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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <p class="show_diff"> Showing changes from revision #37 to #38: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kac-Moody+group'>Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supergroup'>super Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></li> </ul> </div> <h4 id='category_theory'><math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+theory'>category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/higher+category+theory'>higher category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%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/diff/%28infinity%2C1%29-category'>(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/hom-object+in+a+quasi-category'>hom-objects</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/equivalence+in+an+%28infinity%2C1%29-category'>equivalences in</a>/<a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+%28infinity%2C1%29-categories'>of</a> <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sub-%28infinity%2C1%29-category'>sub-(∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/reflective+sub-%28infinity%2C1%29-category'>reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+an+%28infinity%2C1%29-category'>reflective localization</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/opposite+quasi-category'>opposite (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/over-%28infinity%2C1%29-category'>over (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/join+of+quasi-categories'>join of quasi-categories</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-functor'>(∞,1)-functor</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/flat+%28infinity%2C1%29-functor'>exact (∞,1)-functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-functors'>(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+of+%28infinity%2C1%29-presheaves'>(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/fibration+of+quasi-categories'>fibrations</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inner+fibration'>inner fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/right%2Fleft+Kan+fibration'>left/right fibration</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Cartesian+fibration'>Cartesian fibration</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/%28%E2%88%9E%2C1%29-limit'>limit</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/terminal+object+in+a+quasi-category'>terminal object</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>locally presentable</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/essentially+small+%28infinity%2C1%29-category'>essentially small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+small+%28infinity%2C1%29-category'>locally small</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/accessible+%28infinity%2C1%29-category'>accessible</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/idempotent+complete+%28infinity%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/diff/Yoneda+lemma+for+%28infinity%2C1%29-categories'>(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-Grothendieck+construction'>(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor+theorem'>adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/stable+%28infinity%2C1%29-category'>stable (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-topos'>(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+weak+equivalences'>category with weak equivalences</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+category'>model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derivator'>derivator</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/quasi-category'>quasi-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+quasi-categories'>model structure for quasi-categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+for+Cartesian+fibrations'>model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/relation+between+quasi-categories+and+simplicial+categories'>relation to simplicial categories</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homotopy+coherent+nerve'>homotopy coherent nerve</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+model+category'>simplicial model category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/locally+presentable+%28infinity%2C1%29-category'>presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kan+complex'>Kan complex</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+sets'>model structure for Kan complexes</a></li> </ul> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a></li><li><a href='#models'>Models</a></li><li><a href='#related_concepts_and_examples'>Related concepts and Examples</a></li><li><a href='#References'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>An <strong>∞-group</strong> is a <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object</a> in <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a>.</p> <p>Equivalently (by the <a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a>) it is a <a class='existingWikiWord' href='/nlab/show/diff/pointed+object'>pointed</a> <a class='existingWikiWord' href='/nlab/show/diff/connected+object'>connected</a> <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/infinity-groupoid'>groupoid</a>.</p> <p>Under the <a class='existingWikiWord' href='/nlab/show/diff/homotopy+hypothesis'>identification</a> of <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> with <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a> this is known as a grouplike <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>A</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>A_\infty</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>space</a>, for instance.</p> <p>An <strong><math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Lie group</strong> is accordingly a group object in <a class='existingWikiWord' href='/nlab/show/diff/Lie+n-groupoid'>∞-Lie groupoid</a>s. And so on.</p> <h2 id='properties'>Properties</h2> <p>For details see <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>groupoid object in an (∞,1)-category</a>.</p> <h2 id='models'>Models</h2> <p>By</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+simplicial+groups'>model structure on simplicial groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+reduced+simplicial+sets'>model structure on reduced simplicial sets</a></p> </li> </ul> <h2 id='related_concepts_and_examples'>Related concepts and Examples</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braided+2-group'>braided 2-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pointed+connected+groupoid'>pointed connected groupoid</a></p> </li> <li> <p><strong><math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-group</strong>, <a class='existingWikiWord' href='/nlab/show/diff/braided+infinity-group'>braided ∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/automorphism+infinity-group'>automorphism ∞-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/bisection+of+a+Lie+groupoid'>∞-group of bisections</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+of+units'>∞-group of units</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Picard+infinity-group'>Picard ∞-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Brauer+%E2%88%9E-group'>Brauer ∞-group</a></li> </ul> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/p-compact+group'>p-compact group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/center+of+an+infinity-group'>center of an ∞-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/normal+subgroup'>normal morphism of ∞-groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/stabilizer+group'>stabilizer ∞-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/outer+automorphism+infinity-group'>outer automorphism ∞-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Eilenberg-Mac+Lane+object'>Eilenberg-MacLane object</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-gerbe'>∞-gerbe</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%E2%88%9E-group+cohomology'>∞-group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/augmented+infinity-group'>augmented ∞-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+completion'>∞-group completion</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+%E2%88%9E-group'>finite ∞-group</a></p> </li> </ul> </li> <li> <p><span class='newWikiWord'>free infinity-group type<a href='/nlab/new/free+infinity-group+type'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> </ul> <table><thead><tr><th /><th><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-operad'>(∞,1)-operad</a></th><th><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra</a></th><th>grouplike version</th><th>in <a class='existingWikiWord' href='/nlab/show/diff/Top'>Top</a></th><th>generally</th></tr></thead><tbody><tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/A-infinity+operad'>A-∞ operad</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a>, e.g. <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/loop+space+object'>loop space object</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>E-k operad</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/En-algebra'>E-k algebra</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/abelian+infinity-group'>k-monoidal ∞-group</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/iterated+loop+space'>iterated loop space</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/iterated+loop+space+object'>iterated loop space object</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/E-infinity+operad'>E-∞ operad</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/abelian+infinity-group'>abelian ∞-group</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/E-infinity+space'>E-∞ space</a>, if grouplike: <a class='existingWikiWord' href='/nlab/show/diff/infinite+loop+space'>infinite loop space</a> <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/Gamma-space'>∞-space</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/infinite+loop+space+object'>infinite loop space object</a></td></tr> <tr><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/connective+spectrum'>connective spectrum</a></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>≃</mo></mrow><annotation encoding='application/x-tex'>\simeq</annotation></semantics></math> <a class='existingWikiWord' href='/nlab/show/diff/infinite+loop+space+object'>connective spectrum object</a></td></tr> <tr><td style='text-align: left;'><strong><a class='existingWikiWord' href='/nlab/show/diff/stabilization'>stabilization</a></strong></td><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;' /><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></td><td style='text-align: left;'><a class='existingWikiWord' href='/nlab/show/diff/spectrum+object'>spectrum object</a></td></tr> </tbody></table> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/looping'>looping and delooping</a>, <a class='existingWikiWord' href='/nlab/show/diff/stabilization+hypothesis'>stabilization hypothesis</a></li> </ul> <h2 id='References'>References</h2> <p>(For more see also the references at <em><a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>infinity-action</a></em>.)</p> <p>A standard textbook reference on <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groups in the <a class='existingWikiWord' href='/nlab/show/diff/classical+model+structure+on+simplicial+sets'>classical model structure on simplicial sets</a> is</p> <ul> <li id='GoerssJardine'><a class='existingWikiWord' href='/nlab/show/diff/Paul+Goerss'>Paul Goerss</a>, <a class='existingWikiWord' href='/nlab/show/diff/John+Frederick+Jardine'>Rick Jardine</a>, chapter V of <em><a class='existingWikiWord' href='/nlab/show/diff/Simplicial+homotopy+theory'>Simplicial homotopy theory</a></em> <a href='http://www.maths.abdn.ac.uk/~bensondj/papers/g/goerss-jardine/ch-5.dvi'>chapter V</a>.</li> </ul> <p><a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>Group objects in (infinity,1)-categories</a> are the topic of</p> <ul> <li id='Lurie'><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, section 6.1.2 in <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Topos+Theory'>Higher Topos Theory</a></em></li> </ul> <del class='diffdel'><p>Model category presentations of group(oid) objects in <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding='application/x-tex'>\infty Grpd</annotation></semantics></math> by groupoidal <a class='existingWikiWord' href='/nlab/show/diff/complete+Segal+space'>complete Segal spaces</a> are discussed in</p></del><del class='diffdel'> </del><del class='diffdel'><ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Julie+Bergner'>Julia Bergner</a>,</p> <p><em>Adding inverses to diagrams encoding algebraic structures</em>, Homology, Homotopy and Applications 10 (2008), no. 2, 149–174. (<a href='http://arxiv.org/abs/math/0610291'>arXiv:0610291</a>)</p> <p><em>Adding inverses to diagrams II: Invertible homotopy theories are spaces</em>, Homology, Homotopy and Applications, Vol. 10 (2008), No. 2, pp.175-193. (<a href='http://www.intlpress.com/hha/v10/n2/a9/'>web</a>, <a href='http://arxiv.org/abs/0710.2254'>arXiv:0710.2254</a>)</p> </li> </ul></del><del class='diffdel'> </del><p>Discussion from the point of view of <a class='existingWikiWord' href='/nlab/show/diff/category+object+in+an+%28infinity%2C1%29-category'>category objects in an (∞,1)-category</a> is in</p> <ul> <li id='Lurie2'><a class='existingWikiWord' href='/nlab/show/diff/Jacob+Lurie'>Jacob Lurie</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C2%29-Categories+and+the+Goodwillie+Calculus'>(∞,2)-Categories and the Goodwillie Calculus</a></em> (<a href='http://arxiv.org/abs/0905.0462'>arXiv:0905.0462</a>)</li> </ul> <p>The <a class='existingWikiWord' href='/nlab/show/diff/homotopy+theory'>homotopy theory</a> of <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-groups that are <a class='existingWikiWord' href='/nlab/show/diff/n-connected+object+of+an+%28infinity%2C1%29-topos'>n-connected</a> and <a class='existingWikiWord' href='/nlab/show/diff/truncation'>r-truncated</a> for <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>≤</mo><mi>r</mi></mrow><annotation encoding='application/x-tex'>n \leq r</annotation></semantics></math> is discussed in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Antonio+R.+Garz%C3%B3n'>A.R. Garzón</a>, J.G. Miranda, <em>Serre homotopy theory in subcategories of simplicial groups</em>, Journal of Pure and Applied Algebra Volume 147, Issue 2, 24 March 2000, Pages 107-123 (<a href='https://doi.org/10.1016 /S0022-4049(98)00143-1'>doi:10.1016/S0022-4049(98)00143-1</a>)</li> </ul> <p>Discussion of aspects of ordinary <a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a> in relation to <math class='maruku-mathml' display='inline' id='mathml_850f3696d7f88159334b71ca75d2aa7ea7131a68_14' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-group theory:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Roman+Mikhailov'>Roman Mikhailov</a>, <em>Homotopy patterns in group theory</em>, Proceedings of the <a href='https://icm2022.org'>ICM 2022</a> (<a href='https://arxiv.org/abs/2111.00737'>arXiv:2111.00737</a>)</li> </ul> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>$\infty$-groups</a> in <a class='existingWikiWord' href='/nlab/show/diff/homotopy+type+theory'>homotopy type theory</a>:</p> <ul> <li id='BuchholtzDoornRijke18'><a class='existingWikiWord' href='/nlab/show/diff/Ulrik+Buchholtz'>Ulrik Buchholtz</a>, <a class='existingWikiWord' href='/nlab/show/diff/Floris+van+Doorn'>Floris van Doorn</a>, <a class='existingWikiWord' href='/nlab/show/diff/Egbert+Rijke'>Egbert Rijke</a>, <em><a class='existingWikiWord' href='/nlab/show/diff/Higher+Groups+in+Homotopy+Type+Theory'>Higher Groups in Homotopy Type Theory</a></em>, LICS ‘18: Proceedings of the 33rd Annual ACM/IEEE Symposium on Logic in Computer Science (2018) 205-214 [[arXiv:1802.04315](https://arxiv.org/abs/1802.04315), <a href='https://doi.org/10.1145/3209108.3209150'>doi:10.1145/3209108.3209150</a>]</li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p><div class='property'> category: <a class='category_link' href='/nlab/list/%E2%88%9E-groupoid'>∞-groupoid</a></div></p> </div> <div class="revisedby"> <p> Last revised on January 2, 2025 at 08:00:16. 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