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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> A-infinity-category </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/9429/#Item_5" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="homological_algebra">Homological algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a></strong></p> <p>(also <a class="existingWikiWord" href="/nlab/show/nonabelian+homological+algebra">nonabelian homological algebra</a>)</p> <p><em><a class="existingWikiWord" href="/schreiber/show/Introduction+to+Homological+Algebra">Introduction</a></em></p> <p><strong>Context</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enriched category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-additive+category">pre-additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pre-abelian+category">pre-abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+category">Grothendieck category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaves">abelian sheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semi-abelian+category">semi-abelian category</a></p> </li> </ul> <p><strong>Basic definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/kernel">kernel</a>, <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex">complex</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential">differential</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homology">homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+map">chain map</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homotopy">chain homotopy</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+homology+and+cohomology">chain homology and cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-isomorphism">quasi-isomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homological+resolution">homological resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">simplicial homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+homology">generalized homology</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+sequence">exact sequence</a>,</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/long+exact+sequence">long exact sequence</a>, <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+object">injective object</a>, <a class="existingWikiWord" href="/nlab/show/projective+object">projective object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/injective+resolution">injective resolution</a>, <a class="existingWikiWord" href="/nlab/show/projective+resolution">projective resolution</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/flat+resolution">flat resolution</a></p> </li> </ul> </li> </ul> <p><strong>Stable homotopy theory notions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+category">derived category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a>, <a class="existingWikiWord" href="/nlab/show/enhanced+triangulated+category">enhanced triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+model+category">stable model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pretriangulated+dg-category">pretriangulated dg-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E-category">A-∞-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+chain+complexes">(∞,1)-category of chain complexes</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+functor">derived functor</a>, <a class="existingWikiWord" href="/nlab/show/derived+functor+in+homological+algebra">derived functor in homological algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Tor">Tor</a>, <a class="existingWikiWord" href="/nlab/show/Ext">Ext</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+limit">homotopy limit</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+colimit">homotopy colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lim%5E1+and+Milnor+sequences">lim^1 and Milnor sequences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fr-code">fr-code</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+sheaf+cohomology">abelian sheaf cohomology</a></p> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/double+complex">double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Koszul-Tate+resolution">Koszul-Tate resolution</a>, <a class="existingWikiWord" href="/nlab/show/BRST-BV+complex">BRST-BV complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence">spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+filtered+complex">spectral sequence of a filtered complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectral+sequence+of+a+double+complex">spectral sequence of a double complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+spectral+sequence">Grothendieck spectral sequence</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Leray+spectral+sequence">Leray spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Serre+spectral+sequence">Serre spectral sequence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild-Serre+spectral+sequence">Hochschild-Serre spectral sequence</a></p> </li> </ul> </li> </ul> </li> </ul> <p><strong>Lemmas</strong></p> <p><a class="existingWikiWord" href="/nlab/show/diagram+chasing">diagram chasing</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/3x3+lemma">3x3 lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/four+lemma">four lemma</a>, <a class="existingWikiWord" href="/nlab/show/five+lemma">five lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/snake+lemma">snake lemma</a>, <a class="existingWikiWord" href="/nlab/show/connecting+homomorphism">connecting homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/horseshoe+lemma">horseshoe lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Baer%27s+criterion">Baer's criterion</a></p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Schanuel%27s+lemma">Schanuel's lemma</a></p> <p><strong>Homology theories</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/singular+homology">singular homology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cyclic+homology">cyclic homology</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> / <a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal</a>, <a class="existingWikiWord" href="/nlab/show/operadic+Dold-Kan+correspondence">operadic</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Moore+complex">Moore complex</a>, <a class="existingWikiWord" href="/nlab/show/Alexander-Whitney+map">Alexander-Whitney map</a>, <a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+map">Eilenberg-Zilber map</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Zilber+theorem">Eilenberg-Zilber theorem</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cochain+on+a+simplicial+set">cochain on a simplicial set</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+coefficient+theorem">universal coefficient theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/K%C3%BCnneth+theorem">Künneth theorem</a></p> </li> </ul> </div></div> <h4 id="higher_category_theory">Higher category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a></li> </ul> <h2 id="basic_concepts">Basic concepts</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a>, <a class="existingWikiWord" href="/nlab/show/coherence">coherence</a></li> <li><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></li> <li><a class="existingWikiWord" href="/nlab/show/stabilization">looping and suspension</a></li> </ul> <h2 id="basic_theorems">Basic theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/periodic+table">periodic table</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stabilization+hypothesis">stabilization hypothesis</a>-theorem</p> </li> <li> <p><a class="existingWikiWord" href="/michaelshulman/show/exactness+hypothesis">exactness hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/holographic+principle+of+higher+category+theory">holographic principle</a></p> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory+and+physics">higher category theory and physics</a></p> </li> </ul> <h2 id="models">Models</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Theta-space">Theta-space</a></li> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-category">∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2Cn%29-category">(∞,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/n-fold+complete+Segal+space">n-fold complete Segal space</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C2%29-category">(∞,2)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+quasi-category">algebraic quasi-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched category</a></li> <li><a class="existingWikiWord" href="/nlab/show/complete+Segal+space">complete Segal space</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C0%29-category">(∞,0)-category</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/algebraic+Kan+complex">algebraic Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+T-complex">simplicial T-complex</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2CZ%29-category">(∞,Z)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/n-category">n-category</a> = (n,n)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>, <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/1-category">1-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/0-category">0-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-1%29-category">(-1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28-2%29-category">(-2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-poset">n-poset</a> = <a class="existingWikiWord" href="/nlab/show/n-poset">(n-1,n)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/poset">poset</a> = <a class="existingWikiWord" href="/nlab/show/%280%2C1%29-category">(0,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a> = <a class="existingWikiWord" href="/nlab/show/%281%2C2%29-category">(1,2)-category</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/n-groupoid">n-groupoid</a> = (n,0)-category <ul> <li><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/3-groupoid">3-groupoid</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>/<a class="existingWikiWord" href="/nlab/show/decategorification">decategorification</a></li> <li><a class="existingWikiWord" href="/nlab/show/geometric+definition+of+higher+category">geometric definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></li> <li><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/simplicial+model+for+weak+%E2%88%9E-categories">simplicial model for weak ∞-categories</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/complicial+set">complicial set</a></li> <li><a class="existingWikiWord" href="/nlab/show/weak+complicial+set">weak complicial set</a></li> </ul> </li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/algebraic+definition+of+higher+category">algebraic definition of higher category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></li> <li><a class="existingWikiWord" href="/nlab/show/tricategory">tricategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/tetracategory">tetracategory</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Batanin+%E2%88%9E-category">Batanin ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble ∞-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/Grothendieck-Maltsiniotis+%E2%88%9E-categories">Grothendieck-Maltsiniotis ∞-categories</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+category">A-∞ category</a></li> <li><a class="existingWikiWord" href="/nlab/show/triangulated+category">triangulated category</a></li> </ul> </li> </ul> </li> </ul> <h2 id="morphisms">Morphisms</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-morphism">k-morphism</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/transfor">transfor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></li> <li><a class="existingWikiWord" href="/nlab/show/modification">modification</a></li> </ul> </li> </ul> <h2 id="functors">Functors</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/functor">functor</a></li> <li><a class="existingWikiWord" href="/nlab/show/2-functor">2-functor</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/pseudofunctor">pseudofunctor</a></li> <li><a class="existingWikiWord" href="/nlab/show/lax+functor">lax functor</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></li> </ul> <h2 id="universal_constructions">Universal constructions</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-limit">2-limit</a></li> <li><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">(∞,1)-adjunction</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Kan+extension">(∞,1)-Kan extension</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/limit+in+a+quasi-category">(∞,1)-limit</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></li> </ul> <h2 id="extra_properties_and_structure">Extra properties and structure</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/cosmic+cube">cosmic cube</a> <ul> <li><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-category">strict ∞-category</a>, <a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></li> </ul> </li> <li><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></li> </ul> <h2 id="1categorical_presentations">1-categorical presentations</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></li> <li><a class="existingWikiWord" href="/nlab/show/model+category">model category theory</a></li> <li><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#ordinary_linear_categories'>Ordinary linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</a></li> <li><a href='#examples_and_remarks'>Examples (and remarks)</a></li> <li><a href='#more_general_categories'>More general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#for_categories_in_the_sense_of_homological_algebra'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories in the sense of homological algebra</a></li> <li><a href='#for_categories_in_the_wider_sense'>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories in the wider sense</a></li> </ul> </ul> </div> <h2 id="idea">Idea</h2> <p>An <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category</em> is a kind of <a class="existingWikiWord" href="/nlab/show/category">category</a> in which the associativity condition on the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s is relaxed “up to higher coherent homotopy”.</p> <p>The “A” is for Associative and the “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">{}_\infty</annotation></semantics></math>” indicates that associativity is relaxed up to higher homotopies without bound on the degree of the homotopies.</p> <p>In the most widespread use of the word <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories are <em>linear</em> categories in that they have <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a>s that are <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es. These are really models/presentations for <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-categories</a>.</p> <p>If higher coherences in a linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category happen to be equal to identity morphisms (which is encoded by the vanishing of the maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">m_n</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">n\ge3</annotation></semantics></math>, defined below), it becomes the same as a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>. In fact, every linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-equivalent to a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>. In this way, we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories related to <a class="existingWikiWord" href="/nlab/show/dg-category">dg-categories</a> as models for <a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-categories</a> in roughly the same way as <a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-categories</a> relate to <a class="existingWikiWord" href="/nlab/show/simplicially+enriched+category">simplicially enriched categories</a> as models for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-categories</a>: the former is the general incarnation, while the latter is a <a class="existingWikiWord" href="/nlab/show/semi-strict+infinity-category">semi-strictified</a> version.</p> <h3 id="ordinary_linear_categories">Ordinary linear <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</h3> <p>In what is strictly speaking a restrictive sense – which is however widely and conventionally understood in <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> as the standard notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category (see references below) – the <a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a>s of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category are taken to be linear spaces, i.e. <a class="existingWikiWord" href="/nlab/show/module">module</a>s over some <a class="existingWikiWord" href="/nlab/show/ring">ring</a> or <a class="existingWikiWord" href="/nlab/show/field">field</a>, and in fact <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of such modules.</p> <p>Therefore an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category in this standard sense of <a class="existingWikiWord" href="/nlab/show/homological+algebra">homological algebra</a> is a category which is in some way <a class="existingWikiWord" href="/nlab/show/homotopical+enrichment">homotopically enriched</a> over a <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes">category of chain complexes</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ch</mi></mrow><annotation encoding="application/x-tex">Ch</annotation></semantics></math>. Since a category which is <a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> in the ordinary sense of <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> is a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>, there is a close relation between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories and <a class="existingWikiWord" href="/nlab/show/dg-category">dg-categories</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories in this linear sense are a <a class="existingWikiWord" href="/nlab/show/horizontal+categorification">horizontal categorification</a> of the notion of <a class="existingWikiWord" href="/nlab/show/A-infinity-algebra">A-infinity-algebra</a>. As such they are to <a class="existingWikiWord" href="/nlab/show/A-infinity-algebra">A-infinity-algebra</a>s as <a class="existingWikiWord" href="/nlab/show/Lie+infinity-algebroid">Lie infinity-algebroid</a>s are to <a class="existingWikiWord" href="/nlab/show/L-infinity-algebra">L-infinity-algebra</a>s. For this point of view see <a href="#KonsevichSoibelman08">Konsevich-Soibelman 08</a>.</p> <div class="num_defn"> <h6 id="definition">Definition</h6> <p>A category <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> such that</p> <ol> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>,</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(C)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/hom-set">Hom-set</a>s <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_C(X,Y)</annotation></semantics></math> are finite dimensional <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Z</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Z}</annotation></semantics></math>-graded modules</p> </li> <li> <p>for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">X_1,...,X_n</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ob</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ob(C)</annotation></semantics></math> there is a family of linear composition maps (the higher compositions) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>n</mi></msub><mo>:</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>⋯</mi><mo>⊗</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>,</mo><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">m_n : Hom_C(X_0,X_1) \otimes Hom_C(X_1,X_2) \otimes \cdots \otimes Hom_C(X_{n-1},X_n) \to Hom_C(X_0,X_n)</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>−</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n-2</annotation></semantics></math> (homological grading convention is used) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n\geq1</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">m_1</annotation></semantics></math> is the differential on the chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_C(X,Y)</annotation></semantics></math></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">m_n</annotation></semantics></math> satisfy the quadratic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-associativity equation for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">n\geq0</annotation></semantics></math>.</p> </li> </ol> </div> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">m_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">m_2</annotation></semantics></math> will be <a class="existingWikiWord" href="/nlab/show/chain+complex">chain map</a>s but the compositions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>m</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">m_i</annotation></semantics></math> of higher order are not chain maps, nevertheless they are <a class="existingWikiWord" href="/nlab/show/Massey+product">Massey product</a>s.</p> <p>The framework of dg-categories and dg-functors is too narrow for many problems, and it is preferable to consider the wider class of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-functors. Many features of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-functors come from the fact that they form a symmetric closed <a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a>, which is revealed in the language of <a class="existingWikiWord" href="/nlab/show/comonad">comonads</a>.</p> <p>From a higher dimensional perspective <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories are weak <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>-categories with all morphisms invertible. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories can also be viewed as noncommutative formal dg-manifolds with a closed marked subscheme of objects.</p> <h3 id="examples_and_remarks">Examples (and remarks)</h3> <ul> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a> may be regarded as a special case when there are no higher maps (trivial homotopies) of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category.</p> </li> <li> <p>Every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-equivalent to a <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a>.</p> <ul> <li> <p>This is a corollary of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categorical <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>.</p> </li> <li> <p>beware that this statement does not imply that the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories is obsolete (see section 1.8 in Bespalov et al.): in practice it is often easier to work with a given naturally arising <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category than constructing its equivalent <a class="existingWikiWord" href="/nlab/show/dg-category">dg-category</a></p> <ul> <li> <p>for instance when dealing with a <span class="newWikiWord">Fukaya<a href="/nlab/new/Fukaya+A-infinity-category">?</a></span> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category;</p> </li> <li> <p>or when dealing with various constructions on dg-categories, for instance certain quotients,that naturally yield directly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories instead of dg-categories.</p> </li> </ul> </li> </ul> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/path+space">path space</a> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fuk</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Fuk(X)</annotation></semantics></math> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> – a <a class="existingWikiWord" href="/nlab/show/Calabi-Yau+A-%E2%88%9E+category">Calabi-Yau A-∞ category</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity-algebra">algebras</a> are the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories with one object.</p> <ul> <li>For example, the delooping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\Omega{X}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/loop+space">loop space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ω</mi><mi>X</mi></mrow><annotation encoding="application/x-tex">\Omega{X}</annotation></semantics></math> of a topological space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math></li> </ul> </li> </ul> <h3 id="more_general_categories">More general <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</h3> <p>In the widest sense, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category may be used as a term for a category in which the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation constitutes an algebra over an <a class="existingWikiWord" href="/nlab/show/operad">operad</a> which resolves in some sense the associative operad <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ass</mi></mrow><annotation encoding="application/x-tex">Ass</annotation></semantics></math>.</p> <p>One should be aware, though, that this use of the term is not understood by default in the large body of literature concerned with the above linear notion.</p> <p>A less general but non-linear definition is fairly straight forward in any category in which there is a notion of <em>homotopy</em> with the usual properties.</p> <div class="un_defn"> <h6 id="definition_2">Definition</h6> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category is a <a class="existingWikiWord" href="/nlab/show/category+over+an+operad">category over</a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/A-infinity+operad">operad</a>: e.g. the free resolution in the context of dg-operads of the linear associative operad.</p> </div> <div class="num_example"> <h6 id="examples">Examples</h6> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Trimble+n-category">Trimble n-categories</a>;</p> </li> <li> <p>also the classical notion of <a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a> can be interpreted as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> for a suitable Cat-operad.</p> </li> </ul> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a>, <a class="existingWikiWord" href="/nlab/show/associahedron">associahedron</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="for_categories_in_the_sense_of_homological_algebra">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories in the sense of homological algebra</h3> <p>For a short and precise introduction see</p> <ul> <li> <p>B. Keller, <em>Introduction to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-algebras and modules</em> (<a href="http://people.math.jussieu.fr/~keller/publ/ioan.dvi">dvi</a>, <a href="http://people.math.jussieu.fr/~keller/publ/ioan.ps">ps</a>) and Addendum (<a href="http://people.math.jussieu.fr/~keller/publ/addendum.ps">ps</a>), Homology, Homotopy and Applications 3 (2001), 1-35;</p> </li> <li> <p>B. Keller, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math> algebras, modules and functor categories</em>, (<a href="http://people.math.jussieu.fr/~keller/publ/ainffun.dvi">pdf</a>, <a href="http://people.math.jussieu.fr/~keller/publ/ainffun.ps">ps</a>).</p> </li> </ul> <p>and for a <a class="existingWikiWord" href="/nlab/show/Fukaya+category">Fukaya category</a>-oriented introduction see chapter 1 in</p> <ul> <li>P. Seidel, <em>Fukaya category and Picard-Lefschetz theory</em></li> </ul> <p>A very detailed treatment of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories is a recent book</p> <ul> <li> <p>Yu. Bespalov, <a class="existingWikiWord" href="/nlab/show/Volodymyr+Lyubashenko">Volodymyr Lyubashenko</a>, O. Manzyuk, <em>Pretriangulated <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories</em>, Proceedings of the Institute of Mathematics of NAS of Ukraine, vol. 76, Institute of Mathematics of NAS of Ukraine, Kyiv, 2008, 598 (<a href="http://www.math.ksu.edu/~lub/cmcMonad.gz">ps.gz</a>)</p> <ul> <li>notice: the ps.gz file has different page numbers than the printed version, but the numbering of sections and formulae is final. Errata to published version are <a href="http://www.math.ksu.edu/~lub/cmcMoCor.pdf">here</a>.</li> </ul> </li> <li> <p>Oleksandr Manzyk, <em>A-infinity-bimodules and Serre A-infinity-functors</em>, dissertation <a href="https://kluedo.ub.uni-kl.de/files/1910/dissertation.pdf">pdf</a>, <a href="https://kluedo.ub.uni-kl.de/files/1910/dissertation.djvu">djvu</a>; <em>Serre <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math> functors</em>, talk at Categories in geometry and math. physics, Split 2007, slides, <a href="http://www.irb.hr/korisnici/zskoda/manzyukslides.pdf">pdf</a>, work with <a class="existingWikiWord" href="/nlab/show/Volodymyr+Lyubashenko">Volodymyr Lyubashenko</a></p> </li> </ul> <p>The relation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories to <a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a>s is emphasized in the introduction of</p> <ul> <li id="KonsevichSoibelman08"><a class="existingWikiWord" href="/nlab/show/Maxim+Kontsevich">Maxim Kontsevich</a>, <a class="existingWikiWord" href="/nlab/show/Yan+Soibelman">Yan Soibelman</a>, <em>Notes on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-algebras, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories and non-commutative geometry. I</em>, in <a class="existingWikiWord" href="/nlab/show/Anton+Kapustin">Anton Kapustin</a>, <a class="existingWikiWord" href="/nlab/show/Maximilian+Kreuzer">Maximilian Kreuzer</a>, <a class="existingWikiWord" href="/nlab/show/Karl-Georg+Schlesinger">Karl-Georg Schlesinger</a> (eds.) <em>Homological mirror symmetry – New developments and perspectives</em>, Springer 2008 (<a href="http://arxiv.org/abs/math/0606241">arXiv:math/0606241</a>, <a href="https://www.springer.com/de/book/9783540680291">doi:10.1007/978-3-540-68030-7</a>)</li> </ul> <p>which mostly discusses just <a class="existingWikiWord" href="/nlab/show/A-infinity-algebra">A-infinity-algebra</a>s, but points out a generalizations to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories, see the overview on <a href="http://arxiv.org/PS_cache/math/pdf/0606/0606241v2.pdf#page=3">p. 3</a></p> <p>Essentially the authors say that an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category should be a non(-graded-)commutative <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a>/<a class="existingWikiWord" href="/nlab/show/L-infinity-algebroid">L-infinity-algebroid</a>.</p> <p>More <a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a> and <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories is discussed in</p> <ul> <li id="LefevreHasegawa03"> <p>Kenji Lefèvre-Hasegawa, <em>Sur les A-infini catégories</em> (<a href="http://arxiv.org/abs/math/0310337">arXiv:math/0310337</a>)</p> </li> <li> <p><span class="newWikiWord">Bruno Valette<a href="/nlab/new/Bruno+Valette">?</a></span>, <em>Homotopy theory of homotopy algebras</em>, Annales de l’Institut Fourier <strong>70</strong>:2 (2020) 683–738 <a href="https://10.5802/aif.3322">doi</a> <a href="https://zbmath.org/?q=an:1335.18001">Zbl:1335.18001</a> <a href="https://arxiv.org/abs/1411.5533">arXiv:1411.5533</a></p> </li> </ul> <p>See also</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Yong-Geun+Oh">Yong-Geun Oh</a>, <a class="existingWikiWord" href="/nlab/show/Hiro+Lee+Tanaka">Hiro Lee Tanaka</a>, <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories, their <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>-categories, and their localizations</em> (<a href="https://arxiv.org/abs/2003.05806">arXiv:2003.05806</a>)</li> </ul> <h3 id="for_categories_in_the_wider_sense">For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-categories in the wider sense</h3> <p>If one understands <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">A_\infty</annotation></semantics></math>-category as “operadically defined higher category”, then relevant references would include:</p> <ul> <li>Eugenia Cheng, <em>Comparing operadic definitions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-category</em> (<a href="http://arxiv.org/abs/0809.2070">arXiv</a>)</li> </ul> <p>With operads modeled by <a class="existingWikiWord" href="/nlab/show/dendroidal+sets">dendroidal sets</a>, <a class="existingWikiWord" href="/nlab/show/n-categories">n-categories</a> for low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> viewed as objects with an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>−</mo><mn>∞</mn></mrow><annotation encoding="application/x-tex">A-\infty</annotation></semantics></math>-composition operation are discussed in section 5 of</p> <ul> <li>Andor Lucacs, <em>Cyclic Operads, Dendroidal Structures, Higher Categories</em> (<a href="http://igitur-archive.library.uu.nl/dissertations/2011-0211-200314/lukacs.pdf">pdf</a>)</li> </ul> <p>and</p> <ul> <li>Andor Lucacs, <em>Dendroidal weak 2-categories</em> (<a href="http://de.arxiv.org/abs/1304.4278">arXiv:1304.4278</a>)</li> </ul> <p>See also the references at <em><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></em>.</p> </body></html> </div> <div class="revisedby"> <p> Last revised on February 13, 2025 at 18:19:10. 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