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class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/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/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/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/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>symmetric monoidal functor</em> is a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mi>C</mi><mo>→</mo><mi>D</mi></mrow><annotation encoding="application/x-tex">F : C \to D</annotation></semantics></math> between <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal categories</a> that is a <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a> which respects the symmetry on both sides.</p> <h2 id="definition">Definition</h2> <p>A (lax) <a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>:</mo><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>D</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F : (C,\otimes) \to (D, \otimes)</annotation></semantics></math>, with monoidal structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math>, between <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a> is <strong>symmetric</strong> if for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><mi>B</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A,B \in C</annotation></semantics></math> the diagram</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>F</mi><mi>A</mi><mo>⊗</mo><mi>F</mi><mi>B</mi></mtd> <mtd><mover><mo>→</mo><mi>σ</mi></mover></mtd> <mtd><mi>F</mi><mi>B</mi><mo>⊗</mo><mi>F</mi><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mrow><mi>A</mi><mo>,</mo><mi>B</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mrow><mi>B</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>B</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F A \otimes F B &\stackrel{\sigma}{\to}& F B \otimes F A \\ {}^{\mathllap{\nabla_{A,B}}}\downarrow && \downarrow^{\mathrlap{\nabla_{B,A}}} \\ F(A\otimes B) &\stackrel{F(\sigma)}{\to}& F(B \otimes A) } </annotation></semantics></math></div> <p>commutes, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> denotes the symmetry isomorphism both of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> <p>As long as it goes between symmetric monoidal categories a symmetric monoidal functor is the same as a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a>.</p> <h2 id="properties">Properties</h2> <div class="num_prop" id="SymmetricMonoidalFunctorInducesFunctorOnCommutativeMonoids"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a> induces <a class="existingWikiWord" href="/nlab/show/functor">functor</a> on <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoids</a>)</strong></p> <p>A symmetric monoidal functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>τ</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>⟶</mo><mrow><mo>(</mo><msub><mi>𝒞</mi> <mn>2</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>τ</mi> <mn>2</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> \left(\mathcal{C}_1, \otimes_1, \tau_1\right) \longrightarrow \left(\mathcal{C}_2, \otimes_2, \tau_2\right) </annotation></semantics></math></div> <p>between two symmetric monoidal categories canonically preserves <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoids</a> and extends to a functor between categories of commutative monoids (see <a href="geometry+of+physics+--+categories+and+toposes#MonoidsPreservedByLaxMonoidalFunctor">here</a> for more)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mrow><mo>(</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>τ</mi> <mn>1</mn></msub><mo>)</mo></mrow><mo>⟶</mo><mi>CMon</mi><mrow><mo>(</mo><msub><mi>𝒞</mi> <mn>2</mn></msub><mo>,</mo><msub><mo>⊗</mo> <mn>2</mn></msub><mo>,</mo><msub><mi>τ</mi> <mn>2</mn></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> CMon\left(\mathcal{C}_1, \otimes_1, \tau_1\right) \longrightarrow CMon\left(\mathcal{C}_2, \otimes_2, \tau_2\right) </annotation></semantics></math></div></div> <h2 id="examples">Examples</h2> <div class="num_example" id="IdentityFunctorOnCategoryOfChainComplexesOfSuperVectorSpaces"> <h6 id="example">Example</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes+of+super+vector+spaces">category of chain complexes of super vector spaces</a>)</strong></p> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+chain+complexes+of+super+vector+spaces">category of chain complexes of super vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> equipped with the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+chain+complexes">tensor product of chain complexes</a> carries two <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/braidings">braidings</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Deligne</mi></msub></mrow><annotation encoding="application/x-tex">\tau_{Deligne}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mi>Bernst</mi></msub></mrow><annotation encoding="application/x-tex">\tau_{Bernst}</annotation></semantics></math> (<a href="chain+complex+in+super+vector+spaces#SymmetricStructureOnCategoryOfChainComplexesOfSuperVectorSpaces">this Prop.</a>). The <a class="existingWikiWord" href="/nlab/show/identity+functor">identity functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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> carries the structure of a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> symmetric monoidal functor with respect to these two, making them equivalent. By Prop. <a class="maruku-ref" href="#SymmetricMonoidalFunctorInducesFunctorOnCommutativeMonoids"></a> this in turn induces an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence</a> on the catories of commutative monoids, which in this case are <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebras">differential graded-commutative superalgebras</a>, with one of two equivalent grading conventions</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>dgcsAlg</mi> <mi>Deligne</mi></msub><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>dgcsAlg</mi> <mi>Bernstein</mi></msub></mrow><annotation encoding="application/x-tex"> dgcsAlg_{Deligne} \;\simeq\; dgcsAlg_{Bernstein} </annotation></semantics></math></div><div> <p><strong><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">sign rule</a> for <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebras">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><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_1"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_2"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_3"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_4"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_5"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_6"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_7"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_8"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_9"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_10"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_11"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_12"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_13"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_14"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>common in<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_15"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_16"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>discussion of<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_17"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_18"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_19"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_20"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/AKSZ+sigma-models">AKSZ sigma-models</a><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_21"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_22"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>representative<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_23"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_24"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math>references<math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_25"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_26"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_27"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math> <br /><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_28"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_29"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /> <math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_30"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_31"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_32"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_33"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math><br /><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_34"><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 xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_a6811c7bba5c65cdb676334141c04b852e3347fa_35"><semantics><mrow><mphantom><mi>A</mi></mphantom></mrow><annotation encoding="application/x-tex">\phantom{A}</annotation></semantics></math></td></tr> </tbody></table> </div></div> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><strong>symmetric monoidal functor</strong></p> </li> </ul> <h2 id="references">References</h2> <p>An exposition is in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <a href="http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf">Some definitions everyone should know</a>.</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 31, 2023 at 18:14:15. 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