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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="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#left_and_right_actions'>Left and right actions</a></li> <li><a href='#see_also'>See also</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A ternary <a class="existingWikiWord" href="/nlab/show/function">function</a> which simultaneously exhibits an <a class="existingWikiWord" href="/nlab/show/action">action</a> on a <a class="existingWikiWord" href="/nlab/show/set">set</a> from both the left and the right side.</p> <p>Sets with biactions are the <a class="existingWikiWord" href="/nlab/show/bimodule+objects">bimodule objects</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> <a class="existingWikiWord" href="/nlab/show/Set">Set</a>.</p> <h2 id="definition">Definition</h2> <p>Given 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>S</mi></mrow><annotation encoding="application/x-tex">S</annotation></semantics></math> and <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>M</mi><mo>,</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(M, e_M, \mu_M)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N, e_N, \mu_N)</annotation></semantics></math>, a <strong><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>-<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>-biaction</strong> or <strong>two-sided action</strong> is a ternary function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>:</mo><mi>M</mi><mo>×</mo><mi>S</mi><mo>×</mo><mi>N</mi><mo>→</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">\alpha:M \times S \times N \to S</annotation></semantics></math> such that</p> <ul> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\alpha(e_M, s, e_N) = s</annotation></semantics></math></p> </li> <li> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">a \in M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">b \in M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">c \in N</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>d</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">d \in N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha\big(a, \alpha(b, s, c), d\big) = \alpha\big(\mu_M(a, b), s, \mu_N(c, d)\big)</annotation></semantics></math></p> </li> </ul> <h2 id="left_and_right_actions">Left and right actions</h2> <p>The <a class="existingWikiWord" href="/nlab/show/left+action">left <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>-action</a> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_M(a, s) \coloneqq \alpha(a, s, e_N)</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">a \in M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>. It is a left action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\alpha_M(e_M, s) = \alpha(e_M, s, e_N) = s</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>M</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><msub><mi>α</mi> <mi>L</mi></msub><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><mi>b</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>N</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><msub><mi>α</mi> <mi>M</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha_M\big(a, \alpha_L(b, s)\big) = \alpha\big(a, \alpha(b, s, e_N), e_N\big) = \alpha\big(\mu_M(a, b), s, \mu_N(e_N, e_N)\big) = \alpha\big(\mu_M(a, b), s, e_N\big) = \alpha_M\big(\mu_M(a, b), s\big)</annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/right+action">right <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>-action</a> is defined as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha_N(s, c) \coloneqq \alpha(e_M, s, c)</annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">c \in N</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>. It is a right action because</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>s</mi></mrow><annotation encoding="application/x-tex">\alpha_N(s, e_N) = \alpha(e_M, s, e_N) = s</annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>α</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><mi>d</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>M</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><msub><mi>α</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><mi>d</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha_N\big(\alpha_N(s, c), d\big) = \alpha\big(e_M, \alpha(e_M, s, c), d\big) = \alpha\big(\mu_M(e_M, e_M), s, \mu_N(c, d)\big) = \alpha\big(e_M, s, \mu_N(c, d)\big) = \alpha_N\big(s, \mu_N(c, d)\big)</annotation></semantics></math></div> <p>The left <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>-action and right <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>-action satisfy the following identity:</p> <ul> <li>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>S</mi></mrow><annotation encoding="application/x-tex">s \in S</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">a \in M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">c \in N</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mi>M</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><msub><mi>α</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><msub><mi>α</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>α</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha_M\big(a, \alpha_N(s, c)\big) = \alpha_N\big(\alpha_M(a, s), c\big)</annotation></semantics></math>.</li> </ul> <p>This is because when expanded out, the identity becomes:</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 maxsize="1.2em" minsize="1.2em">(</mo><mi>a</mi><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>c</mi><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha\big(a, \alpha(e_M, s, c), e_N\big) = \alpha\big(e_M, \alpha(a, s, e_N), c\big)</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>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><msub><mi>e</mi> <mi>M</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>,</mo><msub><mi>e</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mi>α</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><msub><mi>μ</mi> <mi>M</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>M</mi></msub><mo>,</mo><mi>a</mi><mo stretchy="false">)</mo><mo>,</mo><mi>s</mi><mo>,</mo><msub><mi>μ</mi> <mi>N</mi></msub><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>N</mi></msub><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\alpha\big(\mu_M(a, e_M), s, \mu_N(c, e_N)\big) = \alpha\big(\mu_M(e_M, a), s, \mu_N(e_N, c)\big)</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>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo><mo>=</mo><mi>α</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>s</mi><mo>,</mo><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\alpha(a, s, c) = \alpha(a, s, c)</annotation></semantics></math></div> <h2 id="see_also">See also</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a></p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on May 25, 2022 at 06:17:35. 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