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</li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+diagram'>string diagram</a>, <a class='existingWikiWord' href='/nlab/show/diff/tensor+network'>tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+category'>braided monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/balanced+monoidal+category'>balanced monoidal category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/twist'>twist</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/category+with+duals'>category with duals</a> (list of them)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dualizable+object'>dualizable object</a> (what they have)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/rigid+monoidal+category'>rigid monoidal category</a>, a.k.a. <a class='existingWikiWord' href='/nlab/show/diff/rigid+monoidal+category'>autonomous category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pivotal+category'>pivotal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spherical+category'>spherical category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ribbon+category'>ribbon category</a>, a.k.a. <a class='existingWikiWord' href='/nlab/show/diff/ribbon+category'>tortile category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+closed+category'>compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class='newWikiWord'>monoidal dagger-category<a href='/nlab/new/monoidal+dagger-category'>?</a></span></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+dagger-category'>symmetric monoidal dagger-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dagger-compact+category'>dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/trace'>trace</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/traced+monoidal+category'>traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+category'>closed monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/star-autonomous+category'>star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+monoidal+category'>cartesian monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/semicartesian+monoidal+category'>semicartesian monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+category+with+diagonals'>monoidal category with diagonals</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multicategory'>multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/semisimple+category'>semisimple category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fusion+category'>fusion category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>monoidal functor</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>lax</a>, <a class='existingWikiWord' href='/nlab/show/diff/oplax+monoidal+functor'>oplax</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>strong</a> <a class='existingWikiWord' href='/nlab/show/diff/bilax+monoidal+functor'>bilax</a>, <a class='existingWikiWord' href='/nlab/show/diff/Frobenius+monoidal+functor'>Frobenius</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+functor'>braided monoidal functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+functor'>symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+category'>monoid in a monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+category'>commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/module+over+a+monoid'>module over a monoid</a></p> </li> </ul> <p><strong id='_examples'>Examples</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Day+convolution'>Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coherence+and+strictification+for+monoidal+categories'>coherence theorem for monoidal categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal 2-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+2-category'>braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal bicategory</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+bicategory'>cartesian bicategory</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>little cubes operad</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+double+category'>compact double category</a></p> </li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#idea'>Idea</a></li><li><a href='#definition'>Definition</a></li><li><a href='#Examples'>Examples</a></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a><ul><li><a href='#general'>General</a></li><li><a href='#ReferencesOppositeCategories'>Opposite categories</a></li></ul></li></ul></div> <h2 id='idea'>Idea</h2> <p>Generalizing the classical notion of <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoid</a>, one can define a <em>commutative monoid</em> (or <em>commutative monoid object</em>) <a class='existingWikiWord' href='/nlab/show/diff/internalization'>internal to</a> any <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C,\otimes,I)</annotation></semantics></math>. These are <a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+category'>monoids in a monoidal category</a> whose multiplicative operation is commutative. Classical commutative monoids are of course just commutative monoids in <a class='existingWikiWord' href='/nlab/show/diff/Set'>Set</a> with the <a class='existingWikiWord' href='/nlab/show/diff/cartesian+product'>cartesian product</a>.</p> <h2 id='definition'>Definition</h2> <div class='num_defn' id='MonoidsInMonoidalCategory'> <h6 id='definition_2'>Definition</h6> <p>Given a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_2' xmlns='http://www.w3.org/1998/Math/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/diff/monoid+in+a+monoidal+category'>monoid internal to</a></strong> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_3' xmlns='http://www.w3.org/1998/Math/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/diff/object'>object</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_4' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>e</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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/diff/unit'>unit</a></em>)</p> </li> <li> <p>a morphism <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>μ</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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/diff/associativity'>associativity</a>) the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_7' xmlns='http://www.w3.org/1998/Math/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>A</mi><mo>⊗</mo><mi>μ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow /> <mpadded lspace='-100%width' width='0'><mrow><mi>μ</mi><mo>⊗</mo><mi>A</mi></mrow></mpadded></msup><mo stretchy='false'>↓</mo></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><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><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{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_8' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>a</mi></mrow><annotation encoding='application/x-tex'>a</annotation></semantics></math> is the associator isomorphism of <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_9' xmlns='http://www.w3.org/1998/Math/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/diff/identity+element'>unitality</a>) the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_10' xmlns='http://www.w3.org/1998/Math/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><msub><mrow /> <mpadded lspace='-100%width' width='0'><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><msub><mo>↙</mo> <mpadded width='0'><mi>r</mi></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><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 class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_11' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding='application/x-tex'>\ell</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_12' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_13' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> <p>Moreover, if <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_14' xmlns='http://www.w3.org/1998/Math/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/diff/symmetric+monoidal+category'>symmetric monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_15' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes, 1, \tau)</annotation></semantics></math> with symmetric <a class='existingWikiWord' href='/nlab/show/diff/braiding'>braiding</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_16' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi></mrow><annotation encoding='application/x-tex'>\tau</annotation></semantics></math>, then a monoid <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_17' xmlns='http://www.w3.org/1998/Math/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/diff/commutative+monoid+in+a+symmetric+monoidal+category'>commutative monoid in</a></strong> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_18' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes, 1, \tau)</annotation></semantics></math> if in addition</p> <ul> <li> <p>(commutativity) the following <a class='existingWikiWord' href='/nlab/show/diff/commutative+diagram'>diagram commutes</a></p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_19' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></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><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mpadded lspace='-100%width' width='0'><mi>μ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mpadded width='0'><mi>μ</mi></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><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><span><del class='diffmod'> A</del><ins class='diffmod'> Note</ins><ins class='diffins'> that</ins><ins class='diffins'> this</ins><ins class='diffins'> condition</ins><ins class='diffins'> makes</ins><ins class='diffins'> sense</ins><ins class='diffins'> even</ins><ins class='diffins'> if</ins><ins class='diffins'> the</ins></span><a class='existingWikiWord' href='/nlab/show/diff/homomorphism'><span><del class='diffmod'> homomorphism</del><ins class='diffmod'> braiding</ins></span></a><span><del class='diffdel'> </del><del class='diffdel'> of</del><del class='diffdel'> monoids</del></span><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_20' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><del class='diffmod'><mo stretchy='false'>(</mo></del><ins class='diffmod'><mi>τ</mi></ins><del class='diffdel'><msub><mi>A</mi> <mn>1</mn></msub></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><msub><mi>μ</mi> <mn>1</mn></msub></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><msub><mi>e</mi> <mn>1</mn></msub></del><del class='diffdel'><mo stretchy='false'>)</mo></del><del class='diffdel'><mo>⟶</mo></del><del class='diffdel'><mo stretchy='false'>(</mo></del><del class='diffdel'><msub><mi>A</mi> <mn>2</mn></msub></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><msub><mi>μ</mi> <mn>2</mn></msub></del><del class='diffdel'><mo>,</mo></del><del class='diffdel'><msub><mi>f</mi> <mn>2</mn></msub></del><del class='diffdel'><mo stretchy='false'>)</mo></del></mrow><annotation encoding='application/x-tex'><span><del class='diffmod'> (A_1,</del><ins class='diffmod'> \tau</ins><del class='diffdel'> \mu_1,</del><del class='diffdel'> e_1)\longrightarrow</del><del class='diffdel'> (A_2,</del><del class='diffdel'> \mu_2,</del><del class='diffdel'> f_2)</del></span></annotation></semantics></math><span> is<ins class='diffins'> not</ins><ins class='diffins'> symmetric,</ins><ins class='diffins'> as</ins><ins class='diffins'> in</ins> a<del class='diffdel'> morphism</del></span><ins class='diffins'><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+category'>braided monoidal category</a></ins><ins class='diffins'>. Such a monoid is also called a </ins><ins class='diffins'><strong>braided monoid</strong></ins><ins class='diffins'> in </ins><ins class='diffins'><math class='maruku-mathml' display='inline' id='mathml_cea959a80f2d5c58ee0dd3c39e14281f59aca0ba_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes, 1, \tau)</annotation></semantics></math></ins><ins class='diffins'>.</ins></p><span /><del class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_21' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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></del><ins class='diffmod'><p>A <a class='existingWikiWord' href='/nlab/show/diff/homomorphism'>homomorphism</a> of monoids <math class='maruku-mathml' display='inline' id='mathml_cea959a80f2d5c58ee0dd3c39e14281f59aca0ba_22' xmlns='http://www.w3.org/1998/Math/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></ins> <del class='diffmod'><p>in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_22' xmlns='http://www.w3.org/1998/Math/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/diff/commutative+diagram'>diagrams commute</a></p></del><ins class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_cea959a80f2d5c58ee0dd3c39e14281f59aca0ba_23' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><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></ins> <del class='diffmod'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_23' xmlns='http://www.w3.org/1998/Math/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 /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></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></del><ins class='diffmod'><p>in <math class='maruku-mathml' display='inline' id='mathml_cea959a80f2d5c58ee0dd3c39e14281f59aca0ba_24' xmlns='http://www.w3.org/1998/Math/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/diff/commutative+diagram'>diagrams commute</a></p></ins> <ins class='diffins'><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_cea959a80f2d5c58ee0dd3c39e14281f59aca0ba_25' xmlns='http://www.w3.org/1998/Math/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 /> <mpadded lspace='-100%width' width='0'><mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy='false'>↓</mo></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></ins><ins class='diffins'> </ins><p>and</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_24' xmlns='http://www.w3.org/1998/Math/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><msub><mrow /> <mpadded lspace='-100%width' width='0'><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><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><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 class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_25' xmlns='http://www.w3.org/1998/Math/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/diff/category+of+monoids'>category of monoids</a> in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_26' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, and <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_27' xmlns='http://www.w3.org/1998/Math/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> <h2 id='Examples'>Examples</h2> <div class='num_example'> <h6 id='example'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative rings</a>)</strong></p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_28' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Ab</mi><mo>,</mo><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})</annotation></semantics></math> for the category <a class='existingWikiWord' href='/nlab/show/diff/Ab'>Ab</a> of <a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian groups</a>, equipped with the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+abelian+groups'>tensor product of abelian groups</a> whose <a class='existingWikiWord' href='/nlab/show/diff/tensor+unit'>tensor unit</a> is the additive group of <a class='existingWikiWord' href='/nlab/show/diff/integer'>integers</a>. With the evident <a class='existingWikiWord' href='/nlab/show/diff/braiding'>braiding</a> this is a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric monoidal category</a>.</p> <p>A commutative monoid in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_29' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>Ab</mi><mo>,</mo><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mo>,</mo><mi>ℤ</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})</annotation></semantics></math> is equivalently a <a class='existingWikiWord' href='/nlab/show/diff/ring'>commutative ring</a>.</p> </div> <div class='num_example'> <h6 id='example_2'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative algebras</a>)</strong></p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/category+of+chain+complexes'>category of chain complexes</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_30' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ch</mi><mo stretchy='false'>(</mo><mi>Vect</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch(Vect)</annotation></semantics></math> with its <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+chain+complexes'>tensor product of chain complexes</a> carries a <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> monoidal <a class='existingWikiWord' href='/nlab/show/diff/braiding'>braiding</a> given on elements in definite degree <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_31' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>n \in \mathbb{Z}</annotation></semantics></math> by</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_32' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>τ</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mo>;</mo><mi>v</mi><mo>⊗</mo><mi>W</mi><mo>↦</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><msub><mi>n</mi> <mi>v</mi></msub><msub><mi>n</mi> <mi>w</mi></msub></mrow></msup><mi>w</mi><mo>⊗</mo><mi>v</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \tau \;\colon; v \otimes W \mapsto (-1)^{ n_v n_w } w \otimes v \,. </annotation></semantics></math></div> <p>The corresponding commutative monoid objects are the <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative algebras</a>.</p> </div> <div class='num_example'> <h6 id='example_3'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative superalgebras</a>)</strong></p> <p>The <a class='existingWikiWord' href='/nlab/show/diff/chain+complex+in+super+vector+spaces'>category of chain complexes of super vector spaces</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_33' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Ch</mi><mo stretchy='false'>(</mo><mi>SuperVect</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Ch(SuperVect)</annotation></semantics></math> with its <a class='existingWikiWord' href='/nlab/show/diff/tensor+product+of+chain+complexes'>tensor product of chain complexes</a> carries two <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+category'>symmetric</a> monoidal <a class='existingWikiWord' href='/nlab/show/diff/braiding'>braidings</a> given on elements in definite bidegree <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_34' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>,</mo><mi>σ</mi><mo stretchy='false'>)</mo><mo>∈</mo><mi>ℤ</mi><mo>×</mo><mi>ℤ</mi><mo stretchy='false'>/</mo><mn>2</mn></mrow><annotation encoding='application/x-tex'>(n,\sigma) \in \mathbb{Z} \times \mathbb{Z}/2</annotation></semantics></math> by</p> <ol> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_35' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>Deligne</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>v</mi><mo>⊗</mo><mi>w</mi><mo>↦</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>v</mi></msub><msub><mi>n</mi> <mi>w</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>v</mi></msub><msub><mi>σ</mi> <mi>w</mi></msub><mo stretchy='false'>)</mo></mrow></msup><mi>w</mi><mo>⊗</mo><mi>v</mi></mrow><annotation encoding='application/x-tex'>\tau_{Deligne} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v n_w + \sigma_v \sigma_w) } w \otimes v</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_36' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>τ</mi> <mi>Bernst</mi></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>v</mi><mo>⊗</mo><mi>w</mi><mo>↦</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>v</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>v</mi></msub><mo stretchy='false'>)</mo><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>w</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>w</mi></msub><mo stretchy='false'>)</mo></mrow></msup><mi>w</mi><mo>⊗</mo><mi>v</mi></mrow><annotation encoding='application/x-tex'>\tau_{Bernst} \;\colon\; v \otimes w \mapsto (-1)^{ (n_v + \sigma_v) (n_w + \sigma_w) } w \otimes v</annotation></semantics></math>.</p> </li> </ol> <p>The corresponding commutative monoid objects are the <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative superalgebras</a>.</p> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/signs+in+supergeometry'>sign rule</a> for <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative superalgebras</a></strong> <br /> (different but <a href='chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect'>equivalent</a>)</p> <table><thead><tr><th /><th><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_37' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#TheSignRuleFromInternalization'>Deligne’s convention</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_38' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th><th><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_39' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#SuperOddConvention'>Bernstein’s convention</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_40' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></th></tr></thead><tbody><tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_41' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_42' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>α</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>α</mi> <mi>j</mi></msub><mo>=</mo></mrow><annotation encoding='application/x-tex'> \alpha_i \cdot \alpha_j = </annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_43' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_44' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_45' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>n</mi> <mi>j</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo>⋅</mo><msub><mi>σ</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mrow></msup><msub><mi>α</mi> <mi>j</mi></msub><mo>⋅</mo><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'>(-1)^{ (n_i \cdot n_j + \sigma_i \cdot \sigma_j) } \alpha_j \cdot \alpha_i</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_46' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_47' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_48' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn><msup><mo stretchy='false'>)</mo> <mrow><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>i</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>i</mi></msub><mo stretchy='false'>)</mo><mo>⋅</mo><mo stretchy='false'>(</mo><msub><mi>n</mi> <mi>j</mi></msub><mo>+</mo><msub><mi>σ</mi> <mi>j</mi></msub><mo stretchy='false'>)</mo></mrow></msup><msub><mi>α</mi> <mi>j</mi></msub><mo>⋅</mo><msub><mi>α</mi> <mi>i</mi></msub></mrow><annotation encoding='application/x-tex'> (-1)^{ (n_i + \sigma_i) \cdot (n_j + \sigma_j) } \alpha_j \cdot \alpha_i</annotation></semantics></math><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_49' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_50' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math>common in<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_51' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math> <br /> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_52' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math>discussion of<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_53' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_54' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/supergravity'>supergravity</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_55' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_56' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a class='existingWikiWord' href='/nlab/show/diff/AKSZ+sigma-model'>AKSZ sigma-models</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_57' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> <tr><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_58' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math>representative<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_59' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math> <br /> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_60' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math>references<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_61' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_62' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#BonoraBregolaLechnerPastiTonin87'>Bonora et. al 87</a>,<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_63' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math> <br /><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_64' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#CastellaniDAuriaFre91'>Castellani-D’Auria-Fré 91</a>,<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_65' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><br /> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_66' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#DeligneFreed99'>Deligne-Freed 99</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_67' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td><td style='text-align: left;'><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_68' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='AKSZ+sigma-model#AKSZ'>AKSZ 95</a>,<math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_69' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><br /><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_70' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math><a href='signs+in+supergeometry#CarchediRoytenberg12'>Carchedi-Roytenberg 12</a><math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_71' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding='application/x-tex'>\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> <p>Since the two braidings above are equivalent (<a href='chain+complex+in+super+vector+spaces#EquivalenceTwoSymmetricMonoidalStructuresOnChSuperVect'>this Prop</a>) the corresponding two categories of <a class='existingWikiWord' href='/nlab/show/diff/differential+graded-commutative+algebra'>differential graded-commutative superalgebras</a> are also canonically <a class='existingWikiWord' href='/nlab/show/diff/equivalence+of+categories'>equivalence of categories</a>:</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_72' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ComMon</mi><mrow><mo>(</mo><mi>Ch</mi><mo stretchy='false'>(</mo><mi>SuperVect</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><msub><mi>τ</mi> <mi>Deligne</mi></msub><mo>)</mo></mrow><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><mi>ComMon</mi><mrow><mo>(</mo><mi>Ch</mi><mo stretchy='false'>(</mo><mi>SuperVect</mi><mo stretchy='false'>)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><msub><mi>τ</mi> <mi>Bernst</mi></msub><mo>)</mo></mrow></mrow><annotation encoding='application/x-tex'> ComMon\left( Ch(SuperVect), \otimes, \tau_{Deligne} \right) \;\simeq\; ComMon\left( Ch(SuperVect), \otimes, \tau_{Bernst} \right) </annotation></semantics></math></div><div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_73' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>sdgcAlg</mi> <mi>Deligne</mi></msub><mspace width='thickmathspace' /><mo>≃</mo><mspace width='thickmathspace' /><msub><mi>sdgcAlg</mi> <mi>Bernst</mi></msub></mrow><annotation encoding='application/x-tex'> sdgcAlg_{Deligne} \;\simeq\; sdgcAlg_{Bernst} </annotation></semantics></math></div></div> <div class='num_example'> <h6 id='example_4'>Example</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>commutative ring spectra</a>, <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-infinity rings</a>)</strong></p> <p>Write <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_74' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>SymSpec</mi><mo stretchy='false'>(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mo>∧</mo><mo>,</mo><msub><mi>𝕊</mi> <mi>sym</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(SymSpec(Top_{cg}),\wedge, \mathbb{S}_{sym})</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_75' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>OrthSpec</mi><mo stretchy='false'>(</mo><msub><mi>Top</mi> <mi>cg</mi></msub><mo stretchy='false'>)</mo><mo>,</mo><mo>∧</mo><mo>,</mo><msub><mi>𝕊</mi> <mi>orth</mi></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(OrthSpec(Top_{cg}),\wedge, \mathbb{S}_{orth})</annotation></semantics></math> and <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_76' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mo stretchy='false'>[</mo><msubsup><mi>Top</mi> <mrow><mi>cg</mi><mo>,</mo><mi>fin</mi></mrow> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msubsup><mo>,</mo><msubsup><mi>Top</mi> <mi>cg</mi> <mrow><mo>*</mo><mo stretchy='false'>/</mo></mrow></msubsup><mo stretchy='false'>]</mo><mo>,</mo><mo>∧</mo><mo>,</mo><mi>𝕊</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>([Top^{\ast/}_{cg,fin}, Top^{\ast/}_{cg}], \wedge, \mathbb{S} )</annotation></semantics></math> for the categories, respectively of <a class='existingWikiWord' href='/nlab/show/diff/symmetric+spectrum'>symmetric spectra</a>, <a class='existingWikiWord' href='/nlab/show/diff/orthogonal+spectrum'>orthogonal spectra</a> and <a class='existingWikiWord' href='/nlab/show/diff/excisive+%28%E2%88%9E%2C1%29-functor'>pre-excisive functors</a>, equipped with their <a class='existingWikiWord' href='/nlab/show/diff/symmetric+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a>, whose <a class='existingWikiWord' href='/nlab/show/diff/tensor+unit'>tensor unit</a> is the corresponding standard incarnation of the <a class='existingWikiWord' href='/nlab/show/diff/sphere+spectrum'>sphere spectrum</a>.</p> <p>A commutative monoid in any one of these three categories is equivalently a <a class='existingWikiWord' href='/nlab/show/diff/ring+spectrum'>commutative ring spectrum</a> in the strong sense: via the respective <a class='existingWikiWord' href='/nlab/show/diff/model+structure+on+spectra'>model structure on spectra</a> it represents an <a class='existingWikiWord' href='/nlab/show/diff/E-infinity-ring'>E-infinity ring</a>.</p> </div> <div class='num_example'> <h6 id='example_5'>Example</h6> <p><strong>(in a cocartesian monoidal category)</strong></p> <p>Every object <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_77' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> in a <a class='existingWikiWord' href='/nlab/show/diff/cocartesian+monoidal+category'>cocartesian monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_78' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> becomes a commutative monoid in a unique way: the multiplication must be the <a class='existingWikiWord' href='/nlab/show/diff/codiagonal'>fold map</a> <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_79' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>∇</mo><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>+</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>\nabla \colon A + A \to A</annotation></semantics></math>, and the counit must be the unique map <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_80' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>!</mo><mo lspace='verythinmathspace'>:</mo><mn>0</mn><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>! \colon 0 \to A</annotation></semantics></math>. Similarly every morphism in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_81' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> becomes a morphism of commutative monoid objects, so the category of commutative monoid objects in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_82' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is isomorphic to <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_83' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>.</p> </div> <div class='num_example'> <h6 id='example_6'>Example</h6> <p><strong>(in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_84' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CommMon</mi></mrow><annotation encoding='application/x-tex'>CommMon</annotation></semantics></math>)</strong></p> <p>Since the category <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_85' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CommMon</mi></mrow><annotation encoding='application/x-tex'>CommMon</annotation></semantics></math> of commutative monoids (in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_86' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Set</mi></mrow><annotation encoding='application/x-tex'>Set</annotation></semantics></math>) is cocartesian, the category of commutative monoids in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_87' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>CommMon</mi><mo>,</mo><mo lspace='verythinmathspace' rspace='0em'>+</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(CommMon,+)</annotation></semantics></math> is again <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_88' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CommMon</mi></mrow><annotation encoding='application/x-tex'>CommMon</annotation></semantics></math>. Finite coproducts of commutative monoids are also finite products, so the category of commutative monoids in <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_89' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>CommMon</mi><mo>,</mo><mo>×</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(CommMon,\times)</annotation></semantics></math> is also <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_90' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>CommMon</mi></mrow><annotation encoding='application/x-tex'>CommMon</annotation></semantics></math>.</p> </div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/monoid'>monoid</a>, <a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid'>commutative monoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/monoid+in+a+monoidal+category'>monoid in a monoidal category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/category+of+monoids'>category of monoids</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/module+over+a+monoid'>module over a monoid</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/semilattice+object'>semilattice object</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/commutative+monoid+in+a+symmetric+monoidal+%28infinity%2C1%29-category'>commutative monoid in a symmetric monoidal (infinity,1)-category</a></li> </ul> <h2 id='references'>References</h2> <h3 id='general'>General</h3> <p>Discussion including proof that/when the category of <a class='existingWikiWord' href='/nlab/show/diff/module+object'>module objects</a> is itself <a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+closed+category'>closed symmetric monoidal</a>:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Mark+Hovey'>Mark Hovey</a>, <a class='existingWikiWord' href='/nlab/show/diff/Brooke+Shipley'>Brooke Shipley</a>, <a class='existingWikiWord' href='/nlab/show/diff/Jeff+Smith'>Jeff Smith</a>, pp. 15 in: <em>Symmetric spectra</em>, J. Amer. Math. Soc. <strong>13</strong> (2000) 149-208 [[arXiv:math/9801077](http://arxiv.org/abs/math/9801077), <a href='https://doi.org/10.1090/S0894-0347-99-00320-3'>doi:10.1090/S0894-0347-99-00320-3</a>]</li> </ul> <p>See also:</p> <ul> <li> <p><a href='http://mathoverflow.net/questions/180673/category-of-modules-over-commutative-monoid-in-symmetric-monoidal-category'>MO/180673</a>.</p> </li> <li> <p><a href='#Marty09'>Marty 2009, 1.2 and 1.3</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Martin+Brandenburg'>Martin Brandenburg</a>, Section 4.1 <em>Tensor categorical foundations of algebraic geometry</em>, <a href='http://arxiv.org/abs/1410.1716'>arXiv:1410.1716</a>.</p> </li> </ul> <p>Lecture notes:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Urs+Schreiber'>Urs Schreiber</a>: <em><a href='Introduction+to+Stable+homotopy+theory+--+1-2#AlgebrasAndModules'>Algebras and Modules</a></em>, in <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+Homotopy+Theory'>Introduction to Stable Homotopy Theory</a></em> 1.2: <em><a class='existingWikiWord' href='/nlab/show/diff/Introduction+to+Stable+homotopy+theory+--+1-2'>Structured Spectra</a></em></li> </ul> <h3 id='ReferencesOppositeCategories'>Opposite categories</h3> <p>Discussion of the <a class='existingWikiWord' href='/nlab/show/diff/opposite+category'>opposite categories</a> of commutative monoid objects and regarded as categories of generalized <a class='existingWikiWord' href='/nlab/show/diff/affine+scheme'>affine schemes</a>:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Bertrand+To%C3%ABn'>Bertrand Toën</a>, <a class='existingWikiWord' href='/nlab/show/diff/Michel+Vaqui%C3%A9'>Michel Vaquié</a>, <em>Au-dessous de <math class='maruku-mathml' display='inline' id='mathml_1cf0703e3b80eb57fcbcb345a1bad7d59e889533_91' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Spec</mi><mi>ℤ</mi></mrow><annotation encoding='application/x-tex'>Spec \mathbb{Z}</annotation></semantics></math></em>, Journal of K-Theory <strong>3</strong> 3 (2009) 437-500 [[doi:10.1017/is008004027jkt048](https://doi.org/10.1017/is008004027jkt048)]</p> </li> <li id='Marty09'> <p><a class='existingWikiWord' href='/nlab/show/diff/Florian+Marty'>Florian Marty</a>, <em>Des Ouverts Zariski et des Morphismes Lisses en Géométrie Relative</em>, Ph.D. Toulouse (2009) [[theses:2009TOU30071](https://www.theses.fr/2009TOU30071), <a class='existingWikiWord' href='/nlab/files/Marty-DesOuverts.pdf' title='pdf'>pdf</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Florian+Marty'>Florian Marty</a>, <em>Relative Zariski Open Objects</em>, Journal of K-Theory <strong>10</strong> 1 (2012) 9-39 [[arXiv:0712.3676](https://arxiv.org/abs/0712.3676), <a href='https://doi.org/10.1017/is011012004jkt176'>doi:10.1017/is011012004jkt176</a>]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Abhishek+Banerjee'>Abhishek Banerjee</a>, <em>The relative Picard functor on schemes over a symmetric monoidal category</em>, Bulletin des Sciences Mathématiques <strong>135</strong> 4 (2011) 400-419 [[doi:10.1016/j.bulsci.2011.02.001](https://doi.org/10.1016/j.bulsci.2011.02.001)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Abhishek+Banerjee'>Abhishek Banerjee</a>, <em>On integral schemes over symmetric monoidal categories</em> [[arXiv:1506.04890](https://arxiv.org/abs/1506.04890)]</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Abhishek+Banerjee'>Abhishek Banerjee</a>, <em>Noetherian Schemes over abelian symmetric monoidal categories</em>, International Journal of Mathematics <strong>28</strong> 07 (2017) 1750051 [[doi:1410.3212](https://arxiv.org/abs/1410.3212), <a href='https://doi.org/10.1142/S0129167X17500513'>doi:10.1142/S0129167X17500513</a>]</p> </li> </ul> <p> </p> <p> </p> <p> </p> <p> </p> <p> </p><ins class='diffins'> </ins><ins class='diffins'><p> </p></ins> </div> <div class="revisedby"> <p> Last revised on February 13, 2024 at 10:15:28. 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