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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"> <p class="show_diff"> Showing changes from revision #69 to #70: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <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'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></strong>, <a class='existingWikiWord' href='/nlab/show/diff/model+%28%E2%88%9E%2C1%29-category'>model $\infty$-category</a></p> <p><strong>Definitions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/relative+category'>relative category</a>, <a class='existingWikiWord' href='/nlab/show/diff/homotopical+category'>homotopical category</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibration</a>, <a class='existingWikiWord' href='/nlab/show/diff/cofibration'>cofibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/weak+factorization+system'>weak factorization system</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/resolution'>resolution</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cell+complex'>cell complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+%28as+an+operation%29'>homotopy</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a><math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mspace width='thickmathspace' /></mrow><annotation encoding='application/x-tex'>\;</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Quillen+adjunction'>Quillen adjunction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Quillen+bifunctor'>Quillen bifunctor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/homotopy+Kan+extension'>homotopy Kan extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a>/<a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimit</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+weighted+colimit'>homotopy weighted (co)limit</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coend'>homotopy (co)end</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bousfield-Kan+map'>Bousfield-Kan map</a></p> </li> </ul> <p><strong>Refinements</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+model+category'>monoidal model category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/monoidal+Quillen+adjunction'>monoidal Quillen adjunction</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+model+category'>enriched model category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/enriched+Quillen+adjunction'>enriched Quillen adjunction</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+monoidal+model+category'>monoidal enriched model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+Quillen+adjunction'>simplicial Quillen adjunction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+monoidal+model+category'>simplicial monoidal model category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cofibrantly+generated+model+category'>cofibrantly generated model category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+model+category'>combinatorial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cellular+model+category'>cellular model category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+model+category'>algebraic model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compactly+generated+model+category'>compactly generated model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>proper model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+model+category'>cartesian closed model category</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+cartesian+closed+model+category'>locally cartesian closed model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+functors'>on functor categories (global)</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Reedy+model+structure'>Reedy model structure</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/slice+model+structure'>on slice categories</a></p> </li> <li> <p><span class='newWikiWord'>Bousfield localization<a href='/nlab/new/Bousfield+localization+of+model+categories'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/transferred+model+structure'>transferred model structure</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebraic+fibrant+objects'>on algebraic fibrant objects</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+construction+for+model+categories'>Grothendieck construction for model categories</a></p> </li> </ul> <p><strong>Presentation of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_2' xmlns='http://www.w3.org/1998/Math/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/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+localization'>simplicial localization</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>(∞,1)-categorical hom-space</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable (∞,1)-category</a></p> </li> </ul> <p><strong>Model structures</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Cisinski+model+structure'>Cisinski model structure</a></li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_3' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+for+infinity-groupoids'>for ∞-groupoids</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+topological+spaces'>on topological spaces</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+topological+spaces'>classical model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+compactly+generated+topological+spaces'>on compactly generated spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+Delta-generated+topological+spaces'>on Delta-generated spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+diffeological+spaces'>on diffeological spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Str%C3%B8m+model+structure'>Strøm model structure</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Thomason+model+structure'>Thomason model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+sets'>on simplicial sets</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+semi-simplicial+sets'>on semi-simplicial sets</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/constructive+model+structure+on+simplicial+sets'>constructive model structure</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+left+fibrations'>for right/left fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+groupoids'>model structure on simplicial groupoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cubical+sets'>on cubical sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+strict+omega-groupoids'>on strict ∞-groupoids</a>, <a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure+on+groupoids'>on groupoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+chain+complexes'>on chain complexes</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+abelian+groups'>model structure on cosimplicial abelian groups</a></p> <p>related by the <a class='existingWikiWord' href='/nlab/show/diff/Dold-Kan+correspondence'>Dold-Kan correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+simplicial+sets'>model structure on cosimplicial simplicial sets</a></p> </li> </ul> <p><em>for equivariant <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_4' xmlns='http://www.w3.org/1998/Math/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/diff/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/diff/Borel+model+structure'>coarse model structure on topological G-spaces</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/Borel+model+structure'>Borel model structure</a>)</p> </li> </ul> <p><em>for rational <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_5' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-algebras'>model structure on dgc-algebras</a></li> </ul> <p><em>for rational equivariant <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_6' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+equivariant+chain+complexes'>model structure on equivariant chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+equivariant+dgc-algebras'>model structure on equivariant dgc-algebras</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_7' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+for+homotopy+n-types'>for n-groupoids</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+homotopy+n-types'>for n-types</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure+on+groupoids'>for 1-groupoids</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_8' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+simplicial+groups'>model structure on simplicial groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+reduced+simplicial+sets'>model structure on reduced simplicial sets</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebras</em></p> <p><em>general <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_10' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+monoids+in+a+monoidal+model+category'>on monoids</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>on simplicial T-algebras</a>, on <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a>s</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+a+monad'>on algebas over a monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/diff/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 class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_11' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-algebras'>model structure on differential-graded commutative algebras</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/model+structure+on+dg-algebras'>on dg-algebras</a> and on <a class='existingWikiWord' href='/nlab/show/diff/simplicial+ring'>on simplicial rings</a>/<a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+rings'>on cosimplicial rings</a></p> <p>related by the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+L-infinity+algebras'>for L-∞ algebras</a>: <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-Lie+algebras'>on dg-Lie algebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-coalgebras'>on dg-coalgebras</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+Lie+algebras'>on simplicial Lie algebras</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+spectra'>model structure on spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+ring+spectra'>model structure on ring spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+parameterized+spectra'>model structure on parameterized spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+spectra'>model structure on presheaves of spectra</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_12' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+relative+categories'>on categories with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>Joyal model for quasi-categories</a> (and its <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+cubical+quasicategories'>cubical version</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-categories'>on sSet-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+complete+Segal+spaces'>for complete Segal spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>for Cartesian fibrations</a></p> </li> </ul> <p><em>for stable <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_13' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+dg-categories'>on dg-categories</a></li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_14' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+operads'>on operads</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Segal+operads'>for Segal operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+algebras+over+an+operad'>on algebras over an operad</a>, <a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+dendroidal+sets'>on dendroidal sets</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+dendroidal+complete+Segal+spaces'>for dendroidal complete Segal spaces</a>, <a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+dendroidal+Cartesian+fibrations'>for dendroidal Cartesian fibrations</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_15' xmlns='http://www.w3.org/1998/Math/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/diff/Theta-space'>for (n,r)-categories as ∞-spaces</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+weak+complicial+sets'>for weak ∞-categories as weak complicial sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cellular+sets'>on cellular sets</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/canonical+model+structure'>on higher categories in general</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+strict+omega-categories'>on strict ∞-categories</a></p> </li> </ul> <p><em>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_16' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_17' xmlns='http://www.w3.org/1998/Math/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/diff/model+structure+on+homotopical+presheaves'>on homotopical presheaves</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+presheaves'>on simplicial presheaves</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/global+model+structure+on+simplicial+presheaves'>global model structure</a>/<a class='existingWikiWord' href='/nlab/show/diff/%C4%8Cech+model+structure+on+simplicial+presheaves'>Cech model structure</a>/<a class='existingWikiWord' href='/nlab/show/diff/local+model+structure+on+simplicial+presheaves'>local model structure</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sheaves'>on simplicial sheaves</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+presheaves+of+simplicial+groupoids'>on presheaves of simplicial groupoids</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+sSet-enriched+presheaves'>on sSet-enriched presheaves</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+%282%2C1%29-sheaves'>model structure for (2,1)-sheaves</a>/for stacks</p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#general'>General</a></li><li><a href='#on_connective_dgcalgebras'>On connective dgc-algebras</a><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#SullivanAlgebras'>Cofibrations and Sullivan algebras</a></li><li><a href='#HomComplexes'>Simplicial hom-complexes</a></li><ins class='diffins'><li><a href='#relation_to_simplicial_sets'>Relation to simplicial sets</a></li></ins><li><a href='#relation_to_cosimplicial_commutative_algbras'>Relation to cosimplicial commutative algbras</a></li><li><a href='#CommVsNoncomm'>Commutative vs. non-commutative dg-algebras</a></li></ul></li></ul></li><li><a href='#Unbounded'>Unbounded dg-algebras</a><ul><li><a href='#GradingsAndConventions'>Gradings and conventions</a></li><li><a href='#definition_8'>Definition</a></li><li><a href='#properties_2'>Properties</a><ul><li><a href='#properness'>Properness</a></li><li><a href='#derived_tensor_product'>Derived tensor product</a></li><li><a href='#SimplicialHomObjects'>Derived hom-functor</a></li><li><a href='#DerivedCopowering'>Derived copowering over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math></a></li><li><a href='#DerivedPowering'>Derived powering over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math></a></li><li><a href='#PathObjectsForUnboundedCommutative'>Path objects</a></li><li><a href='#RelationToAInfinityAlgebras'>Relation to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>H \mathbb{Z}</annotation></semantics></math>-algebra spectra</a></li><li><a href='#RelationToEInfinityAlgebras'>Relation to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝔼</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{E}_\infty</annotation></semantics></math>-algebras</a></li></ul></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#on_connective_dgcalgebras_2'>On connective dgc-algebras</a></li><li><a href='#on_noncommutative_dgalgebras'>On non-commutative dg-algebras</a></li><li><a href='#on_unbounded_dgalgebras'>On unbounded dg-algebras</a></li><li><a href='#more'>More</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>A <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure on a category of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>differential graded algebras</a> or more specifically on a <a class='existingWikiWord' href='/nlab/show/diff/dgcAlg'>category of differential graded-commutative algebras</a> tends to present an <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category'>(∞,1)-category</a> of <a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebras</a>.</p> <p>For dg-algebras bounded in negative or positive degrees, the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> asserts that their model category structures are <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalent</a> to the corresponding <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>model structure on (co)simplicial algebras</a>. This case plays a central role in <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a>.</p> <p>The case of model structures on unbounded dg-algebras may be thought of as induced from this by passage to the <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a> modeled on formal duals of the bounded dg-algebras. This is described at <a class='existingWikiWord' href='/nlab/show/diff/dg-geometry'>dg-geometry</a>.</p> <h2 id='general'>General</h2> <p>The category of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>s is that of <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a>s in a <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a>. Accordingly general results on a <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+monoids+in+a+monoidal+model+category'>model structure on monoids in a monoidal model category</a> apply.</p> <p>Below we spell out special cases, such as restricting to <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoids</a> when working over a <a class='existingWikiWord' href='/nlab/show/diff/ground+ring'>ground field</a> of <a class='existingWikiWord' href='/nlab/show/diff/characteristic+zero'>characteristic zero</a>, or restricting to non-negatively graded cochain dg-algebras.</p> <h2 id='on_connective_dgcalgebras'>On connective dgc-algebras</h2> <p>We discuss the projective model structure on <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential non-negatively graded-commutative algebras</a>. This was originally introduced in <a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76</a> as a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> for <a class='existingWikiWord' href='/nlab/show/diff/Dennis+Sullivan'>Dennis Sullivan</a>’s approach to <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a>.</p> <h3 id='definition'>Definition</h3> <div class='num_defn' id='dgcCochainAlgebrasInNonNegativeDegrees'> <h6 id='definition_2'>Definition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_22' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/characteristic+zero'>characteristic zero</a>, write</p> <div class='maruku-equation' id='eq:CategoryOfdgcAlgebras'><span class='maruku-eq-number'>(1)</span><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><mspace width='thickmathspace' /><mo>∈</mo><mspace width='thickmathspace' /><mi>Categories</mi></mrow><annotation encoding='application/x-tex'> dgcAlg^{\geq 0}_{k} \;\in\; Categories </annotation></semantics></math></div> <p>for the <a class='existingWikiWord' href='/nlab/show/diff/category'>category</a> of <a class='existingWikiWord' href='/nlab/show/diff/associative+unital+algebra'>unital</a> <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative algebras</a> over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_24' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> in non-negative degrees, equivalently the category of <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoids</a> in the <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_25' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>Ch</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch^{\geq 0}(k)</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/diff/cochain+complex'>cochain complexes</a> in non-negative degrees, equipped with the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+chain+complexes'>tensor product of chain complexes</a>.</p> </div> <p>(<a href='#GelfandManin96'>Gelfand-Manin 96, V.3.1</a>)</p> <div class='num_example'> <h6 id='example'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial</a> and <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a>)</strong></p> <p>In <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>dgcAlg^{\geq 0}_{k}</annotation></semantics></math> <a class='maruku-eqref' href='#eq:CategoryOfdgcAlgebras'>(1)</a>:</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> is the ground field algebra <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_27' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>;</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/terminal+object'>terminal object</a> is the zero algebra <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>0</mn></mrow><annotation encoding='application/x-tex'>0</annotation></semantics></math> (which is indeed a unital algebra).</p> </li> </ol> </div> <p>(Beware that this is incorrectly stated in <a href='#GelfandManin96'>Gelfand-Manin 96, p. 335</a>)</p> <p>More generally:</p> <div class='num_example'> <h6 id='example_2'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproducts</a> and <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>products</a>)</strong></p> <p>In <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>dgcAlg^{\geq 0}_{k}</annotation></semantics></math> <a class='maruku-eqref' href='#eq:CategoryOfdgcAlgebras'>(1)</a>:</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> is given by the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+algebras'>tensor product of algebras</a>;</p> <p>(see at <em><a href='category+of+monoids#PushoutOfCommutativeMonoids'>pushouts of commutative monoids</a></em>)</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a> is given by <a class='existingWikiWord' href='/nlab/show/diff/direct+sum'>direct sum</a> on underlying <a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded vector spaces</a></p> <p>(since the <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> is a <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a>).</p> </li> </ol> </div> <div class='num_defn' id='dgcCochainAlgebraInNonNegDegreeOfFiniteType'> <h6 id='definition_3'>Definition</h6> <p><strong>(finite type)</strong></p> <p>Say that a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebra</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>A \in dgcAlg^{\geq 0}_k</annotation></semantics></math> (def. <a class='maruku-ref' href='#dgcCochainAlgebrasInNonNegativeDegrees'>1</a>) is of <em><a class='existingWikiWord' href='/nlab/show/diff/finite+type'>finite type</a></em> if its <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>underlying</a> <a class='existingWikiWord' href='/nlab/show/diff/chain+complex'>chain complex</a> is in each degree of <a class='existingWikiWord' href='/nlab/show/diff/finite+number'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/dimension'>dimension</a> as a <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/vector+space'>vector space</a>.</p> </div> <div class='num_defn' id='ProjectiveModelStructureOnCdgAlg'> <h6 id='definition_4'>Definition</h6> <p>Write <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> for the catgory of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebras</a> from def. <a class='maruku-ref' href='#dgcCochainAlgebrasInNonNegativeDegrees'>1</a> equipped with the following <a class='existingWikiWord' href='/nlab/show/diff/class'>classes</a> of morphisms:</p> <ul> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/weak+equivalence'>weak equivalences</a></em> are the <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a>;</p> </li> <li> <p><em><a class='existingWikiWord' href='/nlab/show/diff/fibration'>fibrations</a></em> are the degreewise <a class='existingWikiWord' href='/nlab/show/diff/surjection'>surjections</a>;</p> </li> </ul> </div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, Def. 4.2</a>, <a href='#GelfandManin96'>Gelfand-Manin 96, Def. V.3.3</a>)</p> <div class='num_prop' id='IndeedProjectiveModelStructureOnCdgAlg'> <h6 id='proposition'>Proposition</h6> <p>The category <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from def. <a class='maruku-ref' href='#ProjectiveModelStructureOnCdgAlg'>3</a> is a <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a>, to be called the <em>projective model structure</em>.</p> </div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, Theorem 4.3</a>, <a href='#GelfandManin96'>Gelfand-Manin 96, Theorem V.3.4</a>)</p> <div class='num_remark'> <h6 id='remark'>Remark</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>)</strong></p> <p>Evidently every <a class='existingWikiWord' href='/nlab/show/diff/object'>object</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> (Def. <a class='maruku-ref' href='#ProjectiveModelStructureOnCdgAlg'>3</a>, prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a>) is a <a class='existingWikiWord' href='/nlab/show/diff/fibrant+object'>fibrant object</a>. Therefore these model categories structures are in particular also structures of a <a class='existingWikiWord' href='/nlab/show/diff/category+of+fibrant+objects'>category of fibrant objects</a>.</p> </div> <p>The nature of the cofibrations is discussed <a href='#SullivanAlgebras'>below</a>.</p> <h3 id='properties'>Properties</h3> <h4 id='SullivanAlgebras'>Cofibrations and Sullivan algebras</h4> <div class='num_defn'> <h6 id='definition_5'>Definition</h6> <p><strong>(sphere and disk algebras)</strong></p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k[n]</annotation></semantics></math> for the graded vector space which is the ground field <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> in degree <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> and 0 in all other degrees. For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{N}</annotation></semantics></math>, consider the <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dgc-algebras</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>≔</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> S(n) \coloneqq (\wedge^\bullet k[n], 0) </annotation></semantics></math></div> <p>and for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding='application/x-tex'>n \geq 1</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dgc-algebras</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>≔</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy='false'>(</mo><mi>n</mi><mo>=</mo><mn>0</mn><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>⊕</mo><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mtd> <mtd><mo stretchy='false'>(</mo><mi>n</mi><mo>></mo><mn>0</mn><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow></mrow></mrow><annotation encoding='application/x-tex'> D(n) \coloneqq \left\lbrace \array{ 0 & (n = 0) \\ (\wedge^\bullet (k[n] \oplus k[n-1]), 0) & (n \gt 0) } \right. </annotation></semantics></math></div> <p>for which the differential sends the generator of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k[n-1]</annotation></semantics></math> to that of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k[n]</annotation></semantics></math></p> <p>Write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>i</mi> <mi>n</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>S</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> i_n \colon S(n) \to D(n) </annotation></semantics></math></div> <p>for the obvious morphism that takes the generator in degree <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> to the generator in degree <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math> (and for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n = 0</annotation></semantics></math> it is the unique morphism from the <a class='existingWikiWord' href='/nlab/show/diff/initial+object'>initial object</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(0,0)</annotation></semantics></math>).</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>></mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>n \gt 0</annotation></semantics></math> write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>j</mi> <mi>n</mi></msub><mo lspace='verythinmathspace'>:</mo><mi>k</mi><mo stretchy='false'>[</mo><mn>0</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>D</mi><mo stretchy='false'>(</mo><mi>n</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> j_n \colon k[0] \to D(n) \,. </annotation></semantics></math></div></div> <div class='num_prop'> <h6 id='proposition_2'>Proposition</h6> <p><strong>(generating cofibrations)</strong></p> <p>The sets</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>=</mo><mo stretchy='false'>{</mo><msub><mi>i</mi> <mi>n</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow></msub><mo>∪</mo><mo stretchy='false'>{</mo><mi>k</mi><mo stretchy='false'>[</mo><mn>0</mn><mo stretchy='false'>]</mo><mo>→</mo><mi>S</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>,</mo><mi>S</mi><mo stretchy='false'>(</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>→</mo><mi>k</mi><mo stretchy='false'>[</mo><mn>0</mn><mo stretchy='false'>]</mo><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'> I = \{i_n \}_{n \geq 1} \cup \{k[0] \to S(0), S(0) \to k[0]\} </annotation></semantics></math></div> <p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi><mo>=</mo><mo stretchy='false'>{</mo><msub><mi>j</mi> <mi>n</mi></msub><msub><mo stretchy='false'>}</mo> <mrow><mi>n</mi><mo>></mo><mn>1</mn></mrow></msub></mrow><annotation encoding='application/x-tex'> J = \{j_n \}_{n \gt 1} </annotation></semantics></math></div> <p>are sets of generating cofibrations and acyclic cofibrations, respectively, exhibiting the model category <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a> as a <a class='existingWikiWord' href='/nlab/show/diff/cofibrantly+generated+model+category'>cofibrantly generated model category</a>.</p> </div> <p>review includes (<a href='#Hess06'>Hess 06, p. 6</a>)</p> <p>In this section we describe the cofibrations in the model structure on <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcalg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcalg^{\geq 0}_k)_{proj}</annotation></semantics></math> (def. <a class='maruku-ref' href='#ProjectiveModelStructureOnCdgAlg'>3</a>, prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a>). Notice that it is these that are in the image of the dual <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a>.</p> <p>Before we characterize the cofibrations, first some notation.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> a <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/graded+vector+space'>graded vector space</a> write <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi></mrow><annotation encoding='application/x-tex'>\wedge^\bullet V</annotation></semantics></math> for the <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>Grassmann algebra</a> over it. Equipped with the trivial differential <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding='application/x-tex'>d = 0</annotation></semantics></math> this is a <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dga</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>=</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\wedge^\bullet V, d=0)</annotation></semantics></math>.</p> <p>With <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> our ground field we write <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k,0)</annotation></semantics></math> for the corresponding dg-algebra, the tensor unit for the standard <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal structure</a> on <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>dgAlg</mi></mrow><annotation encoding='application/x-tex'>dgAlg</annotation></semantics></math>. This is the <a class='existingWikiWord' href='/nlab/show/diff/exterior+algebra'>Grassmann algebra</a> on the 0-vector space <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mn>0</mn><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k,0) = (\wedge^\bullet 0, 0)</annotation></semantics></math>.</p> <div class='num_defn'> <h6 id='definition_6'>Definition</h6> <p><strong>(Sullivan algebras)</strong></p> <p>A <strong><a class='existingWikiWord' href='/nlab/show/diff/Sullivan+model'>relative Sullivan algebra</a></strong> is a <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphism</a> of dg-algebras that is an inclusion</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (A,d) \to (A \otimes_k \wedge^\bullet V, d') </annotation></semantics></math></div> <p>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(A,d)</annotation></semantics></math> some dg-algebra and for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> some graded vector space, such that</p> <ul> <li> <p>there is a <a class='existingWikiWord' href='/nlab/show/diff/well-order'>well ordered set</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>J</mi></mrow><annotation encoding='application/x-tex'>J</annotation></semantics></math></p> </li> <li> <p>indexing a basis <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>{</mo><msub><mi>v</mi> <mi>α</mi></msub><mo>∈</mo><mi>V</mi><mo stretchy='false'>|</mo><mi>α</mi><mo>∈</mo><mi>J</mi><mo stretchy='false'>}</mo></mrow><annotation encoding='application/x-tex'>\{v_\alpha \in V| \alpha \in J\}</annotation></semantics></math> of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>;</p> </li> <li> <p>such that with <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mo>=</mo><mi>span</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mi>α</mi></msub><mo stretchy='false'>|</mo><mi>α</mi><mo><</mo><mi>β</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>V_{\lt \beta} = span(v_\alpha | \alpha \lt \beta)</annotation></semantics></math> for all basis elements <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>v</mi> <mi>β</mi></msub></mrow><annotation encoding='application/x-tex'>v_\beta</annotation></semantics></math> we have that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>d</mi><mo>′</mo><msub><mi>v</mi> <mi>β</mi></msub><mo>∈</mo><mi>A</mi><mo>⊗</mo><msup><mo>∧</mo> <mo>•</mo></msup><msub><mi>V</mi> <mrow><mo><</mo><mi>β</mi></mrow></msub><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> d' v_\beta \in A \otimes \wedge^\bullet V_{\lt \beta} \,. </annotation></semantics></math></div></li> </ul> <p>This is called a <strong>minimal</strong> <a class='existingWikiWord' href='/nlab/show/diff/Sullivan+model'>relative Sullivan algebra</a> if in addition the condition</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>α</mi><mo><</mo><mi>β</mi><mo stretchy='false'>)</mo><mo>⇒</mo><mo stretchy='false'>(</mo><mi>deg</mi><msub><mi>v</mi> <mi>α</mi></msub><mo>≤</mo><mi>deg</mi><msub><mi>v</mi> <mi>β</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (\alpha \lt \beta) \Rightarrow (deg v_\alpha \leq deg v_\beta) </annotation></semantics></math></div> <p>holds. For a Sullivan algebra <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> relative to the tensor unit we call the <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dga</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\wedge^\bullet V,d)</annotation></semantics></math> simply a <strong>Sullivan algebra</strong>. And a <strong>minimal Sullivan algebra</strong> if <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>k</mi><mo>,</mo><mn>0</mn><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(k,0) \to (\wedge^\bullet V, d)</annotation></semantics></math> is a minimal relative Sullivan algebra.</p> </div> <div class='num_remark'> <h6 id='remark_2'>Remark</h6> <p>Sullivan algebras were introduced by <a class='existingWikiWord' href='/nlab/show/diff/Dennis+Sullivan'>Dennis Sullivan</a> in his development of <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a>. This is one of the key application areas of the model structure on dg-algebras.</p> </div> <div class='num_remark'> <h6 id='remark_3'>Remark</h6> <p><strong>(<math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>L_\infty</annotation></semantics></math>-algebras)</strong></p> <p>Because they are <a class='existingWikiWord' href='/nlab/show/diff/semifree+dga'>semifree dgas</a>, Sullivan dg-algebras <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\wedge^\bullet V,d)</annotation></semantics></math> are (at least for degreewise finite dimensional <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math>) <a class='existingWikiWord' href='/nlab/show/diff/Chevalley-Eilenberg+algebra'>Chevalley-Eilenberg algebra</a>s of <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞-algebra</a>s.</p> <p>The co-commutative differential co-algebra encoding the corresponding <a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞-algebra</a> is the free cocommutative algebra <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mo>∨</mo> <mo>•</mo></msup><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>\vee^\bullet V^*</annotation></semantics></math> on the degreewise dual of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>V</mi></mrow><annotation encoding='application/x-tex'>V</annotation></semantics></math> with differential <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>D</mi><mo>=</mo><msup><mi>d</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>D = d^*</annotation></semantics></math>, i.e. the one given by the formula</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo stretchy='false'>(</mo><mi>D</mi><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>∨</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>∨</mo><mi>⋯</mi><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>=</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>(</mo><mi>d</mi><mi>ω</mi><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><mi>⋯</mi><mo>,</mo><msub><mi>v</mi> <mi>n</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \omega(D(v_1 \vee v_2 \vee \cdots v_n)) = - (d \omega) (v_1, v_2, \cdots, v_n) </annotation></semantics></math></div> <p>for all <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ω</mi><mo>∈</mo><mi>V</mi></mrow><annotation encoding='application/x-tex'>\omega \in V</annotation></semantics></math> and all <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>v_i \in V^*</annotation></semantics></math>.</p> </div> <div class='num_prop'> <h6 id='proposition_3'>Proposition</h6> <p><strong>(cofibrations are relative Sullivan algebras)</strong></p> <p>The cofibrations in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_{k})_{proj}</annotation></semantics></math> are precisely the <a class='existingWikiWord' href='/nlab/show/diff/retract'>retracts</a> of <a class='existingWikiWord' href='/nlab/show/diff/Sullivan+model'>relative Sullivan algebras</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo><mo>→</mo><mo stretchy='false'>(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo>′</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(A,d) \to (A\otimes_k \wedge^\bullet V, d')</annotation></semantics></math>.</p> <p>Accordingly, the cofibrant objects in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_{k})_{proj}</annotation></semantics></math> are precisely the Sullivan algebras <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mo>∧</mo> <mo>•</mo></msup><mi>V</mi><mo>,</mo><mi>d</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\wedge^\bullet V, d)</annotation></semantics></math></p> </div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, Prop. 7.11</a><a href='#GelfandManin96'>Gelfand-Manin 96., Prop. V.5.4</a>)</p> <h4 id='HomComplexes'>Simplicial hom-complexes</h4> <p>We discuss <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial</a> <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping spaces</a> between <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebras</a>. These <em>almost</em> make the projective model structure <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> from prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a> into a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a>, except that the <a class='existingWikiWord' href='/nlab/show/diff/copower'>tensoring</a>/<a class='existingWikiWord' href='/nlab/show/diff/powering'>powering</a> <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> holds only for <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> or else on <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebras</a> of <a class='existingWikiWord' href='/nlab/show/diff/finite+type'>finite type</a>. Still, this has useful implications, for instance it implies that the <a class='existingWikiWord' href='/nlab/show/diff/reduced+suspension'>reduced suspension</a> and <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a> <a class='existingWikiWord' href='/nlab/show/diff/adjunction'>adjunction</a> on [augmented algebras|augmented]] <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebras</a> is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a>.</p> <div class='num_defn' id='MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'> <h6 id='definition_7'>Definition</h6> <p><strong>(simplicial mapping spaces)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>A,B \in dgcAlg^{\geq 0}_k</annotation></semantics></math> (def. <a class='maruku-ref' href='#dgcCochainAlgebrasInNonNegativeDegrees'>1</a>), let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_92' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'> Maps(A,B) \in sSet </annotation></semantics></math></div> <p>be the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> whose <a class='existingWikiWord' href='/nlab/show/diff/simplex'>n-simplices</a> are the dg-algebra <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphisms</a> from <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_93' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> into the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_94' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> with the de Rham complex of <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+on+simplices'>polynomial differential forms on the n-simplex</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_95' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega_{poly}^\bullet(\Delta^n)</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_96' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy='false'>)</mo> <mi>n</mi></msub><mspace width='thickmathspace' /><mo>≔</mo><mspace width='thickmathspace' /><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mspace width='thickmathspace' /><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> Maps(A,B)_n \;\coloneqq\; Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) </annotation></semantics></math></div> <p>and whose face and degeneracy maps are the obvious ones induced from the fact that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_97' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo lspace='verythinmathspace'>:</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>→</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>\Omega_{poly}^\bullet \colon \Delta^{op} \to dgcAlg^{\geq 0}_k</annotation></semantics></math> is canonically a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+object'>simplicial object</a> in dgc-algebras.</p> <p>We also call this the <em>simplicial <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a></em> from <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_98' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_99' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. This construction naturally extends to a <a class='existingWikiWord' href='/nlab/show/diff/functor'>functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_100' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup><mo>×</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><mo>⟶</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'> Maps(-,-) \;\colon\; (dgcAlg^{\geq 0}_k)^{op} \times dgcAlg^{\geq 0}_k \longrightarrow dgcAlg^{\geq 0}_k </annotation></semantics></math></div> <p>from the <a class='existingWikiWord' href='/nlab/show/diff/product+category'>product category</a> of the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite category</a> of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebras</a> with the category itself.</p> <p>Observe that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_101' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><mi>A</mi><mo>,</mo><mspace width='thickmathspace' /><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mrow /> <mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow></msub><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mrow><mo>(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>A</mi><mspace width='thinmathspace' /><mo>,</mo><mspace width='thinmathspace' /><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> Hom_{dgcAlg^{\geq 0}_k} \left( A, \; \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \;\simeq\; {}_{\Omega^\bullet_{poly}}Hom_{dgcAlg^{\geq 0}_k} \left( \Omega^\bullet_{poly}(\Delta^n) \otimes_k A \,,\, \Omega^\bullet_{poly}(\Delta^n) \otimes_k B \right) \,, </annotation></semantics></math></div> <p>where on the right we have those dg-algebra homomorphism which in addition preserves the left <a class='existingWikiWord' href='/nlab/show/diff/dg-module'>dg-module</a> structure over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_102' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>n</mi></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly}(\Delta^n)</annotation></semantics></math>. This induces for any three <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_103' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>A,B,C \in dgcAlg^{\geq 0}_k</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> homomorphism of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> out of the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>Cartesian product</a> of mapping spaces</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_104' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mo>∘</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>,</mo><mi>C</mi></mrow> <mi>sSet</mi></msubsup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Maps</mi><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>C</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \circ^{sSet}_{A,B,C} \;\colon\; Maps(A,B) \times Maps(B,C) \longrightarrow Maps(A,C) \,. </annotation></semantics></math></div></div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, 5.1</a>)</p> <div class='num_remark'> <h6 id='remark_4'>Remark</h6> <p>The set of 0-simplices of of the <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_105' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Maps(A,B)</annotation></semantics></math> in def. <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a> is <a class='existingWikiWord' href='/nlab/show/diff/natural+isomorphism'>naturally isomorphic</a> to the ordinary <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a> of dg-algebras:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_106' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo>≃</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Maps(A,B)_0 \simeq Hom_{dgcAlg^{\geq 0}_k}(A,B) </annotation></semantics></math></div> <p>and under this identification the two notions of <a class='existingWikiWord' href='/nlab/show/diff/composition'>composition</a> agree.</p> </div> <p>Definition <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a> makes <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_107' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>dgcAlg^{\geq 0}_k</annotation></semantics></math> an <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>-<a class='existingWikiWord' href='/nlab/show/diff/enriched+category'>enriched category</a> (“<a class='existingWikiWord' href='/nlab/show/diff/simplicial+category'>simplicial category</a>”). The follows says that it is also <a class='existingWikiWord' href='/nlab/show/diff/powering'>powered</a>, not over all of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_108' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math>, but over finite simplicial sets:</p> <div class='num_prop' id='PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets'> <h6 id='proposition_4'>Proposition</h6> <p><strong>(powering over finite simplicial sets)</strong></p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_109' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>A, B \in dgcAlg^{\geq 0}_k</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_110' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo></mrow><annotation encoding='application/x-tex'>S \in </annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>, there is a <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural transformation</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_111' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mo>≥</mo></msubsup></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo stretchy='false'>)</mo><mo>⟶</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>Maps</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom_{dgcAlg^{\geq}_k}(A, \Omega^\bullet_{poly}(S) \otimes_k B) \longrightarrow Hom_{sSet}( S, Maps(A,B) ) </annotation></semantics></math></div> <p>from the <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a> of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>dgc-algebras</a> into the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> with the <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+on+simplices'>polynomial differential forms on n-simplices</a> from def. <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a> to the <a class='existingWikiWord' href='/nlab/show/diff/hom-set'>hom-set</a> in <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial sets</a> into the simplicial <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> from def. <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a>.</p> <p>Moreover, this morphism is an <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a> if one of the following conditions holds:</p> <ul> <li> <p><math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_112' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/finite+set'>finite</a> <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_113' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is of <a class='existingWikiWord' href='/nlab/show/diff/finite+type'>finite type</a> (def. <a class='maruku-ref' href='#dgcCochainAlgebraInNonNegDegreeOfFiniteType'>2</a>).</p> </li> </ul> </div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, lemma 5.2</a>)</p> <div class='num_prop'> <h6 id='proposition_5'>Proposition</h6> <p><strong>(pullback powering axiom)</strong></p> <p>Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_114' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo lspace='verythinmathspace'>:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding='application/x-tex'>i \colon V \to W</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_115' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo lspace='verythinmathspace'>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>p \colon X \to Y</annotation></semantics></math> be two <a class='existingWikiWord' href='/nlab/show/diff/morphism'>morphisms</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_116' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup></mrow><annotation encoding='application/x-tex'>dgcAlg^{\geq 0}_k</annotation></semantics></math>. Then their <a class='existingWikiWord' href='/nlab/show/diff/pullback+power'>pullback power</a> with respect to the simplicial <a class='existingWikiWord' href='/nlab/show/diff/compact-open+topology'>mapping space</a> functor (def. <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_117' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>p</mi> <mi>i</mi></msup><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Maps</mi><mo stretchy='false'>(</mo><mi>W</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><mo>⟶</mo><mi>Maps</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><munder><mo>×</mo><mrow><mi>Maps</mi><mo stretchy='false'>(</mo><mi>V</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow></munder><mi>Maps</mi><mo stretchy='false'>(</mo><mi>W</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> p^i \;\colon\; Maps(W,X) \longrightarrow Maps(V,X) \underset{Maps(V,Y)}{\times} Maps(W,Y) </annotation></semantics></math></div> <p>is</p> <ol> <li> <p>a <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a> if <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_118' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> is a cofibration and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_119' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> a fibration in the projective <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure from prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a>;</p> </li> <li> <p>in addition a <a class='existingWikiWord' href='/nlab/show/diff/weak+homotopy+equivalence'>weak homotopy equivalence</a> (i.e. a weak equivalence in the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a>) if at least one of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_120' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_121' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is a weak equivalence in the projective model structure from prop. <a class='maruku-ref' href='#IndeedProjectiveModelStructureOnCdgAlg'>1</a>.</p> </li> </ol> </div> <p>(<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, prop. 5.3</a>)</p> <div class='num_remark'> <h6 id='remark_5'>Remark</h6> <p>Prop. <a class='maruku-ref' href='#PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets'>4</a> <em>would</em> say that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_122' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(dgcAlg^{\geq 0}_k)_{proj}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a> with respect to the simplicial enrichment from def. <a class='maruku-ref' href='#MappingSpaceSimOndgcCochainAlgebrasInNonNegDegrees'>6</a> were it not for the fact that prop. <a class='maruku-ref' href='#PoweringOfdgcCchainAlgebrasInNonNegativeDegreeOverFiniteSimplicialSets'>4</a> gives the <a class='existingWikiWord' href='/nlab/show/diff/powering'>powering</a> only over finite simplicial sets.</p> </div> <ins class='diffins'><h4 id='relation_to_simplicial_sets'>Relation to simplicial sets</h4></ins><ins class='diffins'> </ins><ins class='diffins'><div class='num_prop'> <h6 id='proposition_6'>Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction+between+simplicial+sets+and+connective+dgc-algebras'>Quillen adjunction between simplicial sets and connective dgc-algebras</a>)</strong></p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/PL+de+Rham+complex'>PL de Rham complex</a>-construction is the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> in a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a> between</p> <ul> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/opposite+model+structure'>opposite</a> of the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras'>projective model structure on connective dgc-algebras</a></p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a></p> </li> </ul> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_dde2ecdeeb810cfe68222647073b294b7eb2e427_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo maxsize='1.2em' minsize='1.2em'>(</mo><msubsup><mi>DiffGradedCommAlgebras</mi> <mi>k</mi> <mrow><mo>≥</mo><mn>0</mn></mrow></msubsup><msubsup><mo maxsize='1.2em' minsize='1.2em'>)</mo> <mi>proj</mi> <mi>op</mi></msubsup><munderover><mrow><msub><mo>⊥</mo> <mpadded width='0'><mi>Qu</mi></mpadded></msub></mrow><munder><mo>⟶</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>exp</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></munder><mover><mo>⟵</mo><mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><msubsup><mi>Ω</mi> <mi>PLdR</mi> <mo>•</mo></msubsup><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /></mrow></mover></munderover><msub><mi>SimplicialSets</mi> <mi>Qu</mi></msub></mrow><annotation encoding='application/x-tex'> \big( DiffGradedCommAlgebras^{\geq 0}_{k} \big)^{op}_{proj} \underoverset { \underset {\;\;\; exp \;\;\;} {\longrightarrow} } { \overset {\;\;\;\Omega^\bullet_{PLdR}\;\;\;} {\longleftarrow} } {\bot_{\mathrlap{Qu}}} SimplicialSets_{Qu} </annotation></semantics></math></div></div></ins><ins class='diffins'> </ins><h4 id='relation_to_cosimplicial_commutative_algbras'>Relation to cosimplicial commutative algbras</h4> <p>The <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> gives a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a> to the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+rings'>projective model structure on cosimplicial commutative algebras</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_123' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>cAlg</mi> <mi>k</mi> <mi>Δ</mi></msubsup><msub><mo stretchy='false'>)</mo> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>(cAlg_k^{\Delta})_{proj}</annotation></semantics></math>.</p> <h4 id='CommVsNoncomm'>Commutative vs. non-commutative dg-algebras</h4> <blockquote> <p>this needs harmonization</p> </blockquote> <div class='num_prop'> <h6 id='proposition_6'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_124' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>F</mi><msub><mi>dgcAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>dgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> F dgcAlg_k \to dgAlg_k </annotation></semantics></math></div> <p>from (graded-)commutative <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>s to dg-algebras is the <a class='existingWikiWord' href='/nlab/show/diff/right+adjoint'>right adjoint</a> part of a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen adjunction</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_125' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ab</mi><mo lspace='verythinmathspace'>:</mo><mi>dgAlg</mi><mover><mo>→</mo><mo>←</mo></mover><mi>CdgAlg</mi><mo>:</mo><mi>F</mi></mrow><annotation encoding='application/x-tex'> Ab \colon dgAlg \stackrel{\leftarrow}{\to} CdgAlg : F </annotation></semantics></math></div></div> <blockquote> <p>boundedness?</p> </blockquote> <div class='proof'> <h6 id='proof'>Proof</h6> <p>The forgetful functor clearly preserves fibrations and cofibrations. It has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a>, the free abelianization functor <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_126' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ab</mi></mrow><annotation encoding='application/x-tex'>Ab</annotation></semantics></math>, which sends a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_127' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> to its quotient <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_128' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo stretchy='false'>/</mo><mo stretchy='false'>[</mo><mi>A</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>A/[A,A]</annotation></semantics></math>.</p> </div> <div class='num_theorem'> <h6 id='theorem'>Theorem</h6> <p>Let the ground <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_129' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/characteristic+zero'>characteristic zero</a>. Then every <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_130' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> which has the structure of an <a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over</a> the <a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>E-∞ operad</a> has a <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> morphism <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_131' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>A \to A_c</annotation></semantics></math> to a commutative dg-algebra <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_132' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>A_c</annotation></semantics></math> which is</p> <ul> <li> <p>a morphism of <a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>E-∞ algebras</a> (where <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_133' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mi>c</mi></msub></mrow><annotation encoding='application/x-tex'>A_c</annotation></semantics></math> has the obvious <a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>E-∞ algebras</a> structure)</p> </li> <li> <p>a weak weak equivalence in the model structure on dg-algebras (i.e. a <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> of the underlying cochain complexes).</p> </li> </ul> </div> <p>This is in (<a href='#KrizMay95'>Kriz-May 95, II.1.5</a>).</p> <p>So this says that the weak equivalence classes of the commutative dg-algebras in the model category of all dg-algebras already exhaust the most general non-commutative but homotopy-commutative dg-algebras.</p> <div class='num_remark' id='HomotopyFaithfulnessOfforgettingCommutativity'> <h6 id='remark_6'>Remark</h6> <p>Discussion of a restricted kind of homotopy-faithfulness of the forgetful functor from the homotopy theory of commutative to not-necessarily commutative dg-algebras is in (<a href='#Amrani14'>Amrani 14</a>).</p> </div> <h2 id='Unbounded'>Unbounded dg-algebras</h2> <p>We discuss now the case of unbounded dg-algebras. For these there is no longer the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> available. Instead, these can be understood as arising naturally as function <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_134' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebras in the <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived</a> <a class='existingWikiWord' href='/nlab/show/diff/dg-geometry'>dg-geometry</a> over formal duals of bounded dg-algebras, see <a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>function algebras on ∞-stacks</a>.</p> <h3 id='GradingsAndConventions'>Gradings and conventions</h3> <p>In <a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a> two categorical gradings interact: a <a class='existingWikiWord' href='/nlab/show/diff/cohesive+%28infinity%2C1%29-topos'>cohesive</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_135' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_136' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has a space of <a class='existingWikiWord' href='/nlab/show/diff/k-morphism'>k-morphism</a>s <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_137' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>X_k</annotation></semantics></math> for all non-negative <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_138' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>, and each such has itself a <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>simplicial T-algebra</a></em> of functions with a component in each non-positive degree. But the directions of the face maps are opposite. We recall the grading situation from <a class='existingWikiWord' href='/nlab/show/diff/function+algebras+on+infinity-stacks'>function algebras on ∞-stacks</a>.</p> <p>Functions on a bare <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_139' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groupoid <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_140' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi></mrow><annotation encoding='application/x-tex'>K</annotation></semantics></math>, modeled as a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>, form a <a class='existingWikiWord' href='/nlab/show/diff/cosimplicial+algebra'>cosimplicial algebra</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_141' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>K</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{O}(K)</annotation></semantics></math>, which under the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> identifies with a cochain <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a> (meaning: with positively graded differential) in non-negative degree</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_142' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↓</mo><mo stretchy='false'>↓</mo><mo stretchy='false'>↓</mo><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></msup><msup><mo stretchy='false'>↓</mo> <mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></msup><msup><mo stretchy='false'>↓</mo> <mrow><msub><mo>∂</mo> <mn>2</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↓</mo> <mrow><msub><mo>∂</mo> <mn>0</mn></msub></mrow></msup><msup><mo stretchy='false'>↓</mo> <mrow><msub><mo>∂</mo> <mn>1</mn></msub></mrow></msup></mtd></mtr> <mtr><mtd><msub><mi>K</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mover><mo>↦</mo><mi>𝒪</mi></mover><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>K</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mrow><msubsup><mo>∂</mo> <mn>0</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy='false'>↑</mo> <mrow><msubsup><mo>∂</mo> <mn>1</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy='false'>↑</mo> <mrow><msubsup><mo>∂</mo> <mn>2</mn> <mo>*</mo></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>K</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mrow><msubsup><mo>∂</mo> <mn>0</mn> <mo>*</mo></msubsup></mrow></msup><msup><mo stretchy='false'>↑</mo> <mrow><msubsup><mo>∂</mo> <mn>1</mn> <mo>*</mo></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>K</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mover><mo>↔</mo><mo>∼</mo></mover><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>i</mi></munder><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>i</mi></munder><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mpadded width='0'><mrow><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mi>i</mi></munder><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mi>i</mi></msup><msubsup><mo>∂</mo> <mi>i</mi> <mo>*</mo></msubsup></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \left( \array{ \vdots \\ \downarrow \downarrow \downarrow \downarrow \\ K_2 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \downarrow^{\partial_2} \\ K_1 \\ \downarrow^{\partial_0} \downarrow^{\partial_1} \\ K_0 } \right) \;\;\;\;\; \stackrel{\mathcal{O}}{\mapsto} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \uparrow \uparrow \uparrow \\ \mathcal{O}(K_2) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \uparrow^{\partial_2^*} \\ \mathcal{O}(K_1) \\ \uparrow^{\partial_0^*} \uparrow^{\partial_1^*} \\ \mathcal{O}(K_0) } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \cdots \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_2 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_1 \\ \uparrow^{\mathrlap{\sum_i (-1)^i \partial_i^*}} \\ A_0 \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \vdots } \right) \,. </annotation></semantics></math></div> <p>On the other hand, a representable <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_143' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> has itself a <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+algebras'>simplicial T-algebra</a></em> of functions, which under the monoidal Dold-Kan correspondence also identifies with a cochain dg-algebra, but then necessarily in non-positive degree to match with the above convention. So we write</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_144' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mover><mo>↔</mo><mo>∼</mo></mover><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mn>0</mn></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathcal{O}(X) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \mathcal{O}(X)_0 \\ \uparrow \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \uparrow \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \uparrow \uparrow \uparrow \\ \vdots } \right) \;\;\;\;\; \stackrel{\sim}{\leftrightarrow} \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ 0 \\ \uparrow \\ 0 \\ \uparrow \\ \mathcal{O}(X)_0 \\ \uparrow \\ \mathcal{O}(X)_{-1} \\ \uparrow \\ \mathcal{O}(X)_{-2} \\ \uparrow \\ \vdots } \right) \,. </annotation></semantics></math></div> <p>Taking this together, for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_145' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>X</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>X_\bullet</annotation></semantics></math> a general <a class='existingWikiWord' href='/nlab/show/diff/infinity-stack'>∞-stack</a>, its function algebra is generally an <em>unbounded</em> cochain dg-algebra with mixed contributions as above, the simplicial degrees contributing in the positive direction, and the homological resolution degrees in the negative direction:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_146' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mo>•</mo></msub><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mo>=</mo><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⨁</mo> <mrow><mi>k</mi><mo>−</mo><mi>p</mi><mo>=</mo><mi>q</mi></mrow></munder><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mi>k</mi></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mi>p</mi></mrow></msub></mtd></mtr> <mtr><mtd><mo stretchy='false'>↑</mo></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>3</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>0</mn></msub><msub><mo stretchy='false'>)</mo> <mn>0</mn></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>0</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>1</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>2</mn></mrow></msub><mo>⊕</mo><mi>𝒪</mi><mo stretchy='false'>(</mo><msub><mi>X</mi> <mn>2</mn></msub><msub><mo stretchy='false'>)</mo> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>3</mn></mrow></msub><mo>⊕</mo><mi>⋯</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy='false'>↑</mo> <mi>d</mi></msup></mtd></mtr> <mtr><mtd><mi>⋮</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathcal{O}(X_\bullet) \;\;\;\;\; = \;\;\;\;\; \left( \array{ \vdots \\ \uparrow \\ \bigoplus_{k-p = q} \mathcal{O}(X_k)_{-p} \\ \uparrow \\ \vdots \\ \uparrow^d \\ \mathcal{O}(X_1)_0 \oplus \mathcal{O}(X_2)_{-1} \oplus \mathcal{O}(X_3)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_0 \oplus \mathcal{O}(X_1)_{-1} \oplus \mathcal{O}(X_2)_{-2} \oplus \cdots \\ \uparrow^{d} \\ \mathcal{O}(X_0)_{-1} \oplus \mathcal{O}(X_1)_{-2} \oplus \mathcal{O}(X_2)_{-3}\oplus \cdots \\ \uparrow^{d} \\ \vdots } \right) \,. </annotation></semantics></math></div> <h3 id='definition_8'>Definition</h3> <div class='num_theorem'> <h6 id='theorem_2'>Theorem</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_147' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/characteristic'>characteristic</a> 0 let</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_148' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>cdgAlg</mi><mo>=</mo><mi>CMon</mi><mo stretchy='false'>(</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> cdgAlg = CMon(Ch_\bullet(k)) </annotation></semantics></math></div> <p>be the category of undounded commutative dg-algebras. With fibrations the degreewise surjections and weak equivalences the <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a>s this is a</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></li> </ul> <p>which is</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>proper</a>;</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/combinatorial+model+category'>combinatorial</a>.</p> </li> </ul> </div> <p>The existence of the model structure follows from the general discussion at <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras+over+an+operad'>model structure on dg-algebras over an operad</a>.</p> <p>Properness and combinatoriality is discussed in (<a href='#ToenVezzosi'>ToënVezzosi</a>):</p> <ul> <li> <p>in lemma 2.3.1.1 they state that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_149' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mo>+</mo></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_+</annotation></semantics></math> constitutes the first two items in a triple which they call an <em>HA context</em> .</p> </li> <li> <p>this implies their assumption 1.1.0.4 which asserts properness and combinatoriality</p> </li> </ul> <p>Discussion of cofibrations in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_150' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>dgAlg</mi> <mi>proj</mi></msub></mrow><annotation encoding='application/x-tex'>dgAlg_{proj}</annotation></semantics></math> is in (<a href='#Keller'>Keller</a>).</p> <h3 id='properties_2'>Properties</h3> <h4 id='properness'>Properness</h4> <p>Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_151' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAg_k</annotation></semantics></math> be the projective model structure on commutative unbounded dg-algebras from above.</p> <p>This is a <a class='existingWikiWord' href='/nlab/show/diff/proper+model+category'>proper model category</a>. See MO discussion <a href='http://mathoverflow.net/q/204414/381'>here</a>.</p> <h4 id='derived_tensor_product'>Derived tensor product</h4> <p>Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_152' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAg_k</annotation></semantics></math> be the projective model structure on commutative unbounded dg-algebras from above</p> <div class='num_prop'> <h6 id='proposition_7'>Proposition</h6> <p>For cofibrant <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_153' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>A \in cdgAlg_k</annotation></semantics></math>, the functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_154' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>k</mi><mi>Mod</mi><mo>→</mo><mi>A</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'> A\otimes_k (-) : k Mod \to A Mod </annotation></semantics></math></div> <p>preserves <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a>s.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_155' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>A,B \in cdgAlg_k</annotation></semantics></math>, their <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived</a> <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_156' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>k Mod</annotation></semantics></math> coincides in the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+category'>homotopy category</a> with the derived <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_157' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>k Mod</annotation></semantics></math>: the morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_158' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><munderover><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo> <mi>k</mi> <mi>L</mi></munderover><mi>B</mi><mover><mo>→</mo><mrow /></mover><mi>A</mi><msubsup><mo>⊗</mo> <mi>k</mi> <mi>L</mi></msubsup><mi>B</mi></mrow><annotation encoding='application/x-tex'> A \coprod_k^{L} B \stackrel{}{\to} A \otimes_k^L B </annotation></semantics></math></div> <p>is an isomorphism in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_159' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ho</mi><mo stretchy='false'>(</mo><mi>k</mi><mi>Mod</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ho(k Mod)</annotation></semantics></math>.</p> </div> <p>This follows by the above with (<a href='#ToenVezzosi'>ToënVezzosi, assumption 1.1.0.4, and page 8</a>).</p> <h4 id='SimplicialHomObjects'>Derived hom-functor</h4> <p>The model structure on unbounded dg-algebras is <em>almost</em> a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a>. See the section <em><a href='model+structure+on+dg-algebras+over+an+operad#SimplicialEnrichment'>simplicial enrichment</a></em> at <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras+over+an+operad'>model structure on dg-algebras over an operad</a></em> for details.</p> <div class='num_defn'> <h6 id='definition_9'>Definition</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_160' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/diff/field'>field</a> of <a class='existingWikiWord' href='/nlab/show/diff/characteristic'>characteristic</a> 0. Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_161' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy='false'>(</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly} : sSet \to (cdgAlg_k)^{op}</annotation></semantics></math> be the functor that assigns polynomial <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+on+simplices'>differential forms on simplices</a>.</p> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_162' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><msub><mi>dgcAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>A,B \in dgcAlg_k</annotation></semantics></math> define the <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_163' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>:</mo><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>↦</mo><msub><mi>Hom</mi> <mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> cdgAlg_k(A,B) : ([n] \mapsto Hom_{cdgAlg_k}(A, B \otimes_k \Omega^\bullet_{poly}(\Delta[n])) \,. </annotation></semantics></math></div> <p>This extends to a functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_164' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo>×</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo>→</mo><mi>sSet</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> cdgAlg_k(-,-) : cdgAlg_k^{op} \times cdgAlg_k \to sSet \,. </annotation></semantics></math></div></div> <div class='num_prop'> <h6 id='proposition_8'>Proposition</h6> <p>The functor <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_165' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cdgAlg_k(-,-)</annotation></semantics></math> satisfies the dual of the <a class='existingWikiWord' href='/nlab/show/diff/pushout-product+axiom'>pushout-product axiom</a>: for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_166' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>i : A \to B</annotation></semantics></math> any cofibration in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_167' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_168' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding='application/x-tex'>p : X \to Y</annotation></semantics></math> any fibration, the canonical morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_169' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msup><mi>i</mi> <mo>*</mo></msup><mo>,</mo><msub><mi>p</mi> <mo>*</mo></msub><mo stretchy='false'>)</mo><mo>:</mo><msub><mi>cdgalg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>X</mi><mo stretchy='false'>)</mo><msub><mo>×</mo> <mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow></msub><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>B</mi><mo>,</mo><mi>Y</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (i^*, p_*) : cdgalg_k(A,B) \to cdgAlg_k(A,X) \times_{cdgAlg_k(A,Y)} cdgAlg_k(B,Y) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+fibration'>Kan fibration</a>, which is acyclic if <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_170' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>i</mi></mrow><annotation encoding='application/x-tex'>i</annotation></semantics></math> or <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_171' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>p</mi></mrow><annotation encoding='application/x-tex'>p</annotation></semantics></math> is.</p> </div> <p>This implies in particular that for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_172' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> cofibrant, <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_173' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cdgAlg_k(A,B)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>.</p> <p>The proof works along the lines of (<a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76, prop. 5.3</a>). See also the discussion at <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras+over+an+operad'>model structure on dg-algebras over an operad</a>.</p> <div class='proof'> <h6 id='proof_2'>Proof</h6> <p>We give the proof for a special case. The general case is analogous.</p> <p>We show that for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_174' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> cofibrant, and for any <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_175' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> (automatically fibrant), <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_176' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cdgAlg_k(A,B)</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a>.</p> <p>By a standard fact in <a class='existingWikiWord' href='/nlab/show/diff/rational+homotopy+theory'>rational homotopy theory</a> (due to <a href='#BousfieldGugenheim76'>Bousfield-Gugenheim 76</a>, discussed at <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+on+simplices'>differential forms on simplices</a>) we have that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_177' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><mo stretchy='false'>(</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mo>+</mo></msubsup><msup><mo stretchy='false'>)</mo> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly} : sSet \to (cdgAlg^+_k)^{op}</annotation></semantics></math> is a left <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen functor</a>, hence in particular sends acyclic cofibrations to acyclic cofibrations, hence acyclic <a class='existingWikiWord' href='/nlab/show/diff/monomorphism'>monomorphism</a>s of <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a>s to acyclic fibrations of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded+algebra'>dg-algebra</a>s.</p> <p>Specifically for each <a class='existingWikiWord' href='/nlab/show/diff/horn'>horn</a> inclusion <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_178' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Λ</mi><mo stretchy='false'>[</mo><mi>n</mi><msub><mo stretchy='false'>]</mo> <mi>k</mi></msub><mo>↪</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>\Lambda[n]_k \hookrightarrow \Delta[n]</annotation></semantics></math> we have that the restriction map <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_179' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>→</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Λ</mi><mo stretchy='false'>[</mo><mi>n</mi><msub><mo stretchy='false'>]</mo> <mi>k</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly}(\Delta[n]) \to \Omega^\bullet_{poly}(\Lambda[n]_k)</annotation></semantics></math> is an acyclic fibration in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_180' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mo>*</mo></msubsup></mrow><annotation encoding='application/x-tex'>cdgAlg_k^*</annotation></semantics></math>, hence in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_181' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math>.</p> <p>A <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_182' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-horn in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_183' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cdgAlg_k(A,B)</annotation></semantics></math> is a morphism <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_184' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>→</mo><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Λ</mi><mo stretchy='false'>[</mo><mi>n</mi><msub><mo stretchy='false'>]</mo> <mi>k</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>A \to B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k)</annotation></semantics></math>. A filler for this horn is a lift <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_185' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>σ</mi></mrow><annotation encoding='application/x-tex'>\sigma</annotation></semantics></math> in</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_186' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><mo stretchy='false'>↓</mo></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Λ</mi><mo stretchy='false'>[</mo><mi>n</mi><msub><mo stretchy='false'>]</mo> <mi>k</mi></msub><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ && B \otimes \Omega^\bullet_{poly}(\Delta[n]) \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow \\ A &\to& B \otimes \Omega^\bullet_{poly}(\Lambda[n]_k) } \,. </annotation></semantics></math></div> <p>If <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_187' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is cofibrant, then such a lift does always exist.</p> </div> <div class='num_prop'> <h6 id='proposition_9'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_188' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>cdgAlg</mi></mrow><annotation encoding='application/x-tex'>A \in cdgAlg</annotation></semantics></math> cofibrant, <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_189' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>cdgAlg_k(A,B)</annotation></semantics></math> is the correct <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom-space</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_190' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>ℝ</mi><mi>Hom</mi><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> cdgAlg_k(A,B) \simeq \mathbb{R}Hom(A,B) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_3'>Proof</h6> <p>By the assumption that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_191' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is cofibrant and according to the facts discussed at <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-categorical+hom-space'>derived hom-space</a>, we need to show that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_192' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi><mi>B</mi><mo>:</mo><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo>↦</mo><mi>B</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>n</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> s B : [n] \mapsto B\otimes_k \Omega^\bullet_{poly}(\Delta[n]) </annotation></semantics></math></div> <p>is a <a class='existingWikiWord' href='/nlab/show/diff/simplicial+resolution'>resolution</a>, or <em>simplicial frame</em> for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_193' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. (Notice that every object is fibrant in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_194' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math>).</p> <p>Since polynomial differential forms are acyclic on simplices (discussed <a href='http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices'>here</a>) it follows that</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_195' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>const</mi><mi>B</mi><mo>→</mo><mi>s</mi><mi>B</mi></mrow><annotation encoding='application/x-tex'> const B \to s B </annotation></semantics></math></div> <p>is degreewise a weak equivalence. It remains to show that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_196' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>s A</annotation></semantics></math> is fibrant in the <a class='existingWikiWord' href='/nlab/show/diff/Reedy+model+structure'>Reedy model structure</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_197' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>[</mo><msup><mi>Δ</mi> <mi>op</mi></msup><mo>,</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msub><mo stretchy='false'>]</mo> <mi>Reedy</mi></msub></mrow><annotation encoding='application/x-tex'>[\Delta^{op}, cdgAlg_k]_{Reedy}</annotation></semantics></math>.</p> <p>One finds that the matching object is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_198' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>match</mi><mi>s</mi><mi>B</mi><msub><mo stretchy='false'>)</mo> <mi>k</mi></msub><mo>=</mo><mi>B</mi><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (match s B)_k = B \otimes \Omega^\bullet_{poly}(\partial \Delta[k]) \,. </annotation></semantics></math></div> <p>Therefore <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_199' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>s</mi><mi>B</mi></mrow><annotation encoding='application/x-tex'>s B</annotation></semantics></math> is Reedy fibrant if in each degree the morphism</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_200' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>s</mi><msub><mi>B</mi> <mi>k</mi></msub><mo>→</mo><mo stretchy='false'>(</mo><mi>match</mi><mi>s</mi><mi>B</mi><msub><mo stretchy='false'>)</mo> <mi>k</mi></msub><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mo>∂</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>↪</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (s B_k \to (match s B)_k ) = (\Omega^\bullet_{poly}(\partial \Delta[k] \hookrightarrow \Delta[k])) </annotation></semantics></math></div> <p>is a fibration. But this follows from the fact that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_201' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo>:</mo><mi>sSet</mi><mo>→</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly} : sSet \to cdgAlg_k^{op}</annotation></semantics></math> is a left <a class='existingWikiWord' href='/nlab/show/diff/Quillen+adjunction'>Quillen functor</a> (as discussed at <a class='existingWikiWord' href='/nlab/show/diff/differential+forms+on+simplices'>differential forms on simplices</a>).</p> </div> <h4 id='DerivedCopowering'>Derived copowering over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_202' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math></h4> <p>We discuss a concrete model for the <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_203' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-copowering of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_204' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>(cdgAlg_k)^\circ</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> in terms of an operation of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_205' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/SimpSet'>sSet</a>.</p> <p>First notice a basic fact about ordinary commutative algebras.</p> <div class='num_prop'> <h6 id='proposition_10'>Proposition</h6> <p>In <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_206' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>CAlg_k</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/coproduct'>coproduct</a> is given by the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_207' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_208' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>i</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>i</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow><mo>≃</mo><mrow><mo>(</mo><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mi>A</mi></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>e</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>Id</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr></mtable></mrow><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> \left( \array{ A &\stackrel{i_A}{\to}& A \coprod B &\stackrel{i_B}{\leftarrow}& B } \right) \simeq \left( \array{ A &\stackrel{Id_A \otimes_k e_B}{\to}& A \otimes_k B & \stackrel{e_A \otimes Id_B}{\leftarrow}& B } \right) </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_4'>Proof</h6> <p>We check the <a class='existingWikiWord' href='/nlab/show/diff/universal+construction'>universal property</a> of the coproduct: for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_209' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo>∈</mo><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>C \in CAlg_k</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_210' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>:</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f,g : A,B \to C</annotation></semantics></math> two morphisms, we need to show that there is a unique morphism <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_211' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>(f,g) : A \otimes_k B \to C</annotation></semantics></math> such that the diagram</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_212' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>Id</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>e</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mtd> <mtd><mover><mo>←</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>Id</mi> <mi>B</mi></msub></mrow></mover></mtd> <mtd><mi>B</mi></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mpadded lspace='-100%width' width='0'><mi>f</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy='false'>↓</mo> <mpadded width='0'><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow></mpadded></msup></mtd> <mtd><msub><mo>↙</mo> <mpadded width='0'><mi>g</mi></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>C</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ A &\stackrel{Id_A \otimes e_B}{\to}& A \otimes_k B &\stackrel{e_A \otimes Id_B}{\leftarrow}& B \\ & {}_{\mathllap{f}}\searrow & \downarrow^{\mathrlap{(f,g)}} & \swarrow_{\mathrlap{g}} \\ && C } </annotation></semantics></math></div> <p>commutes. For the left triangle to commute we need that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_213' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> sends elements of the form <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_214' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(a,e_B)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_215' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(a)</annotation></semantics></math>. For the right triangle to commute we need that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_216' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> sends elements of the form <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_217' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(e_A, b)</annotation></semantics></math> to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_218' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>g(b)</annotation></semantics></math>. Since every element of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_219' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>B</mi></mrow><annotation encoding='application/x-tex'>A \otimes_k B</annotation></semantics></math> is a product of two elements of this form</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_220' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>=</mo><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><msub><mi>e</mi> <mi>B</mi></msub><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (a,b) = (a,e_B) \cdot (e_A, b) </annotation></semantics></math></div> <p>this already uniquely determines <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_221' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(f,g)</annotation></semantics></math> to be given on elements by the map</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_222' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy='false'>)</mo><mo>↦</mo><mi>f</mi><mo stretchy='false'>(</mo><mi>a</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>b</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (a,b) \mapsto f(a) \cdot g(b) \,. </annotation></semantics></math></div> <p>That this is indeed an <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_223' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-algebra homomorphism follows from the fact that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_224' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_225' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>g</mi></mrow><annotation encoding='application/x-tex'>g</annotation></semantics></math> are</p> </div> <div class='num_remark'> <h6 id='remark_7'>Remark</h6> <p>For these derivations it is crucial that we are working with commutative algebras.</p> </div> <div class='num_cor'> <h6 id='corollary'>Corollary</h6> <p>We have that the <a class='existingWikiWord' href='/nlab/show/diff/copower'>copower</a>ing of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_226' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> with the map of sets from two points to the single point</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_227' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo>*</mo><mo lspace='thinmathspace' rspace='thinmathspace'>∐</mo><mo>*</mo><mo>→</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mi>A</mi><mo>≃</mo><mo stretchy='false'>(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>A</mi><mover><mo>→</mo><mi>μ</mi></mover><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (* \coprod * \to *) \cdot A \simeq ( A \otimes_k A \stackrel{\mu}{\to} A ) </annotation></semantics></math></div> <p>is the product morphism on <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_228' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. And that the tensoring with the map from the empty set to the point</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_229' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>∅</mi><mo>→</mo><mo>*</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mi>A</mi><mo>≃</mo><mo stretchy='false'>(</mo><mi>k</mi><mover><mo>→</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow></mover><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (\emptyset \to *)\cdot A \simeq (k \stackrel{e_A}{\to} A) </annotation></semantics></math></div> <p>is the unit morphism on <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_230' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math>. Generally, for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_231' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>:</mo><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding='application/x-tex'>f : S \to T</annotation></semantics></math> any map of sets we have that the tensoring</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_232' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>S</mi><mover><mo>→</mo><mi>f</mi></mover><mi>T</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>A</mi><mo>=</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy='false'>|</mo><mi>S</mi><mo stretchy='false'>|</mo></mrow></msup><mo>→</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy='false'>|</mo><mi>T</mi><mo stretchy='false'>|</mo></mrow></msup></mrow><annotation encoding='application/x-tex'> (S \stackrel{f}{\to} T) \cdot A = A^{\otimes_k |S|} \to A^{\otimes_k |T|} </annotation></semantics></math></div> <p>is the morphism between <a class='existingWikiWord' href='/nlab/show/diff/tensor+power'>tensor power</a>s of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_233' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> of the cardinalities of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_234' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_235' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>T</mi></mrow><annotation encoding='application/x-tex'>T</annotation></semantics></math>, respectively, whose component over a copy of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_236' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> on the right corresponding to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_237' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi><mo>∈</mo><mi>T</mi></mrow><annotation encoding='application/x-tex'>t \in T</annotation></semantics></math> is the iterated product <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_238' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy='false'>|</mo><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>{</mo><mi>t</mi><mo stretchy='false'>}</mo><mo stretchy='false'>|</mo></mrow></msup><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>A^{\otimes_k |f^{-1}\{t\}|} \to A</annotation></semantics></math> on as many tensor powers of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_239' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> as there are elements in the preimage of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_240' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>t</mi></mrow><annotation encoding='application/x-tex'>t</annotation></semantics></math> under <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_241' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>.</p> </div> <p>The analogous statements hold true with <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_242' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>CAlg_k</annotation></semantics></math> replaced by <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_243' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math>: for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_244' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>S \in sSet</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_245' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> A \in cdgAlg_k</annotation></semantics></math> we obtain a simplicial cdg-algebra</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_246' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>⋅</mo><mi>A</mi><mo>∈</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup></mrow><annotation encoding='application/x-tex'> S \cdot A \in cdgAlg_k^{\Delta^{op}} </annotation></semantics></math></div> <p>by the ordinary degreewise <a class='existingWikiWord' href='/nlab/show/diff/copower'>copower</a>ing over <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>, using that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_247' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math> has coproducts (equal to the tensor product over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_248' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>).</p> <p>This is equivalently a commutative monoid in simplicial unbounded chain complexes</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_249' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msubsup><mo>≃</mo><mi>CMon</mi><mo stretchy='false'>(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><msup><mo stretchy='false'>)</mo> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> cdgAlg_k^{\Delta^{op}} \simeq CMon(Ch^\bullet(k)^{\Delta^{op}}) \,. </annotation></semantics></math></div> <p>By the logic of the <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> the symmetric <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>lax monoidal</a> <a class='existingWikiWord' href='/nlab/show/diff/Moore+complex'>Moore complex</a> functor (via the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Zilber+map'>Eilenberg-Zilber map</a>) sends this to a commutative <a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a> in non-positively graded cochain complexes in unbounded cochain complexes</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_250' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>CMon</mi><mo stretchy='false'>(</mo><msubsup><mi>Ch</mi> <mo>−</mo> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C^\bullet(S \cdot A) \in CMon(Ch^\bullet_-(Ch^\bullet(k))) \,. </annotation></semantics></math></div> <p>Since the <a class='existingWikiWord' href='/nlab/show/diff/total+complex'>total complex</a> functor <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_251' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tot</mi><mo>:</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>→</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Tot : Ch^\bullet(Ch^\bullet(k)) \to Ch^\bullet(k)</annotation></semantics></math> is itself symmetric lax monoidal (…), this finally yields</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_252' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Tot</mi><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>CMon</mi><mo stretchy='false'>(</mo><msup><mi>Ch</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>k</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> Tot C^\bullet(S \cdot A) \in CMon(Ch^\bullet(k)) \simeq cdgAlg_k </annotation></semantics></math></div> <div class='num_defn'> <h6 id='definition_10'>Definition</h6> <p>Define the functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_253' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi><mo>:</mo><mi>sSet</mi><mo>×</mo><mi>cdgAlg</mi><mo>→</mo><mi>cdgAlg</mi></mrow><annotation encoding='application/x-tex'> CC : sSet \times cdgAlg \to cdgAlg </annotation></semantics></math></div> <p>by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_254' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>Tot</mi><msup><mi>C</mi> <mo>•</mo></msup><mo stretchy='false'>(</mo><mi>S</mi><mo>⋅</mo><mi>A</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> CC(S,A) := Tot C^\bullet(S \cdot A) \,. </annotation></semantics></math></div></div> <div class='num_remark'> <h6 id='remark_8'>Remark</h6> <p>We have</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_255' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi><mo stretchy='false'>(</mo><mi>Y</mi><mo>,</mo><mi>A</mi><msup><mo stretchy='false'>)</mo> <mi>n</mi></msup><mo>:</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>⨁</mo> <mrow><mi>k</mi><mo>≥</mo><mn>0</mn></mrow></munder><mo stretchy='false'>(</mo><msup><mi>A</mi> <mrow><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy='false'>|</mo><msub><mi>Y</mi> <mi>k</mi></msub><mo stretchy='false'>|</mo></mrow></msup><msub><mo stretchy='false'>)</mo> <mrow><mi>n</mi><mo>+</mo><mi>k</mi></mrow></msub></mrow><annotation encoding='application/x-tex'> CC(Y,A)^n := \bigoplus_{k \geq 0} (A^{\otimes_k |Y_k| })_{n+k} </annotation></semantics></math></div></div> <p>This appears essentially (…) as (<a href='#GinotTradlerZeinalian'>GinotTradlerZeinalian, def 3.1.1</a>).</p> <div class='num_prop'> <h6 id='proposition_11'>Proposition</h6> <p>The <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-limit'>(∞,1)-copowering</a> of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_256' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msub><mi>dgcAlg</mi> <mi>k</mi></msub><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup></mrow><annotation encoding='application/x-tex'>(dgcAlg_k)^\circ</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> is modeled by the <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a> of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_257' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi></mrow><annotation encoding='application/x-tex'>CC</annotation></semantics></math>.</p> </div> <p>This follows from (<a href='#GinotTradlerZeinalian'>GinotTradlerZeinalian, theorem 4.2.7</a>), which asserts that the <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a> of this tensoring is the unique <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a>, up to equivalence, satisfying the axioms of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_258' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-copowering.</p> <div class='num_prop'> <h6 id='proposition_12'>Proposition</h6> <p>The functor</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_259' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi><mo>:</mo><mi>sSet</mi><mo>×</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub><mo>→</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> CC : sSet \times cdgAlg_k \to cdgAlg_k </annotation></semantics></math></div> <p>preserves weak equivalences in both arguments.</p> </div> <p>This is essentially due to (<a href='#Pirashvili'>Pirashvili</a>). The full statement is (<a href='#GinotTradlerZeinalian'>GinotTradlerZeinalian, prop. 4.2.1</a>).</p> <div class='num_remark'> <h6 id='remark_9'>Remark</h6> <p>This means that the assumption for the copowering models of higher order <a class='existingWikiWord' href='/nlab/show/diff/Hochschild+cohomology'>Hochschild cohomology</a> are satsified in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_260' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math> which are described in the section <a href='http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#PirashviliHigherOrder'>Pirashvili's higher Hochschild homology</a> is satisfied:</p> <p>this means that for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_261' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><mi>cdgAlg</mi></mrow><annotation encoding='application/x-tex'>A \in cdgAlg</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_262' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>S \in sSet</annotation></semantics></math>, <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_263' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CC</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>CC(S,A)</annotation></semantics></math> is a model for the function <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_264' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-algebra on the <a class='existingWikiWord' href='/nlab/show/diff/free+loop+space+object'>free loop space object</a> of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_265' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>A</mi></mrow><annotation encoding='application/x-tex'>Spec A</annotation></semantics></math>. See the section <a href='http://nlab.mathforge.org/nlab/show/Hochschild+cohomology#OvercdgAlgs'>Higher order Hochschild homology modeled on cdg-algebras</a> for more details.</p> </div> <h4 id='DerivedPowering'>Derived powering over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_266' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math></h4> <div class='num_prop'> <h6 id='claim'>Claim</h6> <p>Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_267' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi><mo>∈</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>S \in \infty Grpd</annotation></semantics></math> be presented by a degreewise finite <a class='existingWikiWord' href='/nlab/show/diff/simplicial+set'>simplicial set</a> (which we denote by the same symbol).</p> <p>Then the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limit</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_268' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>cdgAlg_k</annotation></semantics></math> over the <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_269' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math>-shaped diagram constant on <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_270' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> is given by <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_271' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly}(S)</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_272' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℝ</mi><msub><mrow><munder><mi>lim</mi> <mo>←</mo></munder></mrow> <mi>S</mi></msub><mi>const</mi><mi>k</mi><mo>≃</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \mathbb{R}{\lim_{\leftarrow}}_S const k \simeq \Omega^\bullet_{poly}(S) \,. </annotation></semantics></math></div></div> <div class='proof'> <h6 id='proof_5'>Proof</h6> <p>We show dually that for degreewise finite <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_273' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>S</mi></mrow><annotation encoding='application/x-tex'>S</annotation></semantics></math> the assignment <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_274' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mi>Spec</mi><mi>A</mi><mo stretchy='false'>)</mo><mo>↦</mo><mi>Spec</mi><mo stretchy='false'>(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>A</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(S, Spec A) \mapsto Spec (\Omega^\bullet_{poly}(S) \otimes A)</annotation></semantics></math> models the <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_275' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-copowering in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_276' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow><annotation encoding='application/x-tex'>cdgAlg_k^{op}</annotation></semantics></math>.</p> <p>By the discussion at <a href='http://nlab.mathforge.org/nlab/show/limit+in+a+quasi-category#Tensoring'>(∞,1)-copowering</a> it is sufficient to to establish an equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_277' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup><mo stretchy='false'>(</mo><mi>Spec</mi><mo stretchy='false'>(</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo>≃</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><mo stretchy='false'>(</mo><msubsup><mi>dgcAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><msup><mo stretchy='false'>)</mo> <mo>∘</mo></msup><mo stretchy='false'>(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> (dgcAlg_{k}^{op})^\circ(Spec (\Omega^\bullet_{poly}(S) \otimes A), Spec B) \simeq \infty Grpd(S, (dgcAlg_{k}^{op})^\circ(Spec A, Spec B)) </annotation></semantics></math></div> <p>natural in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_278' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>. Consider a cofibrant model of <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_279' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, which we denote by the same symbol. The we compute with 1-categorical <a class='existingWikiWord' href='/nlab/show/diff/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/diff/end'>coend</a> calculus</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_280' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable columnalign='right left right left right left right left right left' columnspacing='0em' displaystyle='true'><mtr><mtd><mi>sSet</mi><mo stretchy='false'>(</mo><mi>S</mi><mo>,</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo stretchy='false'>(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mi>sSet</mi></msub><mo stretchy='false'>(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>,</mo><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo stretchy='false'>(</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>Hom</mi> <mi>Set</mi></msub><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>,</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy='false'>(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msub><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy='false'>(</mo><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy='false'>(</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>k</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mo stretchy='false'>(</mo><msub><mi>S</mi> <mi>k</mi></msub><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>k</mi><mo>,</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>⋅</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><msup><mi>Δ</mi> <mi>k</mi></msup><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr> <mtr><mtd /> <mtd><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy='false'>(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \begin{aligned} sSet(S, cdgAlg_k^{op}(Spec A,Spec B)) & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{sSet}(S \times \Delta[r], cdgAlg_k^{op}(Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{Set}(S_k \times \Delta[k,r], Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot \int_{[k] \in \Delta} Hom_{cdgAlg_k^{op}}((S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(\int^{[k] \in \Delta} (S_k \times \Delta[k,r]) \cdot Spec \Omega^\bullet_{poly}(\Delta^k) \times Spec A, Spec B)) \\ & \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S \times \Delta[r]) \times Spec A, Spec B)) \end{aligned} \,, </annotation></semantics></math></div> <p>where all steps are <a class='existingWikiWord' href='/nlab/show/diff/isomorphism'>isomorphism</a>s and the dot denotes the ordinary 1-categorical <a class='existingWikiWord' href='/nlab/show/diff/copower'>copower</a>ing of the 1-category <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_281' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>cdgAlg</mi> <mi>op</mi></msup></mrow><annotation encoding='application/x-tex'>cdgAlg^{op}</annotation></semantics></math> over <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a>. In the last step we are using that the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a> commutes with finite limits of dg-algebras. (This is where the finiteness assumption is needed).</p> <p>Now we use that <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_282' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup></mrow><annotation encoding='application/x-tex'>\Omega^\bullet_{poly}</annotation></semantics></math> preserves <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>product</a>s up to <a class='existingWikiWord' href='/nlab/show/diff/quasi-isomorphism'>quasi-isomorphism</a> (as discussed <a href='http://nlab.mathforge.org/nlab/show/differential+forms+on+simplices#Properties'>here</a>)</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_283' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo>×</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>≃</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>⊗</mo><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \Omega^\bullet_{poly}(S \times \Delta[r]) \simeq \Omega^\bullet_{poly}(S) \otimes \Omega_{poly}^\bullet(\Delta[r]) \,. </annotation></semantics></math></div> <p>This being a weak equivalence between fibrant objects and since <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_284' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math> is assumed cofibrant, we have by the <a href='#SimplicialHomObjects'>above discussion</a> of the derived hom-functor (and using the <a class='existingWikiWord' href='/nlab/show/diff/factorization+lemma'>factorization lemma</a>) a weak equivalence</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_285' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>⋯</mi><mo>≃</mo><msup><mo>∫</mo> <mrow><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>∈</mo><mi>Δ</mi></mrow></msup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo>⋅</mo><msub><mi>Hom</mi> <mrow><msubsup><mi>cdgAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow></msub><mo stretchy='false'>(</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>S</mi><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mi>Δ</mi><mo stretchy='false'>[</mo><mi>r</mi><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo><mo>×</mo><mi>Spec</mi><mi>A</mi><mo>,</mo><mi>Spec</mi><mi>B</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \cdots \simeq \int^{[r] \in\Delta} \Delta[r] \cdot Hom_{cdgAlg_k^{op}}(Spec \Omega^\bullet_{poly}(S) \times Spec \Omega^\bullet_{poly}\Delta[r]) \times Spec A, Spec B)) \,. </annotation></semantics></math></div> <p>Since all this is <a class='existingWikiWord' href='/nlab/show/diff/natural+transformation'>natural</a> in <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_286' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi></mrow><annotation encoding='application/x-tex'>B</annotation></semantics></math>, this proves the claim.</p> </div> <h4 id='PathObjectsForUnboundedCommutative'>Path objects</h4> <div class='num_prop'> <h6 id='proposition_13'>Proposition</h6> <p>For <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_287' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>cdgAlg</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'>A \in cdgAlg_k</annotation></semantics></math>, a <a class='existingWikiWord' href='/nlab/show/diff/path+space+object'>path object</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_288' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>P</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mover><mo>→</mo><mi>fib</mi></mover><mi>A</mi><mo>×</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'> A \stackrel{\simeq}{\to} P(A) \stackrel{fib}{\to} A \times A </annotation></semantics></math></div> <p>for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_289' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> is given by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_290' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>P</mi><mo stretchy='false'>(</mo><mi>A</mi><mo stretchy='false'>)</mo><mo>:</mo><mo>=</mo><mi>A</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msubsup><mi>Ω</mi> <mi>poly</mi> <mo>•</mo></msubsup><mo stretchy='false'>(</mo><mi>Δ</mi><mo stretchy='false'>[</mo><mn>1</mn><mo stretchy='false'>]</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> P(A) := A \otimes_k \Omega^\bullet_{poly}(\Delta[1]) </annotation></semantics></math></div></div> <p>This follows along the above lines. The statement appears for instance as (<a href='#Behrend'>Behrend, lemma 1.19</a>).</p> <h4 id='RelationToAInfinityAlgebras'>Relation to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_291' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>H \mathbb{Z}</annotation></semantics></math>-algebra spectra</h4> <p>For every <a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a> <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_292' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> there is the notion of <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectra</a> over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_293' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>. Let <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_294' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo>:</mo><mo>=</mo><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>R := H \mathbb{Z}</annotation></semantics></math> be the <a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+spectrum'>Eilenberg-MacLane spectrum</a> for the <a class='existingWikiWord' href='/nlab/show/diff/integer'>integer</a>s. Then unbounded dg-algebras (over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_295' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>\mathbb{Z}</annotation></semantics></math>) are one model for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_296' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>H \mathbb{Z}</annotation></semantics></math>-algebra spectra.</p> <div class='num_prop'> <h6 id='proposition_14'>Proposition</h6> <p>There is a <a class='existingWikiWord' href='/nlab/show/diff/Quillen+equivalence'>Quillen equivalence</a> between the standard <a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a> structure for <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_297' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>H</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>H \mathbb{Z}</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectra</a> and the model structure on unbounded differential graded algebras.</p> </div> <p>See <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a> for details.</p> <h4 id='RelationToEInfinityAlgebras'>Relation to <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_298' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝔼</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>\mathbb{E}_\infty</annotation></semantics></math>-algebras</h4> <p>Commutative dg-algebras over a field <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_299' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> of characteristic 0 constitute a presentation of <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-infinity algebras</a> over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_300' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> ([Lurie, prop. A.7.1.4.11]).</p> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+chain+complexes'>model structure on chain complexes</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-modules'>model structure on dg-modules</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-operads'>model structure on dg-operads</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras+over+an+operad'>model structure on dg-algebras over an operad</a></p> <ul> <li> <p><strong>model structure on dg-algebras</strong></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-categories'>model structure on dg-categories</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-coalgebras'>model structure on dg-coalgebras</a></p> </li> </ul> <h2 id='references'>References</h2> <h3 id='on_connective_dgcalgebras_2'>On connective dgc-algebras</h3> <p>The cofibrantly generated model structure on <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative algebras</a> is originally due to</p> <ul> <li id='BousfieldGugenheim76'><a class='existingWikiWord' href='/nlab/show/diff/Aldridge+Bousfield'>Aldridge Bousfield</a>, <a class='existingWikiWord' href='/nlab/show/diff/Victor+Gugenheim'>Victor Gugenheim</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/On+PL+deRham+theory+and+rational+homotopy+type'>On PL deRham theory and rational homotopy type</a></em>, Memoirs of the AMS 179 (1976) (<a href='https://bookstore.ams.org/memo-8-179'>ams:memo-8-179</a>)</li> </ul> <p>Textbook account:</p> <ul> <li id='GelfandManin96'><a class='existingWikiWord' href='/nlab/show/diff/Sergei+Gelfand'>Sergei Gelfand</a>, <a class='existingWikiWord' href='/nlab/show/diff/Yuri+Manin'>Yuri Manin</a>, Chapter V of: <em><a class='existingWikiWord' href='/nlab/show/diff/Methods+of+homological+algebra'>Methods of homological algebra</a></em>, transl. from the 1988 Russian (Nauka Publ.) original, Springer 1996. xviii+372 pp. 2nd corrected ed. 2002 (<a href='https://doi.org/10.1007/978-3-662-12492-5'>doi:10.1007/978-3-662-12492-5</a>)</li> </ul> <p>Review:</p> <ul> <li id='Hess06'> <p><a class='existingWikiWord' href='/nlab/show/diff/Kathryn+Hess'>Kathryn Hess</a>, p. 6 of: <em>Rational homotopy theory: a brief introduction</em>, contribution to <em><a href='https://jdc.math.uwo.ca/summerschool/'>Summer School on Interactions between Homotopy Theory and Algebra</a></em>, University of Chicago, July 26-August 6, 2004, Chicago (<a href='http://arxiv.org/abs/math.AT/0604626'>arXiv:math.AT/0604626</a>), chapter in Luchezar Lavramov, <a class='existingWikiWord' href='/nlab/show/diff/J.+Daniel+Christensen'>Dan Christensen</a>, <a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+Mandell'>Michael Mandell</a>, <a class='existingWikiWord' href='/nlab/show/diff/Brooke+Shipley'>Brooke Shipley</a> (eds.), <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007 (<a href='http://dx.doi.org/10.1090/conm/436'>doi:10.1090/conm/436</a>)</p> </li> <li id='Moerman15'> <p><a class='existingWikiWord' href='/nlab/show/diff/Joshua+Moerman'>Joshua Moerman</a>, Section II of: <em>Rational Homotopy Theory</em>, 2015 (<a href='https://www.ru.nl/publish/pages/813282/rational_homotopy_theory.pdf'>pdf</a>, <a class='existingWikiWord' href='/nlab/files/MoermanRationalHomotopyTheory.pdf' title='pdf'>pdf</a>)</p> </li> </ul> <p>This makes use of the general discussion in section 3 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/diff/Kristen+Schemmerhorn'>Kirsten Schemmerhorn</a>, <em>Model categories and simplicial methods</em>, Notes from lectures given at the University of Chicago, August 2004, in: <em>Interactions between Homotopy Theory and Algebra</em>, Contemporary Mathematics 436, AMS 2007(<a href='http://arxiv.org/abs/math.AT/0609537'>arXiv:math.AT/0609537</a>, <a href='http://dx.doi.org/10.1090/conm/436'>doi:10.1090/conm/436</a>)</li> </ul> <p>that obtains the model structure from the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+chain+complexes'>model structure on chain complexes</a>.</p> <h3 id='on_noncommutative_dgalgebras'>On non-commutative dg-algebras</h3> <p>For general <strong>non-commutative</strong> (or rather: not necessarily graded-commutative) dg-algebras a model structure is given in</p> <ul> <li id='Jardine97'><a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>J. F. Jardine</a>, <em><a class='existingWikiWord' href='/nlab/files/JardineModelDG.pdf' title='A Closed Model Structure for Differential Graded Algebras'>A Closed Model Structure for Differential Graded Algebras</a></em>, Cyclic Cohomology and Noncommutative Geometry, Fields Institute Communications, Vol. 17, AMS (1997), 55-58.</li> </ul> <p>This is also the structure used in</p> <ul> <li>J.L. Castiglioni G. Cortiñas, <em>Cosimplicial versus DG-rings: a version of the Dold-Kan correspondence</em> (<a href='http://arxiv.org/abs/math/0306289v2'>arXiv</a>)</li> </ul> <p>where aspects of its relation to the <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+cosimplicial+rings'>model structure on cosimplicial rings</a> is discussed. (See <a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a> for more on this).</p> <h3 id='on_unbounded_dgalgebras'>On unbounded dg-algebras</h3> <p>Discussion of the model structure on unbounded dg-algebras over a field of characteristic 0 is in</p> <ul> <li id='ToenVezzosi'><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+To%C3%ABn'>Bertrand Toën</a>, <a class='existingWikiWord' href='/nlab/show/diff/Gabriele+Vezzosi'>Gabriele Vezzosi</a>, <em>HAG II, geometric stacks and applicatons</em> (<a href='http://arxiv.org/abs/math/0404373v4'>arXiv:math/0404373v4</a>)</li> </ul> <p>A general discussion of algebras over an operad in unbounded chain complexes is in</p> <ul> <li id='Hinich'><a class='existingWikiWord' href='/nlab/show/diff/Vladimir+Hinich'>Vladimir Hinich</a>, <em>Homological algebra of homotopy algebras</em> Communications in algebra, 25(10). 3291-3323 (1997)(<a href='http://arxiv.org/abs/q-alg/9702015'>arXiv:q-alg/9702015</a>, <em>Erratum</em> (<a href='http://arxiv.org/abs/math/0309453'>arXiv:math/0309453</a>))</li> </ul> <p>A survey of some useful facts with an eye towards <a class='existingWikiWord' href='/nlab/show/diff/dg-geometry'>dg-geometry</a> is in</p> <ul> <li id='Behrend'><a class='existingWikiWord' href='/nlab/show/diff/Kai+Behrend'>Kai Behrend</a>, <em>Differential graded schemes I: prefect resolving algebras</em> (<a href='http://arxiv.org/abs/math/0212225'>arXiv:0212225</a>)</li> </ul> <p>Discussion of cofibrations in unbounded dg-algebras are in</p> <ul> <li id='Keller'><a class='existingWikiWord' href='/nlab/show/diff/Bernhard+Keller'>Bernhard Keller</a>, <em><math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_301' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>A_\infty</annotation></semantics></math>-algebras, modules and functor categories</em> (<a href='http://www.math.jussieu.fr/~keller/publ/ainffun.pdf'>pdf</a>)</li> </ul> <h3 id='more'>More</h3> <p>The derived copowering of unbounded commutative dg-algebras over <math class='maruku-mathml' display='inline' id='mathml_2892a7b1a9158878330b0f2ea4c24b75a0c73d80_302' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>sSet</mi></mrow><annotation encoding='application/x-tex'>sSet</annotation></semantics></math> is discussed (somewhat implicitly) in</p> <ul> <li id='GinotTradlerZeinalian'><a class='existingWikiWord' href='/nlab/show/diff/Gr%C3%A9gory+Ginot'>Grégory Ginot</a>, Thomas Tradler, Mahmoud Zeinalian, <em>Derived higher Hochschild homology, topological chiral homology and factorization algebras</em>, (<a href='http://arxiv.org/abs/1011.6483'>arxiv/1011.6483</a>)</li> </ul> <p>The <em>commutative</em> product on the dg-algebra of the higher order Hochschild complex is discussed in</p> <ul id='GinotTradlerZeinalianChenModel'> <li><a class='existingWikiWord' href='/nlab/show/diff/Gr%C3%A9gory+Ginot'>Grégory Ginot</a>, Thomas Tradler, Mahmoud Zeinalian, <em>A Chen model for mapping spaces and the surface product</em> (<a href='http://arxiv.org/PS_cache/arxiv/pdf/0905/0905.2231v1.pdf'>pdf</a>)</li> </ul> <p>The relation to <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-infinity algebras</a> is discussed in</p> <ul> <li id='KrizMay95'> <p><a class='existingWikiWord' href='/nlab/show/diff/Igor+Kr%CC%8Ciz%CC%8C'>Igor Kriz</a> and <a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <em>Operads, algebras, modules and motives</em> , Astérisque No 233 (1995)</p> </li> <li id='Lurie'> <p><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, section 7.1 of <em>Higher algebra</em> (<a href='http://www.math.harvard.edu/~lurie/papers/higheralgebra.pdf'>pdf</a>)</p> </li> </ul> <p>The relation between commutative and non-commutative dgas is further discussed in</p> <ul> <li id='Amrani14'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ilias+Amrani'>Ilias Amrani</a>, <em>Comparing commutative and associative unbounded differential graded algebras over Q from homotopical point of view</em> (<a href='http://arxiv.org/abs/1401.7285'>arXiv:1401.7285</a>)</p> </li> <li id='Amrani14b'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ilias+Amrani'>Ilias Amrani</a>, <em>Rational homotopy theory of function spaces and Hochschild cohomology</em> (<a href='http://arxiv.org/abs/1406.6269'>arXiv:1406.6269</a>)</p> </li> </ul> <p>For more see also at <em><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+dg-algebras+over+an+operad'>model structure on dg-algebras over an operad</a></em>.</p> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy limits</a> and <a class='existingWikiWord' href='/nlab/show/diff/homotopy+limit'>homotopy colimits</a> of dg-algebras is in</p> <ul> <li id='Walter06'><a class='existingWikiWord' href='/nlab/show/diff/Ben+Walter'>Ben Walter</a>, <em>Rational Homotopy Calculus of Functors</em> (<a href='http://arxiv.org/abs/math/0603336'>arXiv:math/0603336</a>)</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p> </div>  <div class="revisedby"> <p> Revision on August 23, 2020 at 09:34:44 by <a href="/nlab/author/Urs+Schreiber" style="color: #005c19">Urs Schreiber</a> See the <a href="/nlab/history/model+structure+on+dg-algebras" style="color: #005c19">history</a> of this page for a list of all contributions to it. </p> </div> <div class="navigation navfoot"> <a href="https://nforum.ncatlab.org/discussion/356/#Item_21">Discuss</a><span class="backintime"><a href="/nlab/revision/diff/model+structure+on+dg-algebras/71" accesskey="F" 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