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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>Actions</title></head> <body> <blockquote> <p>this entry is about the notion of <em>action</em> in algebra (of one algebraic object on another object). For the notion of <em><a class="existingWikiWord" href="/nlab/show/action+functional">action functional</a></em> in <a class="existingWikiWord" href="/nlab/show/physics">physics</a> see there.</p> </blockquote> <hr /> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <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="representation_theory">Representation theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/geometric+representation+theory">geometric representation theory</a></strong></p> <h2 id="ingredients">Ingredients</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</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/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><span class="newWikiWord">Be?linson-Bernstein localization<a href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+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="actions">Actions</h1> <div class='maruku_toc'> <ul> <li><a href='#Idea'>Idea</a></li> <li><a href='#definitions'>Definitions</a></li> <ul> <li><a href='#ActionsOfAGroup'>Actions of a group</a></li> <li><a href='#actions_of_a_monoid'>Actions of a monoid</a></li> <li><a href='#actions_of_a_category'>Actions of a category</a></li> <li><a href='#actions_of_a_group_object'>Actions of a group object</a></li> <li><a href='#actions_of_a_monoid_object'>Actions of a monoid object</a></li> <li><a href='#actions_of_a_set'>Actions of a set</a></li> </ul> <li><a href='#examples'>Examples</a></li> <li><a href='#RelatedConcepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> <ul> <li><a href='#group_actions'>Group actions</a></li> </ul> </ul> </div> <h2 id="Idea">Idea</h2> <p>There are various variants of the notion of something <em>acting</em> on something else. They are all closely related.</p> <p>The simplest concept of an action involves one <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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, acting on another set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and such an action is given by a <a class="existingWikiWord" href="/nlab/show/function">function</a> from the <a class="existingWikiWord" href="/nlab/show/product">product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> act\colon X \times Y \to Y \,. </annotation></semantics></math></div> <p>For fixed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">x \in X</annotation></semantics></math> this produces an <a class="existingWikiWord" href="/nlab/show/endofunction">endofunction</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">act(x,-) \colon Y \to Y</annotation></semantics></math> and hence some “transformation” or “action” on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. In this way the whole of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>.</p> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>↦</mo><mi>act</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x\mapsto act(x,-))</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/currying">curried</a></em> function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>act</mi><mo>^</mo></mover><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><msup><mi>Y</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">\widehat{act}\colon X \to Y^Y</annotation></semantics></math> of the original <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>act</mi></mrow><annotation encoding="application/x-tex">act</annotation></semantics></math>, which maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/function+set">set</a> of <a class="existingWikiWord" href="/nlab/show/endofunctions">endofunctions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>. Quite generally one has these two perspectives on actions.</p> <p>Usually the key aspect of an action of some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> itself carries an algebraic structure, such as being a <a class="existingWikiWord" href="/nlab/show/group">group</a> (or just a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>) or being a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> or an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a>, which is also possessed by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">Y^Y</annotation></semantics></math> and preserved by the curried action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>act</mi><mo>^</mo></mover></mrow><annotation encoding="application/x-tex">\widehat{act}</annotation></semantics></math>. Note that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is any set then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">Y^Y</annotation></semantics></math> is a monoid, and when <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> acts on it one calls it an <a class="existingWikiWord" href="/nlab/show/MSet">X-set</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Y</mi> <mi>Y</mi></msup></mrow><annotation encoding="application/x-tex">Y^Y</annotation></semantics></math> to have a ring/algebra structure, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> must be some sort of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> or <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> with the action by <a class="existingWikiWord" href="/nlab/show/linear+functions">linear functions</a>; then one calls the action also a <em><a class="existingWikiWord" href="/nlab/show/module">module</a></em> or <em><a class="existingWikiWord" href="/nlab/show/representation">representation</a></em>.</p> <p>In terms of the uncurried action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\times Y\to Y</annotation></semantics></math>, the “preservation” condition says roughly speaking that acting consecutively with two elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is the same as first multiplying them and then acting with the result:</p> <div class="maruku-equation" id="eq:ActionProperty"><span class="maruku-eq-number">(1)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>,</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>2</mn></msub><mo>⋅</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> act(x_2,act(x_1,y)) = act(x_2\cdot x_1, y) \,. </annotation></semantics></math></div> <p>To be precise, this is the condition for a <em>left action</em>; a <em>right action</em> is defined dually in terms of a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y\times X\to Y</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has no algebraic structure, or if its relevant structure is commutative, then there is no essential difference between the two; but in general they can be quite different.</p> <p>This <em>action property</em> can also often be identified with a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> property: it characterizes a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> from the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}X</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/category">category</a> (such as <a class="existingWikiWord" href="/nlab/show/Set">Set</a>) of which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is an object. The distinction between left and right actions is mirrored in the variance; acting on the left yields a covariant functor, whereas acting on the right is expressed via contravariance.</p> <p>In this way essentially every kind of <a class="existingWikiWord" href="/nlab/show/functor">functor</a>, <a class="existingWikiWord" href="/nlab/show/n-functor">n-functor</a> and <a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a> may be thought of as defining a generalized kind of action. This perspective on actions is particularly prevalent in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a>, where for instance <a class="existingWikiWord" href="/nlab/show/coends">coends</a> may be thought of as producing <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> of actions in this general functorial sense.</p> <p>Under the <a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a> (or one of its variants), this perspective turns into the perspective where an action of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is some <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y/X</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}X</annotation></semantics></math>, whose <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>:</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></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Y &\longrightarrow& Y/X \\ && \downarrow \\ && \mathbf{B}X } \,. </annotation></semantics></math></div> <p>Here the total space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">Y/X</annotation></semantics></math> of this bundle is typically the “weak” <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> (for instance: <a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a>) of the action, whence the notation. If one thinks of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>X</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}X</annotation></semantics></math> as the <a class="existingWikiWord" href="/nlab/show/classifying+space">classifying space</a> for the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/universal+principal+bundle">universal principal bundle</a>, then this bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo stretchy="false">/</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>X</mi></mrow><annotation encoding="application/x-tex">Y/X \to \mathbf{B}X</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a> which is <a class="existingWikiWord" href="/nlab/show/associated+bundle">associated</a> via the action to this universal bundle. For more on this perspective on actions see at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></em>.</p> <h2 id="definitions">Definitions</h2> <h3 id="ActionsOfAGroup">Actions of a group</h3> <p>An <strong>action</strong> of a <a class="existingWikiWord" href="/nlab/show/group">group</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> on 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>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> in 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">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> 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> on <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>, that is a <a class="existingWikiWord" href="/nlab/show/group+homomorphism">group homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mi>G</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho \colon G \to Aut(S)</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>S</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(S)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of <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> 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> <p>Group actions, especially <a class="existingWikiWord" href="/nlab/show/continuous+function">continuous</a> actions on <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a>, are also known as <em>transformation groups</em> (starting around <a href="#Klein1872">Klein 1872, Sec. 1</a>, see also <a href="#Koszul65">Koszul 65</a> <a href="#Bredon72">Bredon 72</a>, <a href="#tomDieck79">tom Dieck 79</a>, <a href="#tomDieck87">tom Dieck 87</a>). Alternatively, if the group <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 acts is understood, one calls (<a href="#Bredon72">Bredon 72, Ch. II</a>) the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> equipped with an action by <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 <em><a class="existingWikiWord" href="/nlab/show/topological+G-space">topological G-space</a></em> (or <em><a class="existingWikiWord" href="/nlab/show/G-set">G-set</a></em>, <em><a class="existingWikiWord" href="/nlab/show/G-manifold">G-manifold</a></em>, etc., as the case may be).</p> <p>As indicated above, a more abstract but equivalent definition regards the group <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 category (a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>), denoted <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>, with a 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">\ast</annotation></semantics></math>. Then an <em>action</em> 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> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> is equivalently a <a class="existingWikiWord" href="/nlab/show/functor">functor</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><mi>ρ</mi><mo lspace="verythinmathspace">:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \rho \colon \mathbf{B} G \to \mathcal{C} </annotation></semantics></math></div> <p>Here the object <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> of the previous definition is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(\ast)</annotation></semantics></math> of that single object.</p> <p>Concretely, 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">\mathcal{C}</annotation></semantics></math> is a category like <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, then an action is equivalently a <a class="existingWikiWord" href="/nlab/show/function">function</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>S</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>S</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ G \times S &\longrightarrow& S \\ (g,s) &\mapsto& \rho(g)(s) } </annotation></semantics></math></div> <p>which satisfies the <em>action property</em></p> <div class="maruku-equation" id="eq:ActionPropertyOfGroupActions"><span class="maruku-eq-number">(2)</span><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∀</mo><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mi>s</mi></mrow></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex"> \underset{ g_1, g_2, s }{\forall} \;\;\; \rho(g_1 \cdot g_2)(s) \;=\; \rho(g_1) \big( \rho(g_2)(s) \big) </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><munder><mo>∀</mo><mi>s</mi></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>s</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mi>s</mi></mrow><annotation encoding="application/x-tex"> \underset{s}{\forall} \;\;\; \rho(e)(s) \;=\; s </annotation></semantics></math></div> <h3 id="actions_of_a_monoid">Actions of a monoid</h3> <p>More generally we can define an <em>action</em> of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</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> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to be a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">\rho: \mathbf{B} M \to C </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>B</mi></mstyle><mi>M</mi></mrow><annotation encoding="application/x-tex">\mathbf{B} M</annotation></semantics></math> is (again) <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> regarded as a one-object category.</p> <p>The <em>category of actions</em> 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> 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> is then defined to be 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><msup><mi>C</mi> <mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>M</mi></mrow></msup></mrow><annotation encoding="application/x-tex">C^{\mathbf{B} M}</annotation></semantics></math>. When <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 <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Set</mi></mrow><annotation encoding="application/x-tex">Set</annotation></semantics></math> this is called <a class="existingWikiWord" href="/nlab/show/MSet">MSet</a>.</p> <p>Considering this in <a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a> yields the <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> notion of <em><a class="existingWikiWord" href="/nlab/show/action+objects">action objects</a></em>.</p> <h3 id="actions_of_a_category">Actions of a category</h3> <p>One can<sup id="fnref:1"><a href="#fn:1" rel="footnote">1</a></sup> also define an <a class="existingWikiWord" href="/nlab/show/action+of+a+category+on+a+set">action of a category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> on the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> as a functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> 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>, but usually one just calls this a <a class="existingWikiWord" href="/nlab/show/functor">functor</a>.</p> <p>Another perspective on the same situation is: a (small) category is a <a class="existingWikiWord" href="/nlab/show/monad">monad</a> in the category of <a class="existingWikiWord" href="/nlab/show/span">span</a>s in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>. An action of the category is an algebra for this monad. See <a class="existingWikiWord" href="/nlab/show/action+of+a+category+on+a+set">action of a category on a set</a>.</p> <p>On the other hand, an action of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (not <em>in</em> a monoidal category, as above) is called an <a class="existingWikiWord" href="/nlab/show/module+category">module category</a> (also “<a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>”). This notion can be expanded of course to actions <em>in</em> a <a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a>, where in the case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math> as monoidal bicategory it specializes to the notion of module category.</p> <h3 id="actions_of_a_group_object">Actions of a group object</h3> <p>Suppose we have a category, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, with binary <a class="existingWikiWord" href="/nlab/show/product">product</a>s 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">*</annotation></semantics></math>. There is an alternative way of viewing group actions in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, so that we can talk about an action of a <a class="existingWikiWord" href="/nlab/show/group+object">group 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">C</annotation></semantics></math> on an object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, 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>.</p> <p>By the adjointness relation between cartesian product, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mo>?</mo></mrow><annotation encoding="application/x-tex">A\times ?</annotation></semantics></math>, and function set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>?</mo> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">?^A</annotation></semantics></math>, in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, a group homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha: G\to Aut(X)</annotation></semantics></math></div> <p>corresponds to a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">act: G\times X\to X</annotation></semantics></math></div> <p>which will have various properties encoding that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> was a homomorphism of groups:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">act(g_1g_2,x) = act(g_1,act(g_2,x))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">act(1,x) = x</annotation></semantics></math></div> <p>and these can be encoded diagrammatically.</p> <p>Because of this, we can <strong>define</strong> an action 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">C</annotation></semantics></math> on an object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, 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> to be a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>G</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">act: G\times X\to X</annotation></semantics></math></div> <p>satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.</p> <p>The advantage of this is that it does not require the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to have internal automorphism group objects for all objects being considered.</p> <p>As an example, only <a class="existingWikiWord" href="/nlab/show/locally+compact+topological+space">locally compact topological spaces</a> have well-behaved topological automorphism groups, and thus actions of topological spaces on topological spaces must either be restricted to actions on locally compact spaces, or else be defined as above.</p> <p>As another example, within the category of <a class="existingWikiWord" href="/nlab/show/profinite+group">profinite groups</a> viewed as topological groups, not all objects have automorphism groups for which the natural topology is profinite. Thus profinite group actions on (the underlying topological space of) a profinite group must either be given in this form, or else be restricted to actions on profinite groups for which the automorphism group is naturally profinite.</p> <h3 id="actions_of_a_monoid_object">Actions of a monoid object</h3> <p>Suppose we have a category, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, with binary <a class="existingWikiWord" href="/nlab/show/product">product</a>s 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">*</annotation></semantics></math>. There is an alternative way of viewing monoid actions in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, so that we can talk about an action of a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</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>, 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> on an object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, 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>.</p> <p>By the adjointness relation between cartesian product, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>×</mo><mo>?</mo></mrow><annotation encoding="application/x-tex">A\times ?</annotation></semantics></math>, and function set, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>?</mo> <mi>A</mi></msup></mrow><annotation encoding="application/x-tex">?^A</annotation></semantics></math>, in <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, a monoid homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>M</mi><mo>→</mo><mi>End</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha: M\to End(X)</annotation></semantics></math></div> <p>corresponds to a function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>M</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">act: M\times X\to X</annotation></semantics></math></div> <p>which will have various properties encoding that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> was a homomorphism of monoids:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mn>1</mn></msub><msub><mi>m</mi> <mn>2</mn></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mn>1</mn></msub><mo>,</mo><mi>act</mi><mo stretchy="false">(</mo><msub><mi>m</mi> <mn>2</mn></msub><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">act(m_1m_2,x) = act(m_1,act(m_2,x))</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>x</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">act(1,x) = x</annotation></semantics></math></div> <p>and these can be encoded diagrammatically.</p> <p>Because of this, we can <strong>define</strong> an action of a monoid object, <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>, 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> on an object, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, 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> to be a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>M</mi><mo>×</mo><mi>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">act:M\times X\to X</annotation></semantics></math></div> <p>satisfying conditions that certain diagrams (left to the reader) encoding these two rules, commute.</p> <p>The advantage of this is that it does not require the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to have internal endomorphism monoid objects for all objects being considered.</p> <h3 id="actions_of_a_set">Actions of a set</h3> <p>The action of a set on a set was defined above; it consists of a function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>act</mi><mo>:</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">act: X\times Y\to Y</annotation></semantics></math>. This can equivalently be represented by a <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> as its vertices, with its edges labeled by elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, and such that each vertex has exactly one arrow leaving it with each label. (This is a sort of “Grothendieck construction”.) It is also the same as a simple (non halting) <a class="existingWikiWord" href="/nlab/show/deterministic+automaton">deterministic automaton</a>, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> the set of states and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> the set of inputs.</p> <p>That an action is a type of edge labeled quiver can be seen by explicitly giving the product <a class="existingWikiWord" href="/nlab/show/projection">projection</a> functions, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">p_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">p_2</annotation></semantics></math>, of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>×</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\times Y</annotation></semantics></math>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mrow><mspace width="1em"></mspace><msub><mi>p</mi> <mn>1</mn></msub><mspace width="1em"></mspace></mrow></mover><mi>X</mi><mo>×</mo><mi>Y</mi><munderover><mo>⇉</mo><mrow><mspace width="1em"></mspace><mi>act</mi><mspace width="1em"></mspace></mrow><mrow><msub><mi>p</mi> <mn>2</mn></msub></mrow></munderover><mi>Y</mi></mrow><annotation encoding="application/x-tex">X\overset{\quad p_1 \quad}{ \leftarrow}X\times Y\underoverset{\quad act \quad}{p_2}{\rightrightarrows}Y</annotation></semantics></math></div> <p>The shape of this diagram corresponds to that of an edge labeled quiver:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Labels</mi><mover><mo>←</mo><mrow><mspace width="1em"></mspace><mi>label</mi><mspace width="1em"></mspace></mrow></mover><mi>Edges</mi><munderover><mo>⇉</mo><mrow><mspace width="1em"></mspace><mi>target</mi><mspace width="1em"></mspace></mrow><mi>source</mi></munderover><mi>Vertices</mi></mrow><annotation encoding="application/x-tex">Labels\overset{\quad label \quad}{ \leftarrow}Edges\underoverset{\quad target \quad}{source}{\rightrightarrows}Vertices</annotation></semantics></math></div> <p>While the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> has no algebraic structure to be preserved, the action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>act</mi></mrow><annotation encoding="application/x-tex">act</annotation></semantics></math> generates a unique <strong><a class="existingWikiWord" href="/nlab/show/free+category">free category</a> action</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>act</mi> <mo>*</mo></msup><mo>:</mo><msup><mi>X</mi> <mo>*</mo></msup><mo>×</mo><mi>Y</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">act^{*}:X^{*}\times Y\to Y</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">X^{*}</annotation></semantics></math> is 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>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> containing <a class="existingWikiWord" href="/nlab/show/path">paths</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> elements. The monoidal structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">X^*</annotation></semantics></math> is preserved: two actions in succession is equal to the action of the concatenation of their paths.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>act</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msubsup><mi>x</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>,</mo><mi>act</mi><mo stretchy="false">(</mo><msubsup><mi>x</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>act</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><msubsup><mi>x</mi> <mn>2</mn> <mo>*</mo></msubsup><mo>⋅</mo><msubsup><mi>x</mi> <mn>1</mn> <mo>*</mo></msubsup><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">act^{*}(x^{*}_2,act(x^{*}_1,y)) = act^{*}(x^{*}_2\cdot x^{*}_1, y) </annotation></semantics></math></div> <p>An action of a set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in itself is also called a <strong><a class="existingWikiWord" href="/nlab/show/binary+operation">binary operation</a></strong>, and the set <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/magma">magma</a></strong>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/representation">representation</a> is a “linear action”.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a> one considers <a class="existingWikiWord" href="/nlab/show/Hamiltonian+actions">Hamiltonian actions</a>.</p> </li> <li> <p>In <a class="existingWikiWord" href="/nlab/show/topology">topology</a>: <a class="existingWikiWord" href="/nlab/show/topological+G-space">topological G-space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/circle+action">circle action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+action+on+an+n-sphere">group action on an n-sphere</a></p> </li> </ul> </li> </ul> <p>(…)</p> <h2 id="RelatedConcepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/action+object">action object</a></p> </li> <li> <p><strong>action</strong>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a>,</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/conjugation+action">conjugation action</a>, <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diagonal+action">diagonal action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/transitive+action">transitive action</a>, <a class="existingWikiWord" href="/nlab/show/free+action">free action</a>, <a class="existingWikiWord" href="/nlab/show/regular+action">regular action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/proper+action">proper action</a>, <a class="existingWikiWord" href="/nlab/show/properly+discontinuous+action">properly discontinuous action</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/coaction">coaction</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/action+monad">action monad</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> <li> <p><a class="existingWikiWord" href="/nlab/show/quotient">quotient</a>, <a class="existingWikiWord" href="/nlab/show/quotient+stack">quotient stack</a>, <a class="existingWikiWord" href="/nlab/show/quotient+type">quotient type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, <a class="existingWikiWord" href="/nlab/show/invariant+theory">invariant theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant homotopy theory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Borel+model+structure">Borel model structure</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/biaction">biaction</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="group_actions">Group actions</h3> <p>On <a class="existingWikiWord" href="/nlab/show/group+actions">group actions</a>, mostly in <a class="existingWikiWord" href="/nlab/show/TopologicalSpaces">TopologicalSpaces</a>, hence in the form of <a class="existingWikiWord" href="/nlab/show/topological+G-spaces">topological G-spaces</a>:</p> <p>Historical origins:</p> <ul> <li id="Klein1872"> <p><a class="existingWikiWord" href="/nlab/show/Felix+Klein">Felix Klein</a>, <em><a class="existingWikiWord" href="/nlab/show/Vergleichende+Betrachtungen+%C3%BCber+neuere+geometrische+Forschungen">Vergleichende Betrachtungen über neuere geometrische Forschungen</a></em> (1872) Mathematische Annalen volume 43, pages 63–100 1893 (<a href="https://doi.org/10.1007/BF01446615">doi:10.1007/BF01446615</a>)</p> <p>English translation by M. W. Haskell:</p> <p><em>A comparative review of recent researches in geometry</em>, Bull. New York Math. Soc. 2, (1892-1893), 215-249. (<a href="https://projecteuclid.org/journals/bulletin-of-the-american-mathematical-society-new-series/volume-2/issue-10/A-comparative-review-of-recent-researches-in-geometry/bams/1183407629.full">euclid:1183407629</a>, LaTeX version retyped by Nitin C. Rughoonauth: <a href="https://arxiv.org/abs/0807.3161">arXiv:0807.3161</a>)</p> </li> </ul> <p>Introduction of group actions into (<a class="existingWikiWord" href="/nlab/show/quantum+physics">quantum</a>) <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (cf. <em><a class="existingWikiWord" href="/nlab/show/Gruppenpest">Gruppenpest</a></em>):</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Hermann+Weyl">Hermann Weyl</a>, §III.1 in: <em>Gruppentheorie und Quantenmechanik</em>, S. Hirzel, Leipzig (1931), translated by H. P. Robertson: <em>The Theory of Groups and Quantum Mechanics</em>, Dover (1950) [<a href="https://store.doverpublications.com/0486602699.html">ISBN:0486602699</a>, <a href="https://archive.org/details/ost-chemistry-quantumtheoryofa029235mbp/page/n15/mode/2up">ark:/13960/t1kh1w36w</a>]</li> </ul> <p>Textbook accounts:</p> <ul> <li id="Bredon72"> <p><a class="existingWikiWord" href="/nlab/show/Glen+Bredon">Glen Bredon</a>, <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+compact+transformation+groups">Introduction to compact transformation groups</a></em>, Academic Press 1972 (<a href="https://www.elsevier.com/books/introduction-to-compact-transformation-groups/bredon/978-0-12-128850-1">ISBN 9780080873596</a></p> <p>, <a href="http://www.indiana.edu/~jfdavis/seminar/Bredon,Introduction_to_Compact_Transformation_Groups.pdf">pdf</a>)</p> </li> <li id="tomDieck79"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups+and+Representation+Theory">Transformation Groups and Representation Theory</a></em>, Lecture Notes in Mathematics 766, Springer 1979 (<a href="https://link.springer.com/book/10.1007/BFb0085965">doi:10.1007/BFb0085965</a>)</p> </li> <li id="tomDieck87"> <p><a class="existingWikiWord" href="/nlab/show/Tammo+tom+Dieck">Tammo tom Dieck</a>, <em><a class="existingWikiWord" href="/nlab/show/Transformation+Groups">Transformation Groups</a></em>, de Gruyter 1987 (<a href="https://doi.org/10.1515/9783110858372">doi:10.1515/9783110858372</a>)</p> </li> </ul> <p>Lecture notes:</p> <ul> <li id="Koszul65"> <p><a class="existingWikiWord" href="/nlab/show/Jean-Louis+Koszul">Jean-Louis Koszul</a>, <em>Lectures on Groups of Transformations</em>, Tata Institute 1965 (<a href="http://www.math.tifr.res.in/~publ/ln/tifr32.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/KoszulGroupsOfTransformations.pdf" title="pdf">pdf</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Patrick+Morandi">Patrick Morandi</a>, <em>Group actions</em> (<a class="existingWikiWord" href="/nlab/files/MorandiGroupActions.pdf" title="pdf">pdf</a>)</p> </li> </ul> <div class="footnotes"><hr /><ol><li id="fn:1"> <p>One example of this relatively rare usage is <a class="existingWikiWord" href="/nlab/show/William+Lawvere">William Lawvere</a>: <em>Qualitative Distinctions Between Some Toposes of Generalized Graphs</em>, Contemporary Mathematics 92 (1989) in which this sense of action is routinely used. <a href="#fnref:1" rev="footnote">↩</a></p> </li></ol></div></body></html> </div> <div class="revisedby"> <p> Last revised on December 8, 2023 at 11:16:11. 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