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value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <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>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_algebra">Higher 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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#OverOrdinaryRings'>Over ordinary rings</a></li> <li><a href='#over_semirings'>Over semi-rings</a></li> <li><a href='#OverMonoidsInAMonoidalCategory'>Over monoids in a monoidal category</a></li> </ul> <li><a href='#variants'>Variants</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#tannaka_duality'>Tannaka duality</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>Similar to the way <a class="existingWikiWord" href="/nlab/show/modules">modules</a> generalize <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> by adding the operation of taking non-integer multiples, an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-algebra can be thought of as a generalization of a ring <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>, where the operation of taking integer multiples (seen as iterated addition) has been extended to taking arbitrary multiples with coefficients in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>. In the trivial case, a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>-algebra is simply a ring.</p> <h2 id="definition">Definition</h2> <h3 id="OverOrdinaryRings">Over ordinary rings</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>, an <strong>associative unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-algebra</strong> is equivalently:</p> <ul> <li> <p>a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal to</a> the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Mod"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> <mi>Mod</mi> </mrow> <annotation encoding="application/x-tex">R Mod</annotation> </semantics> </math></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</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">\otimes</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> single-<a class="existingWikiWord" href="/nlab/show/object">object</a> <a class="existingWikiWord" href="/nlab/show/enriched+category">category enriched over</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(R Mod, \otimes)</annotation></semantics></math>;</p> </li> <li> <p>a pointed <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebroid">algebroid</a> with a single object;</p> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> equipped with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">p \colon A \otimes A \longrightarrow A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">i \colon R \to A</annotation></semantics></math> satisfying <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/ring">ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/under+category">under</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> such that the corresponding map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">R \to A</annotation></semantics></math> lands in the <a class="existingWikiWord" href="/nlab/show/center">center</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </li> </ul> <p>If there is no danger for confusion, one often says simply ‘associative algebra’, or even only ‘<a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>’.</p> <p>More generally:</p> <ul> <li> <p>a (merely) <strong>associative algebra</strong> need not have a <a class="existingWikiWord" href="/nlab/show/unit">unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo lspace="verythinmathspace">:</mo><mi>R</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">i \colon R \to A</annotation></semantics></math>; that is, it is a <a class="existingWikiWord" href="/nlab/show/semigroup">semigroup</a> instead of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/ring+over+a+ring"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> </mrow> <annotation encoding="application/x-tex">R</annotation> </semantics> </math>-ring</a> is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> the category <a class="existingWikiWord" href="/nlab/show/Bimod"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> <mi>BiMod</mi> </mrow> <annotation encoding="application/x-tex">R BiMod</annotation> </semantics> </math></a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a> equipped with, crucially, the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+bimodules">tensor product of bimodules</a>.</p> </li> </ul> <p>Less generally, a <strong><a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a></strong> (where <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> and <a class="existingWikiWord" href="/nlab/show/unitality">unitality</a> are usually assumed) is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid objecy</a> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> <a class="existingWikiWord" href="/nlab/show/Mod"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>R</mi> <mi>Mod</mi> </mrow> <annotation encoding="application/x-tex">R Mod</annotation> </semantics> </math></a>.</p> <p>For a given ring the algebras form a category, <a class="existingWikiWord" href="/nlab/show/Alg">Alg</a>, and the commutative algebras a subcategory, <a class="existingWikiWord" href="/nlab/show/CommAlg">CommAlg</a>.</p> <h3 id="over_semirings">Over semi-rings</h3> <p>Note that everywhere rings can be replaced by <a class="existingWikiWord" href="/nlab/show/semi-rings">semi-rings</a> in the previous paragraph. For example a commutative associative unital <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℚ</mi> <mo lspace="verythinmathspace" rspace="0em">+</mo></msup></mrow><annotation encoding="application/x-tex">\mathbb{Q}^{+}</annotation></semantics></math>-algebra is nothing more than a commutative semi-ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/semi-ring+homomorphism">semi-ring homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℚ</mi> <mo lspace="verythinmathspace" rspace="0em">+</mo></msup><mo>→</mo><mi>R</mi></mrow><annotation encoding="application/x-tex">\mathbb{Q}^{+} \rightarrow R</annotation></semantics></math>.</p> <h3 id="OverMonoidsInAMonoidalCategory">Over monoids in a monoidal category</h3> <div class="num_defn" id="MonoidsInMonoidalCategory"> <h6 id="definition_2">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>, then a <strong><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid internal to</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">e \;\colon\; 1 \longrightarrow A</annotation></semantics></math> (called the <em><a class="existingWikiWord" href="/nlab/show/unit">unit</a></em>)</p> </li> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mu \;\colon\; A \otimes A \longrightarrow A</annotation></semantics></math> (called the <em>product</em>);</p> </li> </ol> <p>such that</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>a</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>μ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>μ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{id \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes id}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is the associator isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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><mn>1</mn><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ℓ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>μ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>r</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> are the left and right unitor isomorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes , 1)</annotation></semantics></math> has the structure of a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1, B)</annotation></semantics></math> with symmetric <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>, then a monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu, e)</annotation></semantics></math> as above is called a <strong><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1, B)</annotation></semantics></math> if in addition</p> <ul> <li> <p>(commutativity) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>τ</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow></munderover></mtd> <mtd></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>μ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>μ</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,. </annotation></semantics></math></div></li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)</annotation></semantics></math> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>A</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; A_1 \longrightarrow A_2 </annotation></semantics></math></div> <p>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>, such that the following two <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 } </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><mrow><mtable><mtr><mtd><msub><mn>1</mn> <mi>𝒸</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(\mathcal{C}, \otimes,1)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathcal{C}, \otimes, 1)</annotation></semantics></math> for its subcategory of commutative monoids.</p> </div> <div class="num_defn" id="ModulesInMonoidalCategory"> <h6 id="definition_3">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>, and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), then a <strong>left <a class="existingWikiWord" href="/nlab/show/module+object">module object</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">N \in \mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>N</mi><mo>⟶</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\rho \;\colon\; A \otimes N \longrightarrow N</annotation></semantics></math> (called the <em><a class="existingWikiWord" href="/nlab/show/action">action</a></em>);</p> </li> </ol> <p>such that</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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><mn>1</mn><mo>⊗</mo><mi>N</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ℓ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> is the left unitor isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> <li> <p>(action property) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>N</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>a</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>N</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>N</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>ρ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>N</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>ρ</mi></mover></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,, </annotation></semantics></math></div></li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (N_1, \rho_1) \longrightarrow (N_2, \rho_2) </annotation></semantics></math></div> <p>is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>N</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f\;\colon\; N_1 \longrightarrow N_2 </annotation></semantics></math></div> <p>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>, such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</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>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><msub><mi>N</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,. </annotation></semantics></math></div> <p>For the resulting <strong><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></strong> of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules 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> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module homomorphisms between them, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A Mod(\mathcal{C}) \,. </annotation></semantics></math></div> <p>This is naturally a (pointed) <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched category</a> itself.</p> </div> <div class="num_defn" id="TensorProductOfModulesOverCommutativeMonoidObject"> <h6 id="definition_4">Definition</h6> <p>Given a (pointed) <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topological</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>, given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N_1, \rho_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N_2, \rho_2)</annotation></semantics></math> two left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module+objects">module objects</a> (def.<a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), then the <strong><a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_1 \otimes_A N_2</annotation></semantics></math> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><munderover><mphantom><mi>AAAA</mi></mphantom><munder><mo>⟶</mo><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>τ</mi> <mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><mi>A</mi></mrow></msub><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mover></munderover><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mi>coequ</mi></mover><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coequ}{\longrightarrow} N_1 \otimes_A N_2 </annotation></semantics></math></div></div> <div class="num_prop" id="MonoidalCategoryOfModules"> <h6 id="proposition">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>), and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>). If all <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> exist 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>, then the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_A</annotation></semantics></math> from def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a> makes the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A Mod(\mathcal{C})</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod, \otimes_A, A)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> itself.</p> </div> <div class="num_defn" id="AAlgebra"> <h6 id="definition_5">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> as in prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>, then a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu, e)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) is called an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></strong>.</p> </div> <div class="num_prop"> <h6 id="proposition_2">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes, 1)</annotation></semantics></math> as in prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>, and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mu,e)</annotation></semantics></math> (def. <a class="maruku-ref" href="#AAlgebra"></a>), then there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mi>Alg</mi> <mi>comm</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mrow><mi>A</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/category+of+commutative+monoids">category of commutative monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a> of commutative monoids 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> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, hence between commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebras 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> and commutative monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</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> that are equipped with a homomorphism of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">A \longrightarrow E</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#EKMM97">EKMM 97, VII lemma 1.3</a>)</p> <div class="proof"> <h6 id="proof">Proof</h6> <p>In one direction, consider a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> with unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \;\colon\; A \longrightarrow E</annotation></semantics></math> and product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\mu_{E/A} \colon E \otimes_A E \longrightarrow E</annotation></semantics></math>. There is the underlying product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</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>E</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><munderover><mphantom><mi>AAA</mi></mphantom><munder><mo>⟶</mo><mrow></mrow></munder><mover><mo>⟶</mo><mrow></mrow></mover></munderover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mi>coeq</mi></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,. </annotation></semantics></math></div> <p>By considering a diagram of such coequalizer diagrams with middle vertical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_E\circ e_A</annotation></semantics></math>, one find that this is a unit for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_E, e_E \circ e_A)</annotation></semantics></math> is a commutative monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>.</p> <p>Then consider the two conditions on the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \colon A \longrightarrow E</annotation></semantics></math>. First of all this is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module homomorphism, which means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E } </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutes</a>. Moreover it satisfies the unit property</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>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,. </annotation></semantics></math></div> <p>By forgetting the tensor product over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the latter gives</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>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ρ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,, </annotation></semantics></math></div> <p>where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be <a class="existingWikiWord" href="/nlab/show/pasting">pasted</a> to the square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> above, to yield a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</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>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ρ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,. </annotation></semantics></math></div> <p>This shows that the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_A</annotation></semantics></math> is a homomorphism of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)</annotation></semantics></math>.</p> <p>Now for the converse direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu_A, e_A)</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>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><mi>e</mi><msub><mo>′</mo> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_E, e'_E)</annotation></semantics></math> are two commutative monoids in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \;\colon\; A \to E</annotation></semantics></math> a monoid homomorphism. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> inherits a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> structure by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>E</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover><mi>E</mi><mo>⊗</mo><mi>E</mi><mover><mo>⟶</mo><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mover><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,. </annotation></semantics></math></div> <p>By commutativity and associativity it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</annotation></semantics></math> coequalizes the two induced morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>E</mi><munderover><mphantom><mi>AA</mi></mphantom><mo>⟶</mo><mo>⟶</mo></munderover><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E</annotation></semantics></math>. Hence the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> gives a factorization through some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\mu_{E/A}\colon E \otimes_A E \longrightarrow E</annotation></semantics></math>. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_{E/A}, e_E)</annotation></semantics></math> is a commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra.</p> <p>Finally one checks that these two constructions are inverses to each other, up to isomorphism.</p> </div> <h2 id="variants">Variants</h2> <ul> <li> <p>A <a class="existingWikiWord" href="/nlab/show/cosimplicial+algebra">cosimplicial algebra</a> is a <a class="existingWikiWord" href="/nlab/show/cosimplicial+object">cosimplicial object</a> in the category of algebras.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a> is a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> not in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> but in the category of (co)<a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> is an associative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-algebra that has not only the usual binary product induced from the product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi><mo>×</mo><mi>ℝ</mi><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}\times \mathbb{R} \to \mathbb{R}</annotation></semantics></math>, but has a <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>-ary product operation for every <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>n</mi></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}^n \to \mathbb{R}</annotation></semantics></math>.</p> <p>This may be understood as a special case of an <a class="existingWikiWord" href="/nlab/show/algebra+over+a+Lawvere+theory">algebra over a Lawvere theory</a>, here the <a class="existingWikiWord" href="/nlab/show/Lawvere+theory">Lawvere theory</a> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>.</p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/function+algebra">function algebra</a></li> </ul> <h2 id="properties">Properties</h2> <h3 id="tannaka_duality">Tannaka duality</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/noncommutative+algebra">noncommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonassociative+algebra">nonassociative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonunital+algebra">nonunital algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/finitely+generated+algebra">finitely generated algebra</a>, <a class="existingWikiWord" href="/nlab/show/finitely+presented+algebra">finitely presented algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/power-associative+algebra">power-associative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/augmented+algebra">augmented algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitisation+of+C%2A-algebras">unitisation of C*-algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+algebra">differential algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded+algebra">differential graded algebra</a>, <a class="existingWikiWord" href="/nlab/show/A-infinity+algebra">A-infinity algebra</a></p> </li> </ul> <h2 id="references">References</h2> <p>See most references on <em><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></em>.</p> <p>See also:</p> <ul> <li>Wikipedia, <em><a href="https://en.wikipedia.org/wiki/Associative_algebra">Associative algebra</a></em></li> </ul> <p>Discussion in the generality of <a class="existingWikiWord" href="/nlab/show/brave+new+algebra">brave new algebra</a>:</p> <ul> <li id="EKMM97"><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em><a class="existingWikiWord" href="/nlab/show/Rings%2C+modules+and+algebras+in+stable+homotopy+theory">Rings, modules and algebras in stable homotopy theory</a></em>, AMS Mathematical Surveys and Monographs Volume 47 (1997) (<a href="http://www.math.uchicago.edu/~may/BOOKS/EKMM.pdf">pdf</a>)</li> </ul> </body></html> </div> <div class="revisedby"> 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