="/nlab/show/Artin+algebras">Artin algebras</a> (“Weil alebras” in the language of <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>). The category of <em><a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a></em>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>cInfThPoint</mi><mo>↪</mo><msup><mi>sAlg</mi> <mi>op</mi></msup><mo>≃</mo><mo stretchy="false">(</mo><msup><mi>Alg</mi> <mi>op</mi></msup><msup><mo stretchy="false">)</mo> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex"> cInfThPoint \hookrightarrow sAlg^{op} \simeq (Alg^{op})^{\Delta} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> on <a class="existingWikiWord" href="/nlab/show/simplicial+algebras">simplicial algebras</a> on those which are <a class="existingWikiWord" href="/nlab/show/Artin+algebra">Artinian</a> (or “Weil” ): the category of <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a>.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>FormalSpace</mi><mo>≔</mo><msup><mi>Fun</mi> <mi>lex</mi></msup><mo stretchy="false">(</mo><msup><mi>InfThPoints</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> FormalSpace \coloneqq Fun^{lex}(InfThPoints^{op}, Set) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> over <a class="existingWikiWord" href="/nlab/show/infinitesimally+thickened+points">infinitesimally thickened points</a> on those given by <a class="existingWikiWord" href="/nlab/show/left+exact+functors">left exact functors</a>.</p> </div> <p>In (<a href="#Pridham">Pridham</a>) this is def. 1.18.</p> <div class="num_defn" id="DerivedFormalSpace"> <h6 id="definition_6">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DerivedFormalSpace</mi><mo>≔</mo><msup><mi>Fun</mi> <mi>lex</mi></msup><mo stretchy="false">(</mo><msup><mi>cInfThPoints</mi> <mi>op</mi></msup><mo>,</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> DerivedFormalSpace \coloneqq Fun^{lex}(cInfThPoints^{op}, sSet) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the category of <a class="existingWikiWord" href="/nlab/show/simplicial+presheaves">simplicial presheaves</a> over <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> infinitesimally thickened points on those given by <a class="existingWikiWord" href="/nlab/show/left+exact+functors">left exact functors</a>.</p> </div> <p>In (<a href="#Pridham">Pridham</a>) this is def. 1.32.</p> <div class="num_theorem"> <h6 id="theorem">Theorem</h6> <p>There exists a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DerivedFormalSpace</mi></mrow><annotation encoding="application/x-tex">DerivedFormalSpace</annotation></semantics></math> whose</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are those morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/local+morphisms">local morphisms</a> with respect to quasi-smooth maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>E</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">p \colon E \to Y</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</li> </ul> <p>(…)</p> </div> <p>This is (<a href="#Pridham">Pridham, def. 2.7, theorem 2.14</a>)</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</h6> <p>Between quasi-smooth objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DerivedFormalSpace</mi></mrow><annotation encoding="application/x-tex">DerivedFormalSpace</annotation></semantics></math> the weak equivalences are precisely the morphisms which are <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalences">weak homotopy equivalences</a> of simplicial sets over each object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>cInfThPoint</mi></mrow><annotation encoding="application/x-tex">cInfThPoint</annotation></semantics></math>.</p> </div> <p>This is (<a href="#Pridham">Pridham, cor. 2.16</a>).</p> <h3 id="OnDgInfinitesimallyThickenedPoints">On dg formal spaces</h3> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> which is</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/noetherian+ring">noetherian</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+ring">local</a> with</p> <ol> <li> <p>maximal ideal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/residue+field">residue 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>.</p> </li> </ol> </li> </ol> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ArtCAlg</mi> <mi>Λ</mi></msub></mrow><annotation encoding="application/x-tex"> ArtCAlg_{\Lambda} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of commutative <a class="existingWikiWord" href="/nlab/show/local+Artin+algebras">local Artin algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi></mrow><annotation encoding="application/x-tex">\Lambda</annotation></semantics></math> whose residue field is also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <p><a href="#Pridham">Pridham, 1.2</a></p> <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>μ</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\mu = 0</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Λ</mi><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\Lambda = k</annotation></semantics></math> and we just write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ArtCAlg</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ArtCAlg_k \,. </annotation></semantics></math></div></div> <p>In all of the following this is assumed to be the case, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>.</p> <p>(<a href="#Pridham">Pridham, below def. 3.3</a>)</p> <div class="num_defn" id="dgFormalSpace"> <h6 id="definition_8">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>dgArtCAlg</mi> <mi>k</mi></msub><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> dgArtCAlg_k \in Cat </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/category">category</a> 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>-<a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">graded-commutative</a> <a class="existingWikiWord" href="/nlab/show/Artin+algebra">Artin</a> <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/residue+field">residue 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>.</p> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArtCAlg</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> Pro(dgArtCAlg_k) \in Cat </annotation></semantics></math></div> <p>for its category of <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi><mo>≔</mo><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArtCAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> dgFormalSpace \coloneqq Pro(dgArtCAlg_k)^{op} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of that.</p> </div> <p>This is (<a href="#Pridham">Pridham, def. 3.1</a>) following (<a href="#Manetti02">Manetti 02</a>).</p> <div class="num_remark" id="OnProAlg"> <h6 id="remark_3">Remark</h6> <p>While it so happens that every <a class="existingWikiWord" href="/nlab/show/coalgebra">coalgebra</a> and <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> is the <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of its <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a>-<a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a> subalgebras (see at <em><a href="coalgebra#AsFilteredColimits">coalgebra – As filtered colimits</a></em>), this is not in general the case for <a class="existingWikiWord" href="/nlab/show/algebras">algebras</a>. But it follows that the <a class="existingWikiWord" href="/nlab/show/linear+dual">linear dual</a> of a general coalgebra is a <a class="existingWikiWord" href="/nlab/show/filtered+limit">filtered limit</a> of finite-dimensional algebras, hence a <a class="existingWikiWord" href="/nlab/show/pro-object">pro-object</a> in finite dimensional algebras. This is the reason for the appearance of <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> in def. <a class="maruku-ref" href="#dgFormalSpace"></a>.</p> </div> <div class="num_prop" id="ModelStructureOnDgFormalSpace"> <h6 id="proposition_7">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/cofibrantly+generated+model+category">cofibrantly generated</a> <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArtinCAlg</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Pro(dgArtinCAlg_k)</annotation></semantics></math>, def. <a class="maruku-ref" href="#dgFormalSpace"></a> – hence an <a class="existingWikiWord" href="/nlab/show/opposite+model+category">opposite model structure</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpaces</mi></mrow><annotation encoding="application/x-tex">dgFormalSpaces</annotation></semantics></math> – whose <a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are those morphisms that are <a class="existingWikiWord" href="/nlab/show/local+morphisms">local morphisms</a> relative to quasi-smooth maps in the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> over their codomain.</p> </div> <p>This is (<a href="#Pridham">Pridham, prop. 4.36</a>).</p> <div class="num_defn" id="ChevalleyEilenbergAlgebraConstruction"> <h6 id="definition_9">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mover><mo>→</mo><mrow></mrow></mover><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArtinCAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> CE \;\colon\; L_\infty Alg \stackrel{}{\to} Pro(dgArtinCAlg_k)^{op} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/functor">functor</a> which regards an <a class="existingWikiWord" href="/nlab/show/L-infinity+algebra">L-infinity algebra</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">\mathfrak{g}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/dg-coalgebra">dg-coalgebra</a> by prop. <a class="maruku-ref" href="#LInfinityAlgebraIsQuasiFreeDgCoalgebra"></a> and then forms the <a class="existingWikiWord" href="/nlab/show/dual+vector+space">linear dual</a> <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>, the <strong><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CE(\mathfrak{g})</annotation></semantics></math> 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">\mathfrak{g}</annotation></semantics></math> (a pro-dg-algebra according to def. <a class="maruku-ref" href="#dgFormalSpace"></a>).</p> </div> <div class="num_prop" id="LInfinityAlgebraIsFibrantObjectIndgFormalSpace"> <h6 id="proposition_8">Proposition</h6> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras are precisely the <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math> (hence in the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg+cocommutative+coalgebras">model structure on dg cocommutative coalgebras</a>, by <a href="#Pridham">Pridham Cor. 4.56</a>): the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> <a class="existingWikiWord" href="/nlab/show/functor">functor</a> of def. <a class="maruku-ref" href="#ChevalleyEilenbergAlgebraConstruction"></a>,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mover><mo>→</mo><mi>CE</mi></mover><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArtinCAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>≃</mo><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex"> L_\infty Alg \stackrel{CE}{\to} Pro(dgArtinCAlg_k)^{op} \simeq dgFormalSpace </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> onto its <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a>, which are the <a class="existingWikiWord" href="/nlab/show/fibrant+objects">fibrant objects</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CE</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>dgFormalSpace</mi> <mi>fib</mi></msub><mo>↪</mo><mi>dgFormalSpace</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CE \;\colon\; L_\infty Alg \stackrel{\simeq}{\to} dgFormalSpace_{fib} \hookrightarrow dgFormalSpace \,. </annotation></semantics></math></div></div> <p>This is proven inside the proof of (<a href="#Pridham">Pridham, prop. 4.42</a>).</p> <div class="num_remark" id="CategoryOfFibrantObjectsLInfinity"> <h6 id="remark_4">Remark</h6> <p>Prop. <a class="maruku-ref" href="#LInfinityAlgebraIsFibrantObjectIndgFormalSpace"></a> shows in particular that the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math> of prop/def. <a class="maruku-ref" href="#LInfinityAlgebraIsQuasiFreeDgCoalgebra"></a> carries the structure of a <em><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a></em> that presents the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras. Notice that, of course, passing to the full subcategory of fibrant objects does not change the <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> presented by the underlying <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a> in that we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> between the <a class="existingWikiWord" href="/nlab/show/simplicial+localizations">simplicial localizations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><mi>dgFormalSpace</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>L</mi> <mi>W</mi></msub><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_W(dgFormalSpace) \simeq L_W(L_\infty Alg) \,. </annotation></semantics></math></div></div> <p>The following proposition characterizes the structure of this <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>.</p> <div class="num_prop" id="LInfinityAlgebrasAsACategoryOfFibrantObjects"> <h6 id="proposition_9">Proposition</h6> <p>The induced structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math> under the inclusion of prop. <a class="maruku-ref" href="#LInfinityAlgebraIsFibrantObjectIndgFormalSpace"></a> has</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a> are precisely the maps that are <a class="existingWikiWord" href="/nlab/show/quasi-isomorphisms">quasi-isomorphisms</a> on the underlying chain complexes;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fibrations">fibrations</a> include in particular the maps that are surjections on the underlying chain complexes.</p> </li> </ol> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>The first statement is proven in the proof of (<a href="#Pridham">Pridham, prop. 4.42</a>).</p> <p>The second statement follows by (<a href="#Pridham">Pridham, def. 4.34</a>) with the existence of the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpaces</mi></mrow><annotation encoding="application/x-tex">dgFormalSpaces</annotation></semantics></math>.</p> <blockquote> <p>Should spell out how this follows, using lifting.</p> </blockquote> </div> <div class="num_prop" id="OnWEsOnLInfinity"> <h6 id="remarkwarning_5">Remark/Warning</h6> <p>Beware, as in remark <a class="maruku-ref" href="#OndgCoAlgWEs"></a>, that the class of weak equivalences in prop. <a class="maruku-ref" href="#LInfinityAlgebrasAsACategoryOfFibrantObjects"></a> differs from that of those maps on associated <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebras">Chevalley-Eilenberg algebras</a> which are quasi-isos on the underlying chain complexes of the <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> (which instead are the weak equivalences in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-algebras">model structure on dg-algebras</a>, hence in particular those used in Sullivan <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational homotopy theory</a>). Instead the weak equivalences correspond to the maps of CE-algebra that are quasi-isomorphisms only on the chain complexes given by the co-unary component of the differential of the CE-algebra.</p> </div> <div class="num_prop"> <h6 id="proposition_10">Proposition</h6> <p>There is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a> between the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math> hence of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub><mi>Alg</mi></mrow><annotation encoding="application/x-tex">L_\infty Alg</annotation></semantics></math> according to remark <a class="maruku-ref" href="#CategoryOfFibrantObjectsLInfinity"></a>, and the homotopy category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras according to (<a href="#Kontsevich94">Kontsevich 94</a>).</p> </div> <p>(<a href="#Pridham">Pridham, prop. 4.42, see above def. 4.29</a>)</p> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p>If <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 of <a class="existingWikiWord" href="/nlab/show/characteristic">characteristic</a> 0 then there is a <a class="existingWikiWord" href="/nlab/show/zig-zag">zig-zag</a> of <a class="existingWikiWord" href="/nlab/show/Quillen+equivalences">Quillen equivalences</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DerivedFormalSpace</mi></mrow><annotation encoding="application/x-tex">DerivedFormalSpace</annotation></semantics></math>, def. <a class="maruku-ref" href="#DerivedFormalSpace"></a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math>, def. <a class="maruku-ref" href="#dgFormalSpace"></a>, hence an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> between their <a class="existingWikiWord" href="/nlab/show/simplicial+localizations">simplicial localizations</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mi>W</mi></msub><mi>DerivedFormalSpace</mi><mo>≃</mo><msub><mi>L</mi> <mi>W</mi></msub><mi>dgFormalSpace</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> L_W DerivedFormalSpace \simeq L_W dgFormalSpace \,. </annotation></semantics></math></div></div> <p>This is (<a href="#Pridham">Pridham, cor. 4.49</a>).</p> <div class="num_prop"> <h6 id="proposition_12">Proposition</h6> <p>For arbitrary <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>, there is 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><msub><mi>dgLie</mi> <mi>k</mi></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>dgFormalSpace</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dgLie_k \stackrel{\simeq}{\to} dgFormalSpace \,. </annotation></semantics></math></div></div> <p>(<a href="#Pridham">Pridham, theorem 4.55</a>)</p> <div class="num_prop" id="DualizingInclusionOfDGFormalSpaceIntoDgCoalgebras"> <h6 id="proposition_13">Proposition</h6> <p>The inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi><mo>→</mo><msub><mi>dgCoCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> dgFormalSpace \to dgCoCAlg_k </annotation></semantics></math></div> <p>given by sending an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Pro</mi><mo stretchy="false">(</mo><msub><mi>dgArticCAlg</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>≔</mo><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">Pro(dgArticCAlg_k)^{op} \coloneqq dgFormalSpace</annotation></semantics></math>, hence an <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, to its dual dg-coalgebra <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^*</annotation></semantics></math>, is the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> part of a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the model structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math>, prop. <a class="maruku-ref" href="#ModelStructureOnDgFormalSpace"></a>, and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a>, prop. <a class="maruku-ref" href="#ModelStructureOnCocommutativeDGCoalgebras"></a>.</p> </div> <p>(<a href="#Pridham">Pridham, cor. 4.56</a>)</p> <h3 id="OnCosimplicialAlgebras">On cosimplicial algebras (and dual Dold-Kan correspondence)</h3> <p>Also a version of the “<a href="monoidal+Dold-Kan+correspondence#DualDoldKanQuillenEquivalences">dual monoidal Dold-Kan correspondence</a>” gives a Quillen equivalence between two model structures for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras. This is (<a href="#Pridham">Pridham, section 4.4</a>). This we discuss now</p> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>This equivalence has the nice property that starting with the <a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a> and then “denormalizing” it under dual monoidal Dold-Kan to a cosimplicial nilpotent algebra yields manifestly an incarnation of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebra in terms of simplicial complexes of infinitesimal simplices as is implicit in the work of <a class="existingWikiWord" href="/nlab/show/Anders+Kock">Anders Kock</a> in <a class="existingWikiWord" href="/nlab/show/synthetic+differential+geometry">synthetic differential geometry</a>. This is spelled out further in <a class="existingWikiWord" href="/schreiber/show/differential+cohomology+in+a+cohesive+topos">dcct, section 4.5.1</a>.</p> </div> <div class="num_defn" id="CosimplicialDgAlgebras"> <h6 id="definition_10">Definition</h6> <p>Write</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>dg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex"> (dg\hat {\mathcal{C}})^{\Delta} </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial</a> <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg</a>-<a class="existingWikiWord" href="/nlab/show/Artin+algebras">Artin algebras</a> (<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">\mathbb{N}</annotation></semantics></math>-graded).</p> </div> <p>(<a href="#Pridham">Pridham, def. 4.6</a>)</p> <div class="num_prop" id="ModelStructureOnCosimplicialDgAlgebras"> <h6 id="proposition_14">Proposition</h6> <p>The category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>dg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex">(dg\hat{\mathcal{C}})^{\Delta}</annotation></semantics></math> of def. <a class="maruku-ref" href="#CosimplicialDgAlgebras"></a> carries a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure where</p> <p>(…)</p> </div> <p>This is (<a href="#Pridham">Pridham, def. 4.11, prop. 4.12</a>).</p> <div class="num_defn" id="dgdgAlgebras"> <h6 id="definition_11">Definition</h6> <p>Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>DGdg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex"> DGdg\hat \mathcal{C} </annotation></semantics></math></div> <p>for <a class="existingWikiWord" href="/nlab/show/pro-objects">pro-objects</a> in <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> (<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">\mathbb{N}</annotation></semantics></math>-graded) in <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg</a>-<a class="existingWikiWord" href="/nlab/show/Artin+algebras">Artin algebras</a> (<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">\mathbb{N}</annotation></semantics></math>-graded).</p> </div> <p>(<a href="#Pridham">Pridham, def. 4.19</a>)</p> <div class="num_prop"> <h6 id="proposition_15">Proposition</h6> <p>The dual <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a> functor from <a class="existingWikiWord" href="/nlab/show/dg-algebras">dg-algebras</a> to <span class="newWikiWord">cosimplicial algebras<a href="/nlab/new/cosimplicial+algebras">?</a></span> (the inverse equivalence to the <a class="existingWikiWord" href="/nlab/show/normalized+cochain+complex">normalized cochain complex</a> functor)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>D</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>DGdg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mo stretchy="false">(</mo><mi>dg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover><msup><mo stretchy="false">)</mo> <mi>Δ</mi></msup></mrow><annotation encoding="application/x-tex"> D \;\colon\; DGdg\hat \mathcal{C} \to (dg\hat \mathcal{C})^{\Delta} </annotation></semantics></math></div> <p>induces on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>DGdg</mi><mover><mi>𝒞</mi><mo stretchy="false">^</mo></mover></mrow><annotation encoding="application/x-tex">DGdg\hat \mathcal{C}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred model structure</a> from that of prop. <a class="maruku-ref" href="#ModelStructureOnCosimplicialDgAlgebras"></a> and is the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> of a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> with respect to these model structures</p> </div> <p>This is (<a href="#Pridham">Pridham, theorem 4.26</a>).</p> <h2 id="properties">Properties</h2> <h3 id="general">General</h3> <p>We discuss some further properties of the <a href="DefinitionsAndQuillenEquivalences">above</a> model category structures.</p> <div class="num_prop" id="RightProperness"> <h6 id="proposition_16">Proposition</h6> <p>The model category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dgFormalSpace</mi></mrow><annotation encoding="application/x-tex">dgFormalSpace</annotation></semantics></math>, def. <a class="maruku-ref" href="#dgFormalSpace"></a>, is a <a class="existingWikiWord" href="/nlab/show/right+proper+model+category">right proper model category</a>.</p> </div> <p>This observation has been communicated privately by <a class="existingWikiWord" href="/nlab/show/Jonathan+Pridham">Jonathan Pridham</a></p> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>We need to show that the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of a weak equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> along a fibration <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 again a weak equivalence. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> is a fibration, this is automatic, so by factorisation we reduce to the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> is a cofibration. Now, every trivial cofibration is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi></mrow><annotation encoding="application/x-tex">Spf</annotation></semantics></math> of a composition of acyclic small extensions, so we may take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> to be <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi></mrow><annotation encoding="application/x-tex">Spf</annotation></semantics></math> of an acyclic small extension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A \to B</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/kernel">kernel</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi></mrow><annotation encoding="application/x-tex">Spf</annotation></semantics></math> of a quasi-free map <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">A \to R</annotation></semantics></math>, so the pullback is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spf</mi></mrow><annotation encoding="application/x-tex">Spf</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>R</mi><mo stretchy="false">/</mo><mi>IR</mi></mrow><annotation encoding="application/x-tex">R \to R/IR</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>IR</mi><mo>=</mo><mi>I</mi><msub><mover><mo>⊗</mo><mo stretchy="false">^</mo></mover> <mi>A</mi></msub><mi>R</mi></mrow><annotation encoding="application/x-tex">IR =I\hat{\otimes}_A R</annotation></semantics></math>, so <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>R</mi><mo stretchy="false">/</mo><mi>IR</mi></mrow><annotation encoding="application/x-tex">R \to R/IR</annotation></semantics></math> is also an acyclic small extension.</p> </div> <h3 id="HomotopiesAndDerivedHomSpaces">Homotopies and derived hom spaces</h3> <p>In any <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> we have a notion of <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> between <a class="existingWikiWord" href="/nlab/show/1-morphisms">1-morphisms</a>. In any <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> we still have a notion of <a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a>, given by maps into a <a class="existingWikiWord" href="/nlab/show/path+space+object">path space object</a>. So all of the above model category/fibrant object category structures yield models for <a class="existingWikiWord" href="/nlab/show/homotopies">homotopies</a> between morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras.</p> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a> of and hence of <a class="existingWikiWord" href="/nlab/show/right+homotopies">right homotopies</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras (in the category of def. <a class="maruku-ref" href="#LInfinityAlgebraIsQuasiFreeDgCoalgebra"></a>) is for instance in (<a href="#Dolgushev07">Dolgushev 07, section 5</a>).</p> <p>More generally, a description of the full <a class="existingWikiWord" href="/nlab/show/derived+hom+space">derived hom space</a> between two <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras is obtained via remark <a class="maruku-ref" href="#CategoryOfFibrantObjectsLInfinity"></a> from the description of <a href="category+of+fibrant+objects#DerivedHomSpaces">derived hom-spaces in categories of fibrant objects</a>.</p> <h3 id="HomotopyFiberProducts">Homotopy fiber products</h3> <p>Recognizing <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a> in any of the model structure above can be a bit subtle. A recognition principle of <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> over abelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras , hence useful for discussion of <a href="infinity-Lie+algebra+cohomology#Extensions">∞-Lie algebra extensions</a>), is described in (<a href="#FiorenzaRogersSchreiber13">Fiorenza-Rogers-Schreiber 13, theorem 3.1.13</a>).</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+L-%E2%88%9E+algebras+and+dg-Lie+algebras">relation between L-∞ algebras and dg-Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">model structure on dg-coalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">model structure on dg-Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">model structure on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+operad">L-∞ operad</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/deformation+context">deformation context</a></p> </li> </ul> </li> </ul> <h2 id="references">References</h2> <p>Precursors for 2-reduced dg-algebras are dicussed in</p> <ul> <li id="Quillen69"><a class="existingWikiWord" href="/nlab/show/Dan+Quillen">Dan Quillen</a>, <em>Rational homotopy theory</em>, The Annals of Mathematics, Second Series, Vol. 90, No. 2 (Sep., 1969), pp. 205-295 (<a href="http://www.jstor.org/stable/1970725">jstor:1970725</a>, <a href="http://www.math.northwestern.edu/~konter/gtrs/rational.pdf">pdf</a>)</li> </ul> <p>The homotopy-theoretic nature of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras and their relation to deformation problems was then notably amplified in</p> <ul> <li id="Kontsevich94"><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <em>Topics in algebra — deformation theory</em> Lecture Notes (1994) (<a href="http://www.math.brown.edu/&#8764;abrmovic/kontsdef.ps">ps</a>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/structures">structures</a> on <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebras over operads</a> in <a class="existingWikiWord" href="/nlab/show/chain+complexes">chain complexes</a>:</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>The full model structure on <a class="existingWikiWord" href="/nlab/show/dg-coalgebras">dg-coalgebras</a> (in characteristic 0) as a <a class="existingWikiWord" href="/nlab/show/model+structure+for+L-infinity+algebras">model structure for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a> and the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between <a class="existingWikiWord" href="/nlab/show/dg-Lie+algebras">dg-Lie algebras</a> as well as the interpretation in terms of formal <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 (<a class="existingWikiWord" href="/nlab/show/L-infinity+algebras"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <msub><mi>L</mi> <mn>∞</mn></msub> </mrow> <annotation encoding="application/x-tex">L_\infty</annotation> </semantics> </math>-algebras</a>):</p> <ul> <li id="Hinich98"><a class="existingWikiWord" href="/nlab/show/Vladimir+Hinich">Vladimir Hinich</a>, <em>DG coalgebras as formal stacks</em>, Journal of Pure and Applied Algebra <strong>162</strong> 2 (2001) 209-250 [<a href="http://arxiv.org/abs/math/9812034">arXiv:9812034</a>, <a href="https://doi.org/10.1016/S0022-4049(00)00121-3">doi:10.1016/S0022-4049(00)00121-3</a>]</li> </ul> <p>Also:</p> <ul> <li id="Valette14">Theorem 2.1 in: <a class="existingWikiWord" href="/nlab/show/Bruno+Vallette">Bruno Vallette</a>, <em>Homotopy theory of homotopy algebras</em>, Annales de l’Institut Fourier <strong>70</strong>:2 (2020) 683–738 <a href="https://10.5802/aif.3322">doi</a> <a href="https://zbmath.org/?q=an:1335.18001">Zbl:1335.18001</a> <a href="https://arxiv.org/abs/1411.5533">arXiv:1411.5533</a></li> </ul> <p>The structure of a <a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a> on connective <a class="existingWikiWord" href="/nlab/show/finite+type">finite type</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras:</p> <ul> <li id="Rogers18"><a class="existingWikiWord" href="/nlab/show/Christopher+L.+Rogers">Christopher L. Rogers</a>, <em>An explicit model for the homotopy theory of finite type Lie <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>-algebras</em>, Algebr. Geom. Topol. 20 (2020) 1371-1429 (<a href="https://arxiv.org/abs/1809.05999">arXiv:1809.05999</a>, <a href="https://doi.org/10.2140/agt.2020.20.1371">doi:10.2140/agt.2020.20.1371</a>)</li> </ul> <p>(For relation to <a href="#Valette14">Valette 14</a> see <a href="#Rogers18">Rogers 18, below Theorem 5.9</a>)</p> <p>In</p> <ul> <li id="Lurie"><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Formal+Moduli+Problems">Formal Moduli Problems</a></em></li> </ul> <p>the relation to <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 is discussed more in detail.</p> <p>More model category theoretic developments relating various of the previous approaches and generalizing to arbitrary characteristic are in</p> <ul> <li id="Pridham"><a class="existingWikiWord" href="/nlab/show/Jonathan+Pridham">Jonathan Pridham</a>, <em>Unifying derived deformation theories</em>, Adv. Math. 224 (2010), no.3, 772-826 (<a href="http://arxiv.org/abs/0705.0344">arXiv:0705.0344</a>)</li> </ul> <p>in parts based on</p> <ul> <li id="Manetti02"><a class="existingWikiWord" href="/nlab/show/Marco+Manetti">Marco Manetti</a>, <em>Extended deformation functors</em>, Int. Math. Res. Not., (14):719–756, 2002 (<a href="https://arxiv.org/abs/math/9910071">arXiv:math/9910071</a>)</li> </ul> <p>A useful summary of that paper is given in the <a href="http://poisson.phc.unipi.it/~maggiolo/wp-content/uploads/2008/12/WDTII_Pridham.pdf">notes</a>, by Stefano Maggiolo.</p> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/path+space+objects">path space objects</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras is in section 5 of</p> <ul> <li id="Dolgushev07">Vasiliy A. Dolgushev, <em>Erratum to: “A Proof of Tsygan’s Formality Conjecture for an Arbitrary Smooth Manifold”</em> (<a href="http://arxiv.org/abs/math/0703113">arXiv:math/0703113</a>)</li> </ul> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a> of morphisms to abelian <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-algebras (and hence <a href="infinity-Lie+algebra+cohomology#Extensions">∞-Lie algebra extensions</a>) is in section 3.1 of</p> <ul> <li id="FiorenzaRogersSchreiber13"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em><a class="existingWikiWord" href="/schreiber/show/L-%E2%88%9E+algebras+of+local+observables+from+higher+prequantum+bundles">L-∞ algebras of local observables from higher prequantum bundles</a></em> (<a href="http://arxiv.org/abs/1304.6292">arXiv:1304.6292</a>)</li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alexander+Berglund">Alexander Berglund</a>, <em>Rational homotopy theory of mapping spaces via Lie theory for L-infinity algebras</em>, Homology, Homotopy and Applications Volume 17 (2015) Number 2 (<a href="https://arxiv.org/abs/1110.6145">arXiv:1110.6145</a>, <a href="https://dx.doi.org/10.4310/HHA.2015.v17.n2.a16">doi:10.4310/HHA.2015.v17.n2.a16</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 16, 2024 at 12:11:42. 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