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<li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+object">group object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+object">algebra object</a> (associative, <a class="existingWikiWord" href="/nlab/show/Lie+algebra+object">Lie</a>, …)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+object">module object</a>/<a class="existingWikiWord" href="/nlab/show/action+object">action object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+locale">internal locale</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> (<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/internal+infinity-categories+contents">more</a>)</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+site">internal site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+diagram">internal diagram</a></p> </li> </ul> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/algebraic+theories">algebraic theories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/monads">monads</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebras over</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mspace width="thinmathspace"></mspace></mrow><annotation encoding="application/x-tex">\,</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/operads">operads</a></p> </li> </ul> <p><strong><a class="existingWikiWord" href="/nlab/show/categorical+semantics">categorical semantics</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a>, <a class="existingWikiWord" href="/nlab/show/internal+language">internal language</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+category+theory+and+type+theory">relation between category theory and type theory</a></p> </li> </ul> </div></div> <h4 id="group_theory">Group Theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/group+theory">group theory</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></li> <li><a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>, <a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></li> <li><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></li> <li><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></li> <li><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></li> <li><a class="existingWikiWord" href="/nlab/show/progroup">progroup</a></li> <li><a class="existingWikiWord" href="/nlab/show/homogeneous+space">homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+group">general linear group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a>. <a class="existingWikiWord" href="/nlab/show/projective+unitary+group">projective unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+group">symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/finite+group">finite group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>, <a class="existingWikiWord" href="/nlab/show/cyclic+group">cyclic group</a>, <a class="existingWikiWord" href="/nlab/show/braid+group">braid group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/classification+of+finite+simple+groups">classification of finite simple groups</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sporadic+finite+simple+groups">sporadic finite simple groups</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Monster+group">Monster group</a>, <a class="existingWikiWord" href="/nlab/show/Mathieu+group">Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+variety">abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+topological+group">compact topological group</a>, <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+group">locally compact topological group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/maximal+compact+subgroup">maximal compact subgroup</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+group">string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/super+Lie+group">super Lie group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/2-group">2-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a>, <a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/n-group">n-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+groupal+n-groupoid">k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spectrum">spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>, <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a>, <a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+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/group+cohomology">group cohomology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+extension">group extension</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a>, <a class="existingWikiWord" href="/nlab/show/Ext-group">Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a></li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#in_a_cartesian_monoidal_category'>In a cartesian monoidal category</a></li> <li><a href='#in_a_braided_monoidal_category'>In a braided monoidal category</a></li> <li><a href='#in_a_monoidal_category'>In a monoidal category</a></li> <li><a href='#InTermsOfPresheavesOfGroups'>In terms of presheaves of groups</a></li> <li><a href='#AsDataStructure'>As data structure</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <li><a href='#theory'>Theory</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>A <em>group object</em> in a <a class="existingWikiWord" href="/nlab/show/cartesian+category">cartesian</a> <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group">group</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to <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> (see at <em><a class="existingWikiWord" href="/nlab/show/internalization">internalization</a></em> for more on the general idea).</p> <p>Given a non-cartesian <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> one can still make sense of group objects in the <a class="existingWikiWord" href="/nlab/show/formal+duality">dual</a> guise of <a class="existingWikiWord" href="/nlab/show/Hopf+monoids">Hopf monoids</a>, see there for more and see Rem. <a class="maruku-ref" href="#GroupObjectsInGeneralMonoidalCategories"></a> below.</p> <h2 id="definition">Definition</h2> <h3 id="in_a_cartesian_monoidal_category">In a cartesian monoidal category</h3> <p> <div class='num_defn' id='GroupObjectInCartesianCategory'> <h6>Definition</h6> <p><strong>(group object in <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a>)</strong> <br /> A <strong>group object</strong> or <strong>internal group</strong> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> 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">\mathcal{C}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a> (binary <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> and a <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">\ast</annotation></semantics></math>) is</p> <ul> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <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">\mathcal{C}</annotation></semantics></math></p> </li> <li> <p>and <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> in <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">\mathcal{C}</annotation></semantics></math> as follows:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi mathvariant="normal">e</mi><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mo>*</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathrm{e} \,\colon\, \ast \to G</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/inverse+elements">inverse elements</a>:</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(-)^{-1} \,\colon\, G\to G</annotation></semantics></math></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>m</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">m \colon G\times G \to G</annotation></semantics></math>,</p> </li> </ul> </li> </ul> <p>such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><mi>m</mi></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>m</mi><mo>×</mo><mi>id</mi></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mi>m</mi></mtd></mtr> <mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mi>m</mi></mover></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G\times G\times G & \stackrel{id\times m}{\longrightarrow} & G\times G \\ {}^{ \mathllap{ m\times id } } \big\downarrow && \big\downarrow m \\ G\times G & \stackrel{m}{\longrightarrow} & G } </annotation></semantics></math></div> <p>(expressing the fact multiplication is associative),</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi mathvariant="normal">e</mi><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mo lspace="0em" rspace="thinmathspace">id</mo><mo>,</mo><mi mathvariant="normal">e</mi><mo stretchy="false">)</mo></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><munder><mo>↘</mo><mo lspace="0em" rspace="thinmathspace">id</mo></munder></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mi>m</mi></mtd></mtr> <mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><munder><mo>⟶</mo><mi>m</mi></munder></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G & \stackrel{(\mathrm{e},id)}{\longrightarrow} & G\times G \\ {}^{\mathllap{(\id,\mathrm{e})}} \big\downarrow &\underset{\id}{\searrow}& \big\downarrow m \\ G\times G & \underset{m}{\longrightarrow} &G } </annotation></semantics></math></div> <p>(telling us that the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> is a left and right <a class="existingWikiWord" href="/nlab/show/unit+element">unit element</a>), and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>,</mo><mi>id</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mi>id</mi><mo>,</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd><munder><mo>↘</mo><mi mathvariant="normal">e</mi></munder></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mi>m</mi></mtd></mtr> <mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mi>m</mi></mover></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G & \overset{ ((-)^{-1},id) } {\longrightarrow} & G\times G \\ {}^{ \mathllap{ (id,(-)^{-1}) } } \big\downarrow & \underset{\mathrm{e}}{\searrow} & \big\downarrow m \\ G\times G & \stackrel{m}{\longrightarrow} & G } </annotation></semantics></math></div> <p>(telling us that the inverse map really does take an <a class="existingWikiWord" href="/nlab/show/inverse+element">inverse</a>).</p> </div> </p> <p> <div class='num_remark' id='UseOfDiagonalMorphism'> <h6>Remark</h6> <p>The <a class="existingWikiWord" href="/nlab/show/associativity+law">associativity law</a> technically factors through the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">(G\times G)\times G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G\times (G\times G)</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/pairing">pairing</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(f,g)</annotation></semantics></math> denotes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>f</mi><mo>×</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Δ</mi></mrow><annotation encoding="application/x-tex">(f\times g)\circ\Delta</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Δ</mi></mrow><annotation encoding="application/x-tex">\Delta</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagonal+morphism">diagonal morphism</a>.</p> </div> </p> <p> <div class='num_remark'> <h6>Remark</h6> <p>Even if <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> doesn't have <em>all</em> binary products, as long as products with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> (and the terminal object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>) exist, then one can still speak of a group object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <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">\mathcal{C}</annotation></semantics></math>, as above.</p> </div> </p> <h3 id="in_a_braided_monoidal_category">In a braided monoidal category</h3> <p>Notice that the use of <a class="existingWikiWord" href="/nlab/show/diagonal+maps">diagonal maps</a> (Rem. <a class="maruku-ref" href="#UseOfDiagonalMorphism"></a>) in Def. <a class="maruku-ref" href="#GroupObjectInCartesianCategory"></a> precludes direct generalization of this definition of group objects to non-<a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>, where such maps in general do not exist.</p> <p>Hence, while the <a class="existingWikiWord" href="/nlab/show/underlying">underlying</a> <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> may generally be defined in any <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, the internal formulation of existence of <a class="existingWikiWord" href="/nlab/show/inverse+elements">inverse elements</a> typically uses <a class="existingWikiWord" href="/nlab/show/extra+structure">extra structure</a>, such as that of a compatible <a class="existingWikiWord" href="/nlab/show/comonoid+object">comonoid object</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> to substitute for the missing <a class="existingWikiWord" href="/nlab/show/diagonal+maps">diagonal maps</a>.</p> <p>Given this, inverses may be encoded by an <em><a class="existingWikiWord" href="/nlab/show/antipode">antipode</a></em> map and the resulting “monoidal group objects” are known as <em><a class="existingWikiWord" href="/nlab/show/Hopf+monoids">Hopf monoids</a></em>. These subsume and generalize <em><a class="existingWikiWord" href="/nlab/show/Hopf+algebras">Hopf algebras</a></em>, which are widely studied, for instance in their role as <a class="existingWikiWord" href="/nlab/show/quantum+groups">quantum groups</a>.</p> <p>Hopf monoids may be defined in any <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, or more generally any <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a>, where the braiding is used in stating the fact that the comultiplication is a homomorphism of monoid objects.</p> <h3 id="in_a_monoidal_category">In a monoidal category</h3> <p>A surprising fact reported by Tom Leinster is that in the category of sets, a group is the same as a monoid with the extra property that the associativity square</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>×</mo><mi>m</mi></mrow></mover></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>m</mi><mo>×</mo><mi>id</mi></mrow></mpadded></msup><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo><mi>m</mi></mtd></mtr> <mtr><mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mover><mo>⟶</mo><mi>m</mi></mover></mtd> <mtd><mi>G</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G\times G\times G & \stackrel{id\times m}{\longrightarrow} & G\times G \\ {}^{ \mathllap{ m\times id } } \big\downarrow && \big\downarrow m \\ G\times G & \stackrel{m}{\longrightarrow} & G } </annotation></semantics></math></div> <p>is a pullback. Presumably it is also true that a group object in a cartesian monoidal category is the same as a monoid object in that category where the associativity square is a pullback. This suggests that we can define a group object in <em>any</em> monoidal category to be a monoid object where the associativity square is a pullback. The category does not need to have all pullbacks for this definition to parse. However, the usefulness of this generalization remains to be studied.</p> <h3 id="InTermsOfPresheavesOfGroups">In terms of presheaves of groups</h3> <div class="num_prop"> <h6 id="proposition">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, the category of internal groups in <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">\mathcal{C}</annotation></semantics></math> (in the sense of Def. <a class="maruku-ref" href="#GroupObjectInCartesianCategory"></a>) is <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalent</a> to the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the category of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Grp</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Grp^{C^{op}}</annotation></semantics></math> on <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>, spanned by those presheaves whose underlying set part in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Set</mi> <mrow><msup><mi>C</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">Set^{C^{op}}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a>.</p> </div> <p>This is a special case of the general theory of <em><a class="existingWikiWord" href="/nlab/show/structures+in+presheaf+toposes">structures in presheaf toposes</a></em>.</p> <p>It means that the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> from the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Grp</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Func\big(\mathcal{C}^{op}, Grp\big)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/presheaf+category">presheaf category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>𝒞</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">Func\big(\mathcal{C}^{op}, Set\big)</annotation></semantics></math> (obtained by composing with the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>→</mo></mrow><annotation encoding="application/x-tex">\to</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) creates representable group objects from representable objects.</p> <p>We unwind how this works:</p> <p>An object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> in <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">\mathcal{C}</annotation></semantics></math> equipped with internal group structure is identified equivalently with a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> of <a class="existingWikiWord" href="/nlab/show/functors">functors</a> of the form</p> <div class="maruku-equation" id="eq:PresheafWithValuesInGroups"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Grp</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mpadded width="0" lspace="-100%width"><mrow><msup><mrow></mrow> <mrow><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mo>⋅</mo><mo stretchy="false">)</mo></mrow></msup></mrow></mpadded><mo>↗</mo></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↓</mo></mtd></mtr> <mtr><mtd><msup><mi>𝒞</mi> <mi>op</mi></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>Set</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ && Grp \\ & \mathllap{{}^{(G,\cdot)}}\nearrow & \big\downarrow \\ \mathcal{C}^{op} &\underset{y(G)}{\longrightarrow}& Set } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^{op}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <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>, <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> is the category of <a class="existingWikiWord" href="/nlab/show/groups">groups</a> with <a class="existingWikiWord" href="/nlab/show/group+homomorphisms">group homomorphisms</a> between them, and <a class="existingWikiWord" href="/nlab/show/Set">Set</a> is the category of <a class="existingWikiWord" href="/nlab/show/sets">sets</a> with <a class="existingWikiWord" href="/nlab/show/maps">maps</a>/<a class="existingWikiWord" href="/nlab/show/functions">functions</a> between them. Finally,</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>y</mi><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>C</mi></mtd> <mtd><mover><mo>↪</mo><mphantom><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>−</mo></mrow></mphantom></mover></mtd> <mtd><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mi>G</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ y \colon & C &\xhookrightarrow{\phantom{--}}& PSh(C) \\ & G &\mapsto& Hom_C(-,G) } </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> of <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">\mathcal{C}</annotation></semantics></math> into its <a class="existingWikiWord" href="/nlab/show/category+of+presheaves">category of presheaves</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>PSh</mi><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>≔</mo><mspace width="thinmathspace"></mspace><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>Set</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">PSh(C) \,\coloneqq\, Func(C^{op}, Set)</annotation></semantics></math>, which sends each <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/representable+presheaf">representable presheaf</a> that it represents.</p> <p>Since the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> is <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithful</a>, it is natural to leave it notationally implicit and to write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(S)</annotation></semantics></math> (for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">S \in \mathcal{C}</annotation></semantics></math>) as shorthand for</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>y</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) \,. </annotation></semantics></math></div> <p>(This a also referred to as “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> seen at stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math>”, or similar.)</p> <p>Now, the lift <a class="maruku-eqref" href="#eq:PresheafWithValuesInGroups">(1)</a> of such a presheaf of sets to a presheaf of groups equips for each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>S</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">S \in \mathcal{C}</annotation></semantics></math> the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>y</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>Hom</mi> <mi>C</mi></msub><mo stretchy="false">(</mo><mi>S</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G(S) \coloneqq y(G)(S) \coloneqq Hom_C(S,G) </annotation></semantics></math> with an ordinary group <a class="existingWikiWord" href="/nlab/show/structure">structure</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mo>⋅</mo> <mi>S</mi></msub><mo>,</mo><msub><mi mathvariant="normal">e</mi> <mi>S</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\big(G(S), \cdot_S, \mathrm{e}_S\big)</annotation></semantics></math>, in particular with a product operation (a map of sets) of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>⋅</mo> <mi>S</mi></msub><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \cdot_S \,\colon\, G(S) \times G(S) \longrightarrow G(S) \,. </annotation></semantics></math></div> <p>Moreover, since morphisms in <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> are <a class="existingWikiWord" href="/nlab/show/group+homomorphisms">group homomorphisms</a>, it follows that for every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>S</mi><mo>→</mo><mi>T</mi></mrow><annotation encoding="application/x-tex">f \colon S \to T</annotation></semantics></math> in <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> we get a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mo>⋅</mo> <mi>S</mi></msub></mrow></mover></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo maxsize="1.2em" minsize="1.2em">↑</mo><mpadded width="0"><mrow><msup><mo></mo><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mpadded></mtd> <mtd></mtd> <mtd><mo maxsize="1.2em" minsize="1.2em">↑</mo><mpadded width="0"><mrow><msup><mo></mo><mrow><mi>G</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow></msup></mrow></mpadded></mtd></mtr> <mtr><mtd><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mo>×</mo><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mo>⋅</mo> <mi>T</mi></msub></mrow></munder></mtd> <mtd><mi>G</mi><mo stretchy="false">(</mo><mi>T</mi><mo stretchy="false">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G(S) \times G(S) &\stackrel{\cdot_S}{\to}& G(S) \\ \big\uparrow\mathrlap{^{G(f)\times G(f)}} && \big\uparrow\mathrlap{^{G(f)}} \\ G(T) \times G(T) &\underset{\cdot_T}{\longrightarrow}& G(T) \mathrlap{\,.} } </annotation></semantics></math></div> <p>Taken together this means that there is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo stretchy="false">(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>y</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> y(G \times G) \longrightarrow y(G) </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/representable+presheaves">representable presheaves</a>. By the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>, this <em>uniquely</em> comes from a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⋅</mo><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\cdot \colon G \times G \to G</annotation></semantics></math> in <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">\mathcal{C}</annotation></semantics></math>, which is the product of the group structure on the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> that we are after.</p> <p>etc.</p> <h3 id="AsDataStructure">As data structure</h3> <p>In the language of <a class="existingWikiWord" href="/nlab/show/dependent+type+theory">dependent type theory</a> (using the notation for <a class="existingWikiWord" href="/nlab/show/dependent+pair+types">dependent pair types</a> <em><a class="existingWikiWord" href="/nlab/show/dependent+functions+and+dependent+pairs+--+table">here</a></em>) the type of group data structures is:</p> <p><img src="/nlab/files/GroupDataType-230121.jpg" width="740" /></p> <h2 id="Examples">Examples</h2> <ul> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Sets">Sets</a> is a <a class="existingWikiWord" href="/nlab/show/group">group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/TopologicalSpaces">TopologicalSpaces</a> is a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/SimplicialSets">SimplicialSets</a> is a <a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a> is an <a class="existingWikiWord" href="/nlab/show/H-group">H-group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> is a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a> is a <a class="existingWikiWord" href="/nlab/show/supergroup">super Lie group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Grp">Grp</a> is an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> (using the <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a>).</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> is an abelian group again.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> is a strict <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/Grpd">Grpd</a> is a strict <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn></mrow><annotation encoding="application/x-tex">2</annotation></semantics></math>-group again.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/CRing">CRing</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo></mo><mi>op</mi></msup></mrow><annotation encoding="application/x-tex">^{op}</annotation></semantics></math> is a commutative <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a>.</p> </li> <li> <p>A group object in a <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> is a <a class="existingWikiWord" href="/nlab/show/group+functor">group functor</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/schemes">schemes</a> is a <a class="existingWikiWord" href="/nlab/show/group+scheme">group scheme</a>.</p> </li> <li> <p>A group object in an <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> is a <a class="existingWikiWord" href="/nlab/show/cogroup+object">cogroup object</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/G-sets">G-sets</a>/<a class="existingWikiWord" href="/nlab/show/G-spaces">G-spaces</a> is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/equivariant+group">equivariant group</a>, namely a <a class="existingWikiWord" href="/nlab/show/semidirect+product+group">semidirect product group</a>.</p> </li> <li> <p>A group object in <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a> is a <a class="existingWikiWord" href="/nlab/show/group+stack">group stack</a>.</p> </li> </ul> <h2 id="theory">Theory</h2> <p>The basic results of elementary group theory apply to group objects in any category with finite products. (Arguably, it is precisely the <em>elementary</em> results that apply in any such category.)</p> <p>The theory of group objects is an example of a <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object+in+an+%28%E2%88%9E%2C1%29-category">monoid object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <strong>group object</strong>, <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+action">group action</a>, <a class="existingWikiWord" href="/nlab/show/action+object">action object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>, <a class="existingWikiWord" href="/nlab/show/groupoid+object">groupoid object</a>, <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinity-groupoid">infinity-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/infinity-groupoid+object">infinity-groupoid object</a>, <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/ring+object">ring object</a></p> </li> </ul> <h2 id="References">References</h2> <p>The general definition of internal groups seems to have first been formulated in:</p> <ul> <li id="Grothendieck61"><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, p. 104 (7 of 21) of: <a class="existingWikiWord" href="/nlab/show/FGA">FGA</a> <em>Techniques de construction et théorèmes d’existence en géométrie algébrique III: préschémas quotients</em>, Séminaire Bourbaki: années 1960/61, exposés 205-222, Séminaire Bourbaki, no. 6 (1961), Exposé no. 212, (<a href="http://www.numdam.org/item/?id=SB_1960-1961__6__99_0">numdam:SB_1960-1961__6__99_0</a>, <a href="http://www.numdam.org/item/SB_1960-1961__6__99_0.pdf">pdf</a>, English translation: <a href="https://translations.thosgood.com/fga/fga3.iii.xml">web version</a>)</li> </ul> <p>following the general principle of <a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> formulated in:</p> <ul> <li id="Grothendieck60"><a class="existingWikiWord" href="/nlab/show/Alexander+Grothendieck">Alexander Grothendieck</a>, p. 340 (3 of 23) in: <a class="existingWikiWord" href="/nlab/show/FGA">FGA</a> <em>Technique de descente et théorèmes d’existence en géométrie algébriques. II: Le théorème d’existence en théorie formelle des modules</em>, Séminaire Bourbaki : années 1958/59 - 1959/60, exposés 169-204, Séminaire Bourbaki, no. 5 (1960), Exposé no. 195 (<a href="http://www.numdam.org/item/SB_1958-1960__5__369_0">numdam:SB_1958-1960__5__369_0</a>, <a href="http://www.numdam.org/item/SB_1958-1960__5__369_0.pdf">pdf</a>, English translation: <a href="https://translations.thosgood.com/fga/fga3.ii.xml">web version</a>)</li> </ul> <p>reviewed in:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Barbara+Fantechi">Barbara Fantechi</a>, <a class="existingWikiWord" href="/nlab/show/Lothar+G%C3%B6ttsche">Lothar Göttsche</a>, <a class="existingWikiWord" href="/nlab/show/Luc+Illusie">Luc Illusie</a>, <a class="existingWikiWord" href="/nlab/show/Steven+L.+Kleiman">Steven L. Kleiman</a>, <a class="existingWikiWord" href="/nlab/show/Nitin+Nitsure">Nitin Nitsure</a>, <a class="existingWikiWord" href="/nlab/show/Angelo+Vistoli">Angelo Vistoli</a>, Section 2.2 of: <em>Fundamental algebraic geometry. Grothendieck’s <a class="existingWikiWord" href="/nlab/show/FGA+explained">FGA explained</a></em>, Mathematical Surveys and Monographs <strong>123</strong>, Amer. Math. Soc. (2005) [<a href="http://www.ams.org/mathscinet-getitem?mr=2007f:14001">MR2007f:14001</a>, <a href="https://bookstore.ams.org/surv-123-s">ISBN:978-0-8218-4245-4</a>, <a href="http://indico.ictp.it/event/a0255/other-view?view=ictptimetable">lecture notes</a>]</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/internalization">internalization</a>, <a class="existingWikiWord" href="/nlab/show/H-spaces">H-spaces</a>, <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a>, <a class="existingWikiWord" href="/nlab/show/group+objects">group objects</a> in <a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a> and introducing the <a class="existingWikiWord" href="/nlab/show/Eckmann-Hilton+argument">Eckmann-Hilton argument</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <em>Structure maps in group theory</em>, Fundamenta Mathematicae 50 (1961), 207-221 (<a href="https://www.impan.pl/en/publishing-house/journals-and-series/fundamenta-mathematicae/all/50/2/94854/structure-maps-in-group-theory">doi:10.4064/fm-50-2-207-221</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <em>Group-like structures in general categories I multiplications and comultiplications</em>, Math. Ann. 145, 227–255 (1962) (<a href="https://doi.org/10.1007/BF01451367">doi:10.1007/BF01451367</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Beno+Eckmann">Beno Eckmann</a>, <a class="existingWikiWord" href="/nlab/show/Peter+Hilton">Peter Hilton</a>, <em>Group-like structures in general categories III primitive categories</em>, Math. Ann. <strong>150</strong> 165–187 (1963) (<a href="https://doi.org/10.1007/BF01470843">doi:10.1007/BF01470843</a>)</p> </li> </ul> <p>With emphasis of the role of the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, chapter III, section 6 in: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Springer (1971)</li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Michael+Boardman">John Michael Boardman</a>, <em>Algebraic objects in categories</em>, Chapter 7 of: <em>Stable Operations in Generalized Cohomology</em> [<a href="https://math.jhu.edu/~wsw/papers2/math/28a-boardman-stable.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Boardman-StableOperations.pdf" title="pdf">pdf</a>] in: <a class="existingWikiWord" href="/nlab/show/Ioan+Mackenzie+James">Ioan Mackenzie James</a> (ed.) <em><a class="existingWikiWord" href="/nlab/show/Handbook+of+Algebraic+Topology">Handbook of Algebraic Topology</a></em> Oxford (1995) [<a href="https://doi.org/10.1016/B978-0-444-81779-2.X5000-7">doi:10.1016/B978-0-444-81779-2.X5000-7</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Magnus+Forrester-Barker">Magnus Forrester-Barker</a>, <em>Group Objects and Internal Categories</em> [<a href="https://arxiv.org/abs/math/0212065">arXiv:math/0212065</a>]</p> </li> </ul> <p>In the broader context of internalization via <a class="existingWikiWord" href="/nlab/show/sketches">sketches</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Michael+Barr">Michael Barr</a>, <a class="existingWikiWord" href="/nlab/show/Charles+Wells">Charles Wells</a>, Section 4.1 of: <em><a class="existingWikiWord" href="/nlab/show/Toposes%2C+Triples%2C+and+Theories">Toposes, Triples, and Theories</a></em>, Originally published by: Springer-Verlag, New York, 1985, republished in: Reprints in <a href="http://www.tac.mta.ca/tac/reprints/articles/12/tr12abs.html">Theory and Applications of Categories, No. 12 (2005) pp. 1-287</a></li> </ul> <p>With focus on internalization in <a class="existingWikiWord" href="/nlab/show/sheaf+toposes">sheaf toposes</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a>, <a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, Section II.7 of: <em><a class="existingWikiWord" href="/nlab/show/Sheaves+in+Geometry+and+Logic">Sheaves in Geometry and Logic</a></em>, Springer 1992 (<a href="https://link.springer.com/book/10.1007/978-1-4612-0927-0">doi:10.1007/978-1-4612-0927-0</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 7, 2024 at 13:17:25. 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