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style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#Properties'>Properties</a></li> <li><a href='#Models'>Models</a></li> <li><a href='#examples'>Examples</a></li> <ul> <li><a href='#simplicial_1algebras'>Simplicial 1-algebras</a></li> <li><a href='#homotopy_algebras'>Homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras</a></li> <li><a href='#simplicial_theories'>Simplicial theories</a></li> <li><a href='#structuresheaves'>Structure-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves</a></li> <li><a href='#EInfty'>Symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-algebras</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In as far as an <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> or <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> is nothing but a <a class="existingWikiWord" href="/nlab/show/small+category">small category</a> with finite <a class="existingWikiWord" href="/nlab/show/product">product</a>s and an <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> for the theory a product-preserving <a class="existingWikiWord" href="/nlab/show/functor">functor</a> to <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, the notion has an evident generalization to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> and in particular to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> theory.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>An <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Lawvere theory</strong> is (given by a syntactic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category that is) an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> with finite <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-product</a>s. An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebra for the theory is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo></mrow><annotation encoding="application/x-tex">C \to </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> that preserves these products.</p> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebras+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebras over an (∞,1)-algebraic theory</a> is the full <a class="existingWikiWord" href="/nlab/show/sub-%28%E2%88%9E%2C1%29-category">sub-(∞,1)-category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op}) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">C^{op}</annotation></semantics></math> on the product-preserving <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functors</p> </div> <p>In a full <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category theoretic context this appears as <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, def. 5.5.8.8</a>. A definition in terms of <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a> and the <a class="existingWikiWord" href="/nlab/show/model+structure+on+sSet-categories">model structure on sSet-categories</a> to present <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories is in <a href="(http://www.math.muni.cz/~rosicky/papers/infty6.pdf">Ros</a>. The introduction of that article lists further and older occurences of this definition.</p> <h2 id="Properties">Properties</h2> <div class="num_prop" id="PropertiesOfInfinityCategoriesOfAlgebras"> <h6 id="proposition">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> with finite <a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">products</a>. Then</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-functor">accessible</a> <a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">localization</a> of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_{(\infty,1)}(C^{op})</annotation></semantics></math> (on the <a class="existingWikiWord" href="/nlab/show/opposite+%28%E2%88%9E%2C1%29-category">opposite</a>).</p> <p>So in particular it is a <a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable (∞,1)-category</a>.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">inft</mo><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">Alg_{(\inft)}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/compactly+generated+%28%E2%88%9E%2C1%29-category">compactly generated (∞,1)-category</a>.</p> </li> <li> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>j</mi><mo>:</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>→</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">j : C^{op} \to PSh_{(\infty,1)}(C^{op})</annotation></semantics></math> factors through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C)</annotation></semantics></math>.</p> </li> <li> <p>The full subcategory <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C)</annotation></semantics></math> is stable under <a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a>s.</p> </li> </ul> </div> <p>This is <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, prop. 5.5.8.10</a>.</p> <h2 id="Models">Models</h2> <p>There are various <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C) \hookrightarrow PSh_{(\infty,1)}(C^{op})</annotation></semantics></math>.</p> <p>Recall that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh_{(\infty,1)}(C^{op})</annotation></semantics></math> itself is modeled by the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">[</mo><mi>T</mi><mo>,</mo><mi>sSet</mi><msup><mo stretchy="false">]</mo> <mo>∘</mo></msup><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> PSh_{(\infty,1)}(C^{op}) \simeq [T, sSet]^\circ \,, </annotation></semantics></math></div> <p>where we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> and have on the right the <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor+category">enriched functor category</a> with the projective or injective model structure, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup></mrow><annotation encoding="application/x-tex">(-)^\circ</annotation></semantics></math> denoting the full enriched subcategory on fibrant-cofibrant objects.</p> <p>This says in particular that every weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>T</mi><mo>→</mo><mn>∞</mn><mi mathvariant="normal">Grp</mi></mrow><annotation encoding="application/x-tex">f : T \to \infty \mathrm{Grp}</annotation></semantics></math> is equivalent to a <em>rectified</em> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>T</mi><mo>→</mo><mi>KanCplx</mi></mrow><annotation encoding="application/x-tex">F : T \to KanCplx</annotation></semantics></math>. And <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in PSh_{(\infty,1)}(C^{op})</annotation></semantics></math> belongs to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C)</annotation></semantics></math> if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> preserves finite products <em>weakly</em> in that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>c</mi> <mi>i</mi></msub><mo>∈</mo><mi>C</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{c_i \in C\}</annotation></semantics></math> a finite collection of objects, the canonical natural morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo>×</mo><mi>⋯</mi><mo>×</mo><msub><mo lspace="0em" rspace="thinmathspace">c</mo> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>⋯</mi><mo>×</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> F(c_1 \times \cdots\times \c_n) \to F(c_1) \times \cdots \times F(c_n) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> of <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>es.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> is an ordinary category with products, hence an ordinary <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>, then such a functor is called a <strong><a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></strong>. There is a model category structure on these (see there).</p> <p>We now look at model category structure on <em>strictly</em> product preserving functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">C \to sSet</annotation></semantics></math>, which gives an equivalent model for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Alg</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Alg_{(\infty,1)}(C)</annotation></semantics></math>. See <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a>.</p> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a> with finite <a class="existingWikiWord" href="/nlab/show/product">product</a>s, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sTAlg</mi><mo>⊂</mo><mi>Func</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sTAlg \subset Func(C,sSet)</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to <a class="existingWikiWord" href="/nlab/show/sSet">sSet</a> on those functors that preserve these products.</p> <p>Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">sAlg(C)</annotation></semantics></math> carries the structure of a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">sAlg(C)_{proj}</annotation></semantics></math> where the weak equivalences and the fibrations are objectwise those in the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>.</p> </div> <p>This is due to (<a href="#Quillen">Quillen</a>).</p> <p>The inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>:</mo><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>sPSh</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub></mrow><annotation encoding="application/x-tex">i : sAlg(C) \hookrightarrow sPSh(C^{op})_{proj}</annotation></semantics></math> into the projective <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+presheaves">model structure on simplicial presheaves</a> evidently preserves fibrations and acylclic fibrations and gives a <a class="existingWikiWord" href="/nlab/show/Quillen+adjunction">Quillen adjunction</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mover><munder><mo>↪</mo><mi>i</mi></munder><mo>←</mo></mover><mi>sPSh</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sAlg(C)_{proj} \stackrel{\leftarrow}{\underset{i}{\hookrightarrow}} sPSh(C^{op}) \,. </annotation></semantics></math></div> <div class="num_prop"> <h6 id="proposition_3">Proposition</h6> <p>The total right <a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>i</mi><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>sPSh</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}i : Ho(sAlg(C)_{proj}) \to Ho(sPSh(C^{op})_{proj}) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a> and an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∈</mo><mi>sPSh</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F \in sPSh(C^{op})</annotation></semantics></math> belongs to the <a class="existingWikiWord" href="/nlab/show/essential+image">essential image</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mi>i</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}i</annotation></semantics></math> precisely if it preserves products up to <a class="existingWikiWord" href="/nlab/show/weak+homotopy+equivalence">weak homotopy equivalence</a>.</p> </div> <p>This is due to (<a href="#Bergner">Bergner</a>).</p> <p>It follows that the natural <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>sAlg</mi><mo stretchy="false">(</mo><mi>C</mi><msub><mo stretchy="false">)</mo> <mi>proj</mi></msub><msup><mo stretchy="false">)</mo> <mo>∘</mo></msup><mover><mo>→</mo><mrow></mrow></mover><msub><mi>PSh</mi> <mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (sAlg(C)_{proj})^\circ \stackrel{}{\to} PSh_{(\infty,1)}(C^{op}) </annotation></semantics></math></div> <p>is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">equivalence</a>.</p> <p>A comprehensive statement of these facts is in <a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">HTT, section 5.5.9</a>.</p> <h2 id="examples">Examples</h2> <h3 id="simplicial_1algebras">Simplicial 1-algebras</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/syntactic+category">syntactic category</a> of) an ordinary <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> (a <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>) let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mi>Alg</mi></mrow><annotation encoding="application/x-tex">T Alg</annotation></semantics></math> be the category of its ordinary algebras, the ordinary product-preserving functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex">T \to Set</annotation></semantics></math>.</p> <p>We may regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category and consider its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras. By the above discussion, these are modeled by product-presering functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">T \to sSet</annotation></semantics></math>. But this are equivalently <a class="existingWikiWord" href="/nlab/show/simplicial+object">simplicial object</a>s in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>T</mi><mo>,</mo><mi>sSet</mi><msub><mo stretchy="false">]</mo> <mo>×</mo></msub><mo>≃</mo><mi>T</mi><msup><mi>Alg</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [T, sSet]_\times \simeq T Alg^{\Delta^{op}} \,. </annotation></semantics></math></div> <p>There is a standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> and we find that simplicial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-1-algebras model <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras.</p> <h3 id="homotopy_algebras">Homotopy <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-algebras</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> an ordinary Lawvere theory, there is also a model category structure on ordinary functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>→</mo><mi>sSet</mi></mrow><annotation encoding="application/x-tex">T \to sSet</annotation></semantics></math> that preserve the products only up to weak equivalence. Such functors are called <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s.</p> <p>This model structure is equivalent to the <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> (see <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a> for details) but has the advantage that it is a left <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>.</p> <h3 id="simplicial_theories">Simplicial theories</h3> <p>There is a notion of <em>simplicial algebraic theory</em> that captures some class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebraic theories. For the moment see section 4 of (<a href="#Rezk">Rezk</a>)</p> <h3 id="structuresheaves">Structure-<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaves</h3> <p>A pre<a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a> is a (multi-sorted) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebraic theory. A <em>structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-sheaf</em> on an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}</annotation></semantics></math> in the sense of <a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a>es is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-algebra over this theory</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo>:</mo><mi>𝒯</mi><mo>→</mo><mi>𝒳</mi></mrow><annotation encoding="application/x-tex"> \mathcal{O} : \mathcal{T} \to \mathcal{X} </annotation></semantics></math></div> <p>in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒳</mi></mrow><annotation encoding="application/x-tex">\mathcal{X}</annotation></semantics></math> – a special one satisfying extra conditions that make it indeed behave like a sheaf of <em>function algebras</em> .</p> <h3 id="EInfty">Symmetric monoidal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">E_\infty</annotation></semantics></math>-algebras</h3> <p>There is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(2,1)</annotation></semantics></math>-algebraic theory whose algebras in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29Cat">(∞,1)Cat</a> are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-categories">symmetric monoidal (∞,1)-categories</a>. Hence monoids in these algebras are <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a>s (see <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+%28%E2%88%9E%2C1%29-category">monoid in a monoidal (∞,1)-category</a>).</p> <p>This is in (<a href="#Cranch">Cranch</a>). For more details see <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-algebraic+theory+of+E-infinity+algebras">(2,1)-algebraic theory of E-infinity algebras</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> / <a class="existingWikiWord" href="/nlab/show/essentially+algebraic+theory">essentially algebraic theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/2-Lawvere+theory">2-Lawvere theory</a></p> </li> <li> <p><strong>algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-theory</strong> / <a class="existingWikiWord" href="/nlab/show/essentially+algebraic+%28%E2%88%9E%2C1%29-theory">essentially algebraic (∞,1)-theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+Lawvere+theory">simplicial Lawvere theory</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="references">References</h2> <p>The model structure presentation for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras goes back all the way to</p> <ul id="Quillen"> <li><a class="existingWikiWord" href="/nlab/show/Dan+Quillen">Dan Quillen</a>, <em>Homotopical Algebra</em> Lectures Notes in Mathematics 43, SpringerVerlag, Berlin, (1967)</li> </ul> <p>A characterization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras in terms of <a class="existingWikiWord" href="/nlab/show/sifted+colimit">sifted colimit</a>s is given in</p> <ul> <li>J. Rosicky <em>On homotopy varieties</em> (<a href="http://www.math.muni.cz/~rosicky/papers/infty6.pdf">pdf</a>)</li> </ul> <p>using the incarnation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories as <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+categories">simplicially enriched categories</a>.</p> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categorical perspective on these homotopy-algebraic theories is given in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Andre+Joyal">Andre Joyal</a>, <em>The theory of quasi-categories and its applications</em>, lectures at CRM Barcelona February 2008, draft <a href="http://www.crm.cat/HigherCategories/hc2.pdf#page=44">hc2.pdf</a>_</li> </ul> <p>from page 44 on.</p> <p>A detailed account in the context of a general theory of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a> is the context of section 5.5.8 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <em><a class="existingWikiWord" href="/nlab/show/Higher+Topos+Theory">Higher Topos Theory</a></em> .</li> </ul> <p>The <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> <a class="existingWikiWord" href="/nlab/show/presentable+%28infinity%2C1%29-category">presentations</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras is studied in</p> <ul id="Rezk"> <li><a class="existingWikiWord" href="/nlab/show/Charles+Rezk">Charles Rezk</a>, <em>Every homotopy theory of simplicial algebras admits a proper model</em> (<a href="http://arxiv.org/abs/math/0003065">math/0003065</a>) ,</li> </ul> <p>where it is shown that every such model is <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalent</a> to a left <a class="existingWikiWord" href="/nlab/show/proper+model+category">proper model category</a>. The article uses a monadic definition of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebras.</p> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a>s and their strictification is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Bernard+Badzioch">Bernard Badzioch</a>, <em>Algebraic theories in homotopy theory</em> Annals of Mathematics, 155 (2002), 895-913 (<a href="http://www.jstor.org/stable/3062135">JSTOR</a>)</li> </ul> <p>and for multi-sorted theories in</p> <ul id="Bergner"> <li><a class="existingWikiWord" href="/nlab/show/Julie+Bergner">Julie Bergner</a>, <em>Rigidification of algebras over multi-sorted theories</em> , Algebraic and Geometric Topoogy 7, 2007.</li> </ul> <p>A discussion of <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a>-structures in terms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-algebraic theories is in</p> <ul id="Cranch"> <li><a class="existingWikiWord" href="/nlab/show/James+Cranch">James Cranch</a>, <em>Algebraic Theories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Categories</em> (<a href="http://arxiv.org/abs/1011.3243">arXiv</a>)</li> </ul> <p>See also</p> <ul> <li><a href="http://mathoverflow.net/questions/118500/what-is-a-simplicial-commutative-ring-from-the-point-of-view-of-homotopy-theory">MO discussion</a></li> </ul> <p>Expressed as a higher form of <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+D.+Berman">John D. Berman</a>, <em>Higher Lawvere theories</em> (<a href="https://arxiv.org/abs/1903.02991">arXiv:1903.02991</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on June 18, 2023 at 15:52:53. 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