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1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 12.2-11.5 36.6-20.7 43.4-36.4 6.7-15.7-13.7-14-21.3-7.2-9.1 8-11.9 20.5-23.6 25.1 7.5-23.7 31.8-37.6 38.4-61.4 2-7.3-.8-29.6-13-19.8-14.5 11.6-6.6 37.6-23.3 49.2z"/> <path fill="#193c78" d="M86.3 47.5c0-13-10.2-27.6-5.8-40.4 2.8-8.4 14.1-10.1 17-1 3.8 11.6-.3 26.3-1.8 38 11.7-.7 10.5-16 14.8-24.3 2.1-4.2 5.7-9.1 11-6.7 6 2.7 7.4 9.2 6.6 15.1-2.2 14-12.2 18.8-22.4 27-3.4 2.7-8 6.6-5.9 11.6 2 4.4 7 4.5 10.7 2.8 7.4-3.3 13.4-16.5 21.7-16 14.6.7 12 21.9.9 26.2-5 1.9-10.2 2.3-15.2 3.9-5.8 1.8-9.4 8.7-15.7 8.9-6.1.1-9-6.9-14.3-9-14.4-6-33.3-2-44.7-14.7-3.7-4.2-9.6-12-4.9-17.4 9.3-10.7 28 7.2 35.7 12 2 1.1 11 6.9 11.4 1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> group completion (changes) </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/diff/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/5991/#Item_2" 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"> <p class="show_diff"> Showing changes from revision #15 to #16: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <blockquote> <p>Not to be confused with <em><a class='existingWikiWord' href='/nlab/show/diff/completion+of+a+group'>completion of a group</a></em>.</p> </blockquote> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='algebra'>Algebra</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/diff/universal+algebra'>universal algebra</a></p> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/2-Lawvere+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%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/diff/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+a+Lawvere+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-algebra+over+an+%28infinity%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/diff/associated+infinity-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%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/diff/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/diff/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/A-infinity-ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/A-infinity-space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/C_%E2%88%9E-algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/L-infinity-algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/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/diff/model+structure+on+simplicial+algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/diff/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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> <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='monoid_theory'>Monoid theory</h4> <div class='hide'> <p><strong>monoid theory</strong> in <a class='existingWikiWord' href='/nlab/show/diff/algebra'>algebra</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>infinity-monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+category'>monoid object</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+%28infinity%2C1%29-category'>monoid object in an (infinity,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/semiring'>semiring</a>, <a class='existingWikiWord' href='/nlab/show/diff/rig'>rig</a>, <a class='existingWikiWord' href='/nlab/show/diff/ring'>ring</a>, <a class='existingWikiWord' href='/nlab/show/diff/associative+unital+algebra'>associative unital algebra</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>Mon</a>, <a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>CMon</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>monoid homomorphism</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trivial+monoid'>trivial monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/submonoid'>submonoid</a>, <span class='newWikiWord'>quotient monoid<a href='/nlab/new/quotient+monoid'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/divisor'>divisor</a>, <span class='newWikiWord'>multiple<a href='/nlab/new/multiple'>?</a></span>, <span class='newWikiWord'>quotient element<a href='/nlab/new/quotient+element'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/inverse'>inverse element</a>, <a class='existingWikiWord' href='/nlab/show/diff/unit'>unit</a>, <a class='existingWikiWord' href='/nlab/show/diff/irreducible+element'>irreducible element</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ideal+in+a+monoid'>ideal in a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+ideal+of+a+monoid'>principal ideal in a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoid</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+commutative+monoids'>tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cancellative+monoid'>cancellative monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/GCD+monoid'>GCD monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unique+factorization+monoid'>unique factorization monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/B%C3%A9zout+monoid'>Bézout monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/principal+ideal+monoid'>principal ideal monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/absorption+monoid'>absorption monoid</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/zero-divisor'>zero divisor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/integral+monoid'>integral monoid</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/free+monoid'>free monoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/free+commutative+monoid'>free commutative monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/graphic+category'>graphic monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/action'>monoid action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module+over+a+monoid'>module over a monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+a+monoid'>localization of a monoid</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/endomorphism'>endomorphism monoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+commutative+monoid'>super commutative monoid</a></p> </li> </ul> <div> <p> <a href='/nlab/edit/monoid+theory+-+contents'>Edit this sidebar</a> </p> </div></div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='idea'>Idea</h2> <p>The <a class='existingWikiWord' href='/nlab/show/diff/forgetful+functor'>forgetful functor</a> <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>U</mi></mrow><annotation encoding='application/x-tex'>U</annotation></semantics></math> from <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a> to <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoids</a> has a <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left adjoint</a> <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>. This is called <em>group completion</em>. A standard presentation of the group completion is the <em><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+group+of+a+commutative+monoid'>Grothendieck group of a commutative monoid</a></em>. As such group completion plays a central role in the definition of <a class='existingWikiWord' href='/nlab/show/diff/K-theory'>K-theory</a>.</p> <p>More generally in <a class='existingWikiWord' href='/nlab/show/diff/%28infinity%2C1%29-category+theory'>(∞,1)-category theory</a> and <a class='existingWikiWord' href='/nlab/show/diff/higher+algebra'>higher algebra</a> there is the <a class='existingWikiWord' href='/nlab/show/diff/left+adjoint'>left</a> <a class='existingWikiWord' href='/nlab/show/diff/adjoint+%28infinity%2C1%29-functor'>adjoint (∞,1)-functor</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>K</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>CMon</mi> <mn>∞</mn></msub><mo stretchy='false'>(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy='false'>)</mo><mo>⟶</mo><msub><mi>AbGrp</mi> <mn>∞</mn></msub><mo stretchy='false'>(</mo><mn>∞</mn><mi>Grpd</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> K \;\colon\; CMon_\infty(\infty Grpd) \longrightarrow AbGrp_\infty(\infty Grpd) </annotation></semantics></math></div> <p>to the inclusion of <a class='existingWikiWord' href='/nlab/show/diff/abelian+infinity-group'>abelian ∞-groups</a> (<a class='existingWikiWord' href='/nlab/show/diff/connective+spectrum'>connective spectra</a>) into <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category'>commutative ∞-monoids</a> in <a class='existingWikiWord' href='/nlab/show/diff/Infinity-Grpd'>∞Grpd</a> (<a class='existingWikiWord' href='/nlab/show/diff/E-infinity+space'>E-∞ spaces</a>). This may be called <em><math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-group completion</em>.</p> <p>This serves to define <a class='existingWikiWord' href='/nlab/show/diff/K-theory+of+a+symmetric+monoidal+%28%E2%88%9E%2C1%29-category'>algebraic K-theory of symmetric monoidal (∞,1)-categories</a>.</p> <p>In <a class='existingWikiWord' href='/nlab/show/diff/algebraic+topology'>algebraic topology</a> a key role was played by models of this process at least on <a class='existingWikiWord' href='/nlab/show/diff/H-space'>H-spaces</a> (<a href='#Quillen71'>Quillen 71</a>, <a href='#May'><span><del class='diffmod'> May,</del><ins class='diffmod'> May</ins><ins class='diffins'> 1974,</ins> def. 1.3</span></a>). If <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/diff/topological+monoid'>topological monoid</a>, let <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>B</mi><mi>N</mi></mrow><annotation encoding='application/x-tex'>B N</annotation></semantics></math> denotes its <a class='existingWikiWord' href='/nlab/show/diff/bar+construction'>bar construction</a> (“<a class='existingWikiWord' href='/nlab/show/diff/classifying+space'>classifying space</a>”) and <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_7' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ω</mi><mi>B</mi><mi>N</mi></mrow><annotation encoding='application/x-tex'>\Omega B N</annotation></semantics></math> the <a class='existingWikiWord' href='/nlab/show/diff/loop+space'>loop space</a> of that. Then this</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi><mo>⟶</mo><mi>Ω</mi><mi>B</mi><mi>N</mi></mrow><annotation encoding='application/x-tex'> N \longrightarrow \Omega B N </annotation></semantics></math></div> <p>represents the group completion of <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_9' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>N</mi></mrow><annotation encoding='application/x-tex'>N</annotation></semantics></math> (<a href='#Quillen71'>Quillen 71, section 9</a>, <a href='#May'><span><del class='diffmod'> May,</del><ins class='diffmod'> May</ins><ins class='diffins'> 1974,</ins> theorem 1.6</span></a>). This crucially enters the construction of the <a class='existingWikiWord' href='/nlab/show/diff/K-theory+of+a+permutative+category'>K-theory of a permutative category</a>.</p> <p>According to (<a href='#DwyerKan80'>Dwyer-Kan 80, prop. 3.7, prop. 9.2, remark 9.7</a>) this indeed gives a model for the total <a class='existingWikiWord' href='/nlab/show/diff/derived+functor'>derived functor</a> of 1-categorical group completion.</p> <h2 id='examples'>Examples</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group-completed+configuration+space+of+points'>group-completed configuration space of points</a></li> </ul> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Grothendieck+group'>Grothendieck group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/localization+of+a+monoid'>localization of a monoid</a></p> </li> </ul> <h2 id='references'>References</h2> <p>Classical accounts:</p> <ul> <li id='BarrattPriddy72'> <p><a class='existingWikiWord' href='/nlab/show/diff/Michael+Barratt'>Michael Barratt</a>, <a class='existingWikiWord' href='/nlab/show/diff/Stewart+Priddy'>Stewart Priddy</a>, <em>On the homology of non-connected monoids and their associated groups</em>, Commentarii Mathematici Helvetici, <strong>47</strong> 1 (1972) 1–14 [[doi:10.1007/BF02566785](https://link.springer.com/article/10.1007/BF02566785), <a href='https://eudml.org/doc/139496'>eudml:139496</a>]</p> <blockquote> <p>(on the <a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin+ring'>Pontrjagin ring</a>-structure under group completion of <a class='existingWikiWord' href='/nlab/show/diff/topological+monoid'>topological monoids</a>)</p> </blockquote> </li> <li id='Quillen71'> <p><a class='existingWikiWord' href='/nlab/show/diff/Daniel+Quillen'>Daniel Quillen</a>: <em>On the group completion of a simplicial monoid</em>, Appendix Q in: <a class='existingWikiWord' href='/nlab/show/diff/Eric+Friedlander'>Eric M. Friedlander</a>, <a class='existingWikiWord' href='/nlab/show/diff/Barry+Mazur'>Barry Mazur</a>: <em>Filtrations on the homology of algebraic varieties</em>, Memoirs of the AMS <strong>529</strong> 110 (1994) 89-105 [[doi:10.1090/memo/0529](https://doi.org/10.1090/memo/0529), <a href='http://www.maths.ed.ac.uk/~aar/papers/quillencomp.pdf'>pdf</a>]</p> </li> <li id='May'> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, <em><math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_10' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>E</mi> <mn>∞</mn></msub></mrow><annotation encoding='application/x-tex'>E_\infty</annotation></semantics></math>-Spaces, group completions, and permutative categories</em>, in: <em>New Developments in Topology</em>, Cambridge University Press (1974) 61-94 [[doi:10.1017/CBO9780511662607.008](https://doi.org/10.1017/CBO9780511662607.008), <a href='http://www.math.uchicago.edu/~may/PAPERS/13.pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Dusa+McDuff'>Dusa McDuff</a>, <a class='existingWikiWord' href='/nlab/show/diff/Graeme+Segal'>Graeme Segal</a>: <em>Homology fibrations and the “group-completion” theorem</em>, Inventiones mathematicae <strong>31</strong> (1976) 279-284 [[doi:10.1007/BF01403148](https://doi.org/10.1007/BF01403148)]</p> <blockquote> <p>(on the <a class='existingWikiWord' href='/nlab/show/diff/Pontrjagin+ring'>Pontrjagin ring</a>-structure under group completion of <a class='existingWikiWord' href='/nlab/show/diff/topological+monoid'>topological monoids</a>)</p> </blockquote> </li><ins class='diffins'> </ins><ins class='diffins'><li> <p><a class='existingWikiWord' href='/nlab/show/diff/Peter+May'>Peter May</a>, §4 in: <em>Infinite loop space theory</em>, Bull. Amer. Math. Soc. <strong>83</strong> 4 (1977) 456-494 [[pdf](http://www.math.uchicago.edu/~may/PAPERS/18.pdf), <a href='https://doi.org/10.1090/S0002-9904-1977-14318-8'>doi:10.1090/S0002-9904-1977-14318-8</a>]</p> </li></ins> <li id='DwyerKan80'> <p><a class='existingWikiWord' href='/nlab/show/diff/William+Dwyer'>William Dwyer</a>, <a class='existingWikiWord' href='/nlab/show/diff/Daniel+Kan'>Daniel Kan</a>, <em>Simplicial localization of categories</em>, Journal of pure and applied algebra <strong>17</strong> (1980) 267-284</p> </li> </ul> <p>and specifically concerning <a class='existingWikiWord' href='/nlab/show/diff/configuration+space+of+points'>configuration spaces of points</a>:</p> <ul> <li id='Segal73'> <p><a class='existingWikiWord' href='/nlab/show/diff/Graeme+Segal'>Graeme Segal</a>, <em>Configuration-spaces and iterated loop-spaces</em>, Invent. Math. <strong>21</strong> (1973) 213-221 [[doi:10.1007/BF01390197](https://doi.org/10.1007/BF01390197), <a href='http://dodo.pdmi.ras.ru/~topology/books/segal.pdf'>pdf</a>, MR 0331377]</p> </li> <li id='Okuyama05'> <p><a class='existingWikiWord' href='/nlab/show/diff/Shingo+Okuyama'>Shingo Okuyama</a>: <em>The space of intervals in a Euclidean space</em>, Algebr. Geom. Topol. <strong>5</strong> (2005) 1555-1572 [[arXiv:math/0511645](https://arxiv.org/abs/math/0511645), <a href='https://doi.org/10.2140/agt.2005.5.1555'>doi:10.2140/agt.2005.5.1555</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kazuhisa+Shimakawa'>Kazuhisa Shimakawa</a>: <em>Labeled configuration spaces and group completions</em>, Forum Mathematicum (2007) 353-364 [[doi:10.1515/FORUM.2007.014](https://doi.org/10.1515/FORUM.2007.014), <a class='existingWikiWord' href='/nlab/files/Shimakawa-ConfigGroupCompletion.pdf' title='pdf'>pdf</a>]</p> </li><ins class='diffins'> </ins><ins class='diffins'><li id='Kallel24'> <p><a class='existingWikiWord' href='/nlab/show/diff/Sadok+Kallel'>Sadok Kallel</a>, §4.2.3 in: <em>Configuration spaces of points: A user’s guide</em>, <em><a class='existingWikiWord' href='/nlab/show/diff/Encyclopedia+of+Mathematical+Physics+2nd+ed'>Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [[arXiv:2407.11092](https://arxiv.org/abs/2407.11092)]</p> </li></ins> </ul> <p>Discussion of <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-group completion:</p> <ul> <li id='BunkeTamme12'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Bunke'>Ulrich Bunke</a>, <a class='existingWikiWord' href='/nlab/show/diff/Georg+Tamme'>Georg Tamme</a>, section 2.1 of <em>Regulators and cycle maps in higher-dimensional differential algebraic K-theory</em> (<a href='http://arxiv.org/abs/1209.6451'>arXiv:1209.6451</a>)</p> </li> <li id='Nikolaus13'> <p><a class='existingWikiWord' href='/nlab/show/diff/Thomas+Nikolaus'>Thomas Nikolaus</a>: <em>Algebraic K-Theory of <math class='maruku-mathml' display='inline' id='mathml_fcaedc2a994b35e6697b3d7a98eb05291be0f84b_12' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>∞</mn></mrow><annotation encoding='application/x-tex'>\infty</annotation></semantics></math>-Operads</em> (<a href='http://arxiv.org/abs/1303.2198'>arXiv:1303.2198</a>)</p> </li> <li id='BunkeNikolausVoelkl13'> <p><a class='existingWikiWord' href='/nlab/show/diff/Ulrich+Bunke'>Ulrich Bunke</a>, <a class='existingWikiWord' href='/nlab/show/diff/Thomas+Nikolaus'>Thomas Nikolaus</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michael+V%C3%B6lkl'>Michael Völkl</a>, def. 6.1 in <em>Differential cohomology theories as sheaves of spectra</em> (<a href='http://arxiv.org/abs/1311.3188'>arXiv:1311.3188</a>)</p> </li> </ul> <p>and specifically its monoidal properties:</p> <ul> <li id='GepnerGrothNikolaus13'><a class='existingWikiWord' href='/nlab/show/diff/David+Gepner'>David Gepner</a>, <a class='existingWikiWord' href='/nlab/show/diff/Moritz+Groth'>Moritz Groth</a>, <a class='existingWikiWord' href='/nlab/show/diff/Thomas+Nikolaus'>Thomas Nikolaus</a>, <em>Universality of multiplicative infinite loop space machines</em> [[arXiv:1305.4550](http://arxiv.org/abs/1305.4550)]</li> </ul> <p> </p> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 25, 2024 at 15:23:12. 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