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float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="algebra">Algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></strong></p> <p><a class="existingWikiWord" href="/nlab/show/universal+algebra">universal algebra</a></p> <h2 id="algebraic_theories">Algebraic theories</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/2-algebraic+theory">2-algebraic theory</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory">(∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monad">monad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monad">(∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/operad">operad</a> / <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-operad">(∞,1)-operad</a></p> </li> </ul> <h2 id="algebras_and_modules">Algebras and modules</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+a+monad">algebra over a monad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad">∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+algebraic+theory">algebra over an algebraic theory</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory">∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a></p> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad">∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/associated+bundle">associated bundle</a>, <a class="existingWikiWord" href="/nlab/show/associated+%E2%88%9E-bundle">associated ∞-bundle</a></p> </li> </ul> <h2 id="higher_algebras">Higher algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category">monoid in an (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category">commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smash+product+of+spectra">smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+smash+product+of+spectra">symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ring+spectrum">ring spectrum</a>, <a class="existingWikiWord" href="/nlab/show/module+spectrum">module spectrum</a>, <a class="existingWikiWord" href="/nlab/show/algebra+spectrum">algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+ring">A-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+space">A-∞ space</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/C-%E2%88%9E+algebra">C-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+ring">E-∞ ring</a>, <a class="existingWikiWord" href="/nlab/show/E-%E2%88%9E+algebra">E-∞ algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-module+bundle">(∞,1)-module bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/deformation+theory">deformation theory</a></li> </ul> </li> </ul> <h2 id="model_category_presentations">Model category presentations</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+T-algebras">model structure on simplicial T-algebras</a> / <a class="existingWikiWord" href="/nlab/show/homotopy+T-algebra">homotopy T-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+operads">model structure on operads</a></p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+algebras+over+an+operad">model structure on algebras over an operad</a></p> </li> </ul> <h2 id="geometry_on_formal_duals_of_algebras">Geometry on formal duals of algebras</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></p> </li> </ul> <h2 id="theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Deligne+conjecture">Deligne conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/delooping+hypothesis">delooping hypothesis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/higher+algebra+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="monoid_theory">Monoid theory</h4> <div class="hide"><div> <p><strong>monoid theory</strong> in <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, <a class="existingWikiWord" href="/nlab/show/infinity-monoid">infinity-monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a>, <a class="existingWikiWord" href="/nlab/show/monoid+object+in+an+%28infinity%2C1%29-category">monoid object in an (infinity,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semiring">semiring</a>, <a class="existingWikiWord" href="/nlab/show/rig">rig</a>, <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, <a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Mon">Mon</a>, <a class="existingWikiWord" href="/nlab/show/CMon">CMon</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+homomorphism">monoid homomorphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/trivial+monoid">trivial monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/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/inverse+element">inverse element</a>, <a class="existingWikiWord" href="/nlab/show/unit">unit</a>, <a class="existingWikiWord" href="/nlab/show/irreducible+element">irreducible element</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ideal+in+a+monoid">ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+in+a+monoid">principal ideal in a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/tensor+product+of+commutative+monoids">tensor product of commutative monoids</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/GCD+monoid">GCD monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unique+factorization+monoid">unique factorization monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/B%C3%A9zout+monoid">Bézout monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+ideal+monoid">principal ideal monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/zero+divisor">zero divisor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integral+monoid">integral monoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a>, <a class="existingWikiWord" href="/nlab/show/free+commutative+monoid">free commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graphic+monoid">graphic monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+action">monoid action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+a+monoid">localization of a monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+monoid">endomorphism monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#elementary_definition'>Elementary definition</a></li> <li><a href='#inamonoidalcategory'>In a monoidal category</a></li> <li><a href='#in_terms_of_string_diagrams'>In terms of string diagrams</a></li> <li><a href='#AsAOneObjectCategory'>As a one-object category</a></li> <li><a href='#as_a_strict_monoidal_category'>As a strict monoidal category</a></li> <li><a href='#monoids_over_an_operad'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>-Monoids over an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Operad</a></li> </ul> <li><a href='#properties'> Properties</a></li> <ul> <li><a href='#finite_products_and_sums'>Finite products (and sums)</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#remarks_on_notation'>Remarks on notation</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>In <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, by a <em>monoid</em> one means a collection (<a class="existingWikiWord" href="/nlab/show/set">set</a>) of elements equipped with a <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> (a “multiplication operation”) which is <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> and has a <a class="existingWikiWord" href="/nlab/show/unit+element">unit element</a>.</p> <p>Hence monoid structure on a set is a fairly rudimentary form of algebraic <a class="existingWikiWord" href="/nlab/show/structure">structure</a> which <a class="existingWikiWord" href="/nlab/show/underlying">underlies</a> many familiar <a class="existingWikiWord" href="/nlab/show/structures">structures</a> considered <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, such as that of <em><a class="existingWikiWord" href="/nlab/show/groups">groups</a></em> (which are monoids with all <a class="existingWikiWord" href="/nlab/show/inverse+elements">inverse elements</a>) and <em><a class="existingWikiWord" href="/nlab/show/rings">rings</a></em> (which are <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> compatibly equipped with a <em>second</em> monoid structure).</p> <p>Therefore, in the algebraic literature monoids are, conversely, often called <em>unital <a class="existingWikiWord" href="/nlab/show/semi-groups">semi-groups</a></em>. The root “mono-” in “monoid” refers to the single <a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a> (cf. <em><a class="existingWikiWord" href="/nlab/show/duoid">duoid</a></em> and <em><a class="existingWikiWord" href="/nlab/show/dioid">dioid</a></em>).</p> <p>The terminology of <em><a class="existingWikiWord" href="/nlab/show/magmas">magmas</a></em> is meant to invoke this rudimemtary but foundational nature of basic algebraic structures: Monoids are precisely the <a class="existingWikiWord" href="/nlab/show/unital+magma">unital</a> <a class="existingWikiWord" href="/nlab/show/associative+magmas">associative magmas</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of the notion of monoids is that of <em><a class="existingWikiWord" href="/nlab/show/monads">monads</a></em> whose ubiquitous role in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a> (together with their <a href="adjoint+functor#RelationBetweenAdjunctionsAndMonads">associated</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a>) is hard to overstate.</p> <h2 id="definitions">Definitions</h2> <h3 id="elementary_definition">Elementary definition</h3> <p>The classical definition of monoids in <a class="existingWikiWord" href="/nlab/show/Sets">Sets</a>, as a <a class="existingWikiWord" href="/nlab/show/unital+magma">unital</a> <a class="existingWikiWord" href="/nlab/show/associative+magma">associative magma</a>:</p> <p> <div class='num_defn' id='MonoidsInSets'> <h6>Definition</h6> <p>A <strong>monoid</strong> is a <a class="existingWikiWord" href="/nlab/show/set">set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> equipped with the <a class="existingWikiWord" href="/nlab/show/structure">structure</a> if</p> <ol> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a></strong>) a <a class="existingWikiWord" href="/nlab/show/map">map</a> from the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the set with itself to itself</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mtext>-</mtext><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>M</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex"> (\text{-}) \cdot (\text{-}) \,\colon\, M \times M \to M </annotation></semantics></math></div></li> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a></strong>) an <a class="existingWikiWord" href="/nlab/show/element">element</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">1 \in M</annotation></semantics></math></div></li> </ol> <p>such that the following <a class="existingWikiWord" href="/nlab/show/equations">equations</a> are satisfied</p> <ol> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/associativity">associativity</a></strong>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi><mspace width="thinmathspace"></mspace><mo>∈</mo><mspace width="thinmathspace"></mspace><mi>M</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>x</mi><mo>⋅</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>z</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⋅</mo><mi>z</mi><mo stretchy="false">)</mo><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> x, y, z \,\in\, M \;\;\;\;\; \vdash \;\;\;\;\; (x \cdot y) \cdot z \;=\; x \cdot (y \cdot z) \mathrlap{\,,} </annotation></semantics></math></div></li> </ol> <ul> <li> <p>(<strong><a class="existingWikiWord" href="/nlab/show/unitality">unitality</a></strong>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>M</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⊢</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⋅</mo><mi>x</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>⋅</mo><mn>1</mn><mpadded width="0"><mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow></mpadded></mrow><annotation encoding="application/x-tex"> x \in M \;\;\;\;\; \vdash \;\;\;\;\; 1 \cdot x \;=\; x \;=\; x \cdot 1 \mathrlap{\,.} </annotation></semantics></math></div></li> </ul> <p></p> </div> </p> <h3 id="inamonoidalcategory">In a monoidal category</h3> <p>See <em><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></em>.</p> <h3 id="in_terms_of_string_diagrams">In terms of string diagrams</h3> <p>The data of a monoid may be written in <a class="existingWikiWord" href="/nlab/show/string+diagrams">string diagrams</a> as:</p> <p><img src="/nlab/files/monoid-data-labeled.png" alt="String diagrams of the monoid data (for "Monoid")" /></p> <p>Thanks to the distinctive shapes, one can usually omit the labels:</p> <p><img src="/nlab/files/monoid-data-unlabeled.png" alt="String diagrams of the monoid data, unlabeled (for "Monoid")" /></p> <p>The axioms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>η</mi><mo>⊗</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mn>1</mn> <mi>M</mi></msub><mo>=</mo><mi>μ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>M</mi><mo>⊗</mo><mi>η</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu \cdot (\eta \otimes M) = 1_M = \mu \cdot (M \otimes \eta)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>M</mi><mo>⊗</mo><mi>μ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>μ</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>μ</mi><mo>⊗</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mu \cdot (M \otimes \mu) = \mu \cdot (\mu \otimes M)</annotation></semantics></math> then appear as:</p> <p><img src="/nlab/files/monoid-axioms-unlabeled.png" alt="String diagrams of the monoid axioms (for "Monoid")" /></p> <h3 id="AsAOneObjectCategory">As a one-object category</h3> <p>Equivalently, and more efficiently, we may say that a (classical) monoid is the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> of a <a class="existingWikiWord" href="/nlab/show/category">category</a> with a single <a class="existingWikiWord" href="/nlab/show/object">object</a>, equipped with the structure of its unit element and composition.</p> <p>More tersely, one may say that a monoid <em>is</em> a category with a single object, or more precisely (to get the proper morphisms and <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>-morphisms) a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> category with a single object. But taking this too literally may create conflicts in notation. To avoid this, for a given monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}M</annotation></semantics></math> for the corresponding category with single 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">\bullet</annotation></semantics></math> and with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> as its <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a>: the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo>=</mo><msub><mi>Hom</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow></msub><mo stretchy="false">(</mo><mo>•</mo><mo>,</mo><mo>•</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">M = Hom_{\mathbf{B}M}(\bullet, \bullet)</annotation></semantics></math>. This realizes every monoid as a monoid of <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphisms</a>.</p> <p>Similarly, a monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(C,\otimes,I)</annotation></semantics></math> may be defined as the <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</a> of 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>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> with a single object, equipped with its composition and identity-assigning morphisms; and so on, as in the classical (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Set</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{Set}</annotation></semantics></math>-enriched) case.</p> <p>For more on this see also <a class="existingWikiWord" href="/nlab/show/group">group</a>.</p> <h3 id="as_a_strict_monoidal_category">As a strict monoidal category</h3> <p>An alternate way to view a monoid as a category is as a discrete <a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math> where the elements of the monoid are the objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>C</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{C}</annotation></semantics></math>, the binary operation of the monoid provides the tensor product bifunctor, and the identity of the monoid is the unit object. Preordered monoids then yield (non-discrete) strict monoidal categories with the morphisms witnessing the preorder in the usual way.</p> <h3 id="monoids_over_an_operad"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>-Monoids over an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Operad</h3> <p>The notion of <em>associative monoids</em> discussed above are controled by the <a class="existingWikiWord" href="/nlab/show/associative+operad">associative operad</a>. More generally in <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/operad">operad</a> or <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-operad">(infinity,1)-operad</a>, one can consider <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math>-monoids</strong>. (<a href="#Lurie">Lurie, def. 2.4.2.1</a>)</p> <p>These are closely related to <a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-algebras+over+an+%28infinity%2C1%29-operad">(infinity,1)-algebras over an (infinity,1)-operad</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi></mrow><annotation encoding="application/x-tex">\mathcal{O}</annotation></semantics></math> (<a href="#Lurie">Lurie, prop. 2.4.2.5</a>).</p> <h2 id="properties"> Properties</h2> <h3 id="finite_products_and_sums">Finite products (and sums)</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> be a monoid, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">M^*</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/free+monoid">free monoid</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with canonical function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>h</mi><mo>:</mo><mi>M</mi><mo>→</mo><msup><mi>M</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">h:M \to M^*</annotation></semantics></math> taking the elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> to the generators in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">M^*</annotation></semantics></math>. The finite product operation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid+homomorphism">monoid homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>M</mi> <mo>*</mo></msup><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\prod_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M</annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">M^*</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, where:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>ϵ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\prod_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 1</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\prod_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>⋅</mo><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\left(\prod_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) \cdot \left(\prod_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \prod_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)</annotation></semantics></math></div> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is written additively <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><mo>,</mo><mn>0</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(+, 0)</annotation></semantics></math> instead of multiplicatively <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋅</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\cdot, 1)</annotation></semantics></math>, the operation is called finite sum, and is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>:</mo><msup><mi>M</mi> <mo>*</mo></msup><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\sum_{i = 0}^{\mathrm{len}(-) - 1}(-)(i):M^* \to M</annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>M</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">M^*</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, where:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>ϵ</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\sum_{i = 0}^{\mathrm{len}(\epsilon) - 1} \epsilon(i) = 0</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mi>h</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">\sum_{i = 0}^{\mathrm{len}(h(a)) - 1} (h(a))(i) = a</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>a</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>+</mo><mrow><mo>(</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mi>b</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mo>=</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mrow><mi mathvariant="normal">len</mi><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo>−</mo><mn>1</mn></mrow></munderover><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\left(\sum_{i = 0}^{\mathrm{len}(a) - 1} a(i)\right) + \left(\sum_{i = 0}^{\mathrm{len}(b) - 1} b(i)\right) = \sum_{i = 0}^{\mathrm{len}(a b) - 1} (a b)(i)</annotation></semantics></math></div> <h2 id="examples">Examples</h2> <ul> <li>A monoid in which every element has an inverse is a <a class="existingWikiWord" href="/nlab/show/group">group</a>. For that reason monoids are often known (especially outside category theory) as <em>semi-group</em>s. (But this term is often extended to monoids without identities, that is to sets equipped with any associative operation.)</li> <li>The set of <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphisms</a> of a given object in a category has a canonical monoid structure given by composition.</li> </ul> <h2 id="remarks_on_notation">Remarks on notation</h2> <p>It can be important to distinguish between a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-tuply monoidal structure and the corresponding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-tuply degenerate category, even though there is a map identifying them. The issue appears here for instance when discussing the universal <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>-bundle in its groupoid incarnation. This is</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>→</mo><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> G \to \mathbf{E}G \to \mathbf{B}G </annotation></semantics></math></div> <p>(where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>=</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G = G//G</annotation></semantics></math> is the action groupoid of <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> acting on itself). On the left we crucially have <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> as a monoidal 0-category, on the right as a once-degenerate 1-category. Without this notation we cannot even <em>write down</em> the universal <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>-bundle!</p> <p>Or take the important difference between group <a class="existingWikiWord" href="/nlab/show/representation">representations</a> and group 2-algebras, the former being functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to Vect</annotation></semantics></math>, the latter functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">G \to Vect</annotation></semantics></math>. Both these are very important.</p> <p>Or take an abelian group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and a codomain like <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>2</mn><mi>Vect</mi></mrow><annotation encoding="application/x-tex">2Vect</annotation></semantics></math>. Then there are 3 different things we can sensibly consider, namely 2-functors</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mn>2</mn><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> A \to 2Vect </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi><mo>→</mo><mn>2</mn><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}A \to 2Vect </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>A</mi><mo>→</mo><mn>2</mn><mi>Vect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2A \to 2Vect \,. </annotation></semantics></math></div> <p>All of these concepts are different, and useful. The first one is an object in the group 3-algebra of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The second is a pseudo-representation of the group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. The third is a representations of the 2-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>A</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}A</annotation></semantics></math>. We have notation to distinguish this, and we should use it.</p> <p>Finally, writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> for the 1-object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-groupoid version of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-monoid <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> makes notation behave nicely with respect to nerves, because then realization bars <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mo>⋅</mo><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|\cdot|</annotation></semantics></math> simply commute with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math>s in the game: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">|</mo><mo>=</mo><mi>B</mi><mo stretchy="false">|</mo><mi>G</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|\mathbf{B}G| = B|G|</annotation></semantics></math>.</p> <p>This behavior under nerves shows also that, generally, writing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> gives the right intuition for what an expression means. For instance, what’s the “geometric” reason that a group representation is an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\rho : \mathbf{B}G \to Vect</annotation></semantics></math>? It’s because this is, literally, equivalently thought of as the corresponding classifying map of the vector bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> which is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-associated to the universal <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>-bundle:</p> <p>the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math>-associated vector bundle to the universal <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>-bundle is, in its groupoid incarnations,</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>V</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ V \\ \downarrow \\ V//G \\ \downarrow \\ \mathbf{B}G } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> is the vector space that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is representing on, and this is classified by the representation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\rho : \mathbf{B}G \to Vect</annotation></semantics></math> in that this is the pullback of the universal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>-bundle</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>V</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><msub><mi>Vect</mi> <mo>*</mo></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd> <mtd><mover><mo>→</mo><mi>ρ</mi></mover></mtd> <mtd><mi>Vect</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ V//G &\to& Vect_* \\ \downarrow && \downarrow \\ \mathbf{B}G &\stackrel{\rho}{\to}& Vect } \,, </annotation></semantics></math></div> <p>In summary, it is important to make people understand that groups can be identified with one-object groupoids. But next it is important to make clear that not everything that can be identified should be, for instance concerning the crucial difference between the category in which <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> lives and the 2-category in which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> lives.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+symmetric+proset">monoidal symmetric proset</a></p> </li> <li> <p><strong>monoid</strong>, internal monoid/monoid object,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a>, <a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a>, <a class="existingWikiWord" href="/nlab/show/braided+monoidal+groupoid">braided monoidal groupoid</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+groupoid">symmetric monoidal groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-groupoid">k-tuply monoidal n-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra+in+an+%28%E2%88%9E%2C1%29-category">monoid object in a (∞,1)-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/topological+monoid">topological monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/comonoid">comonoid</a>, <a class="existingWikiWord" href="/nlab/show/cocommutative+comonoid">cocommutative comonoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/duoid">duoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/group+object">group object</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> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/A3-type">A3-type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+setoid">monoidal setoid</a></p> </li> </ul> <div> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/algebra">algebraic</a> <a class="existingWikiWord" href="/nlab/show/mathematical+structure">structure</a></th><th><a class="existingWikiWord" href="/nlab/show/oidification">oidification</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magma">magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/magmoid">magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+set">pointed</a> <a class="existingWikiWord" href="/nlab/show/magma">magma</a> with an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/setoid">setoid</a>/<a class="existingWikiWord" href="/nlab/show/Bishop+set">Bishop set</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magma">unital magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroup">quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasigroupoid">quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loop+%28algebra%29">loop</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/loopoid">loopoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/category">category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/anti-involution">anti-involutive</a> <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dagger+category">dagger category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroup">associative quasigroup</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+quasigroupoid">associative quasigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/group">group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magma">flexible magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magma">alternative magma</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+monoid">absorption monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/absorption+category">absorption category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+monoid">cancellative monoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cancellative+category">cancellative category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rig">rig</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/CMon">CMon</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+ring">nonunital ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+ring">nonassociative ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>-<a class="existingWikiWord" href="/nlab/show/enriched+magmoid">enriched</a> <a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ring">ring</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/ringoid">ringoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/unital+magmoid">unital</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear</a> <a class="existingWikiWord" href="/nlab/show/semicategory">semicategory</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/associative+unital+algebra">associative unital algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+algebra">C-star algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/C-star+category">C-star category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/differential+algebroid">differential algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+algebra">flexible algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/flexible+magmoid">flexible</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+algebra">alternative algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/alternative+magmoid">alternative</a> <a class="existingWikiWord" href="/nlab/show/linear+magmoid">linear magmoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+poset">monoidal poset</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-poset">2-poset</a></td></tr> <tr><td style="text-align: left;"><span class="newWikiWord">strict monoidal groupoid<a href="/nlab/new/strict+monoidal+groupoid">?</a></span></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+%282%2C1%29-category">strict (2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-group">strict 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-groupoid">strict 2-groupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+monoidal+category">strict monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+groupoid">monoidal groupoid</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a>/<a class="existingWikiWord" href="/nlab/show/bigroupoid">bigroupoid</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-category">2-category</a>/<a class="existingWikiWord" href="/nlab/show/bicategory">bicategory</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>Beware that the term “monoid” was first used by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Arthur+Cayley">Arthur Cayley</a>, <em>Second and Third Memoirs on Skew Surfaces</em>, Otherwise Scrolls, Phil. Trans. (1863 and 1869)</li> </ul> <p>for certain <a class="existingWikiWord" href="/nlab/show/surfaces">surfaces</a>, quite unrelated to the modern meaning of the term.</p> <p>Instead, what are now called <em>monoids</em> (<a class="existingWikiWord" href="/nlab/show/unital+magma">unital</a> <a class="existingWikiWord" href="/nlab/show/associative+magmas">associative magmas</a>) were called <em>groupoids</em> (now clashing with the modern use of <em><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></em>) by</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Garrett+Birkhoff">Garrett Birkhoff</a>, <em>Hausdorff Groupoids</em>, Annals of Mathematics, Second Series <strong>35</strong> 2 (1934) 351-360 [<a href="https://www.jstor.org/stable/1968437">jstor:1968437</a>, <a href="https://doi.org/10.2307/1968437">doi:10.2307/1968437</a>]</li> </ul> <p>The modern terminology “monoid” for unital associative magmas is (according to <a href="#Hollings09">Hollings 2009, p. 529</a>) due to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Nicolas+Bourbaki">Nicolas Bourbaki</a>, <a class="existingWikiWord" href="/nlab/show/%C3%89l%C3%A9ments+de+Math%C3%A9matique">Éléments de Mathématique</a> (1943)</li> </ul> <p>For more on the history of the notion:</p> <ul> <li id="Hollings09"> <p>Christopher Hollings, <em>The Early Development of the Algebraic Theory of Semigroups</em>, Archive for History of Exact Sciences <strong>63</strong> (2009) 497–536 [<a href="https://doi.org/10.1007/s00407-009-0044-3">doi:10.1007/s00407-009-0044-3</a>]</p> </li> <li> <p>Math.SE, <em><a href="https://mathoverflow.net/q/338281/381">Who invented Monoid?</a></em></p> </li> </ul> <p>Exposition of basics of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>, Section 2: <em><a href="geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra">Basic notions of categorical algebra</a></em></li> </ul> <p>Properties of monoids expressed through properties of their <a class="existingWikiWord" href="/nlab/show/toposes">toposes</a> of <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a>:</p> <ul> <li>Jens Hemelaer, Morgan Rogers, <em>Monoid Properties as Invariants of Toposes of Monoid Actions</em>, <a href="https://arxiv.org/abs/2004.10513">arXiv:2004.10513</a>.</li> </ul> <p>Formalization of <a class="existingWikiWord" href="/nlab/show/monoid+objects">monoid objects</a> as <a class="existingWikiWord" href="/nlab/show/mathematical+structures">mathematical structures</a> in <a class="existingWikiWord" href="/nlab/show/proof+assistants">proof assistants</a>:</p> <p>in a context of plain <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Mart%C3%ADn+Escard%C3%B3">Martín Escardó</a>, <em><a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html#magmasandmonoids">The Types of Magmas and Monoids</a></em>, §4 in: <em>Introduction to Univalent Foundations of Mathematics with Agda</em> [<a href="https://arxiv.org/abs/1911.00580">arXiv:1911.00580</a>, <a href="https://www.cs.bham.ac.uk/~mhe/HoTT-UF-in-Agda-Lecture-Notes/HoTT-UF-Agda.html">webpage</a>]</li> </ul> <p>in a context of <a class="existingWikiWord" href="/nlab/show/cubical+type+theory">cubical</a> <a class="existingWikiWord" href="/nlab/show/Agda">Agda</a>:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/1lab">1lab</a>: <em><a href="https://1lab.dev/Algebra.Monoid.html">Algebra.Monoid</a></em></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on July 10, 2024 at 18:52:02. 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