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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> <ul> <li><a href='#the_2category_of_symmetric_monoidal_categories'>The 2-category of symmetric monoidal categories</a></li> <li><a href='#ModelsForConnectiveSpectra'>As models for connective spectra</a></li> <li><a href='#relation_to_categories'>Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-categories</a></li> <li><a href='#as_algebras_over_the_little_cubes_operad'>As algebras over the little <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-cubes operad</a></li> <li><a href='#tannaka_duality'>Tannaka duality</a></li> <li><a href='#grothendieck_ring'>Grothendieck ring</a></li> <li><a href='#internal_logic'>Internal logic</a></li> <li><a href='#permutations'>Permutations</a></li> </ul> <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 category</em> is a category with a product operation – a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> – for which the product is <em>as commutative as possible</em>.</p> <p>The point is that there are different degrees to which higher categorical products may be commutative. While a bare <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> is either commutative or not, a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> may be a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> – which already means that the order of products may be reversed up to some <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> – without being <em>symmetric</em> monoidal – which means that changing the order of a product twice, from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \otimes b</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi><mo>⊗</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">b \otimes a</annotation></semantics></math> back to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⊗</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a \otimes b</annotation></semantics></math>, indeed does yield a result <a class="existingWikiWord" href="/nlab/show/equality">equal</a> to the original.</p> <p>For higher monoidal categories there are accordingly ever more shades of the notion of “commutativity” of the monoidal product. This is described in detail at <a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a>.</p> <p>In general, the term <em>symmetric monoidal</em> is used for the maximally commutative case. See for instance <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a>. Notably, a symmetric monoidal <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> is, under the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>, the same as a <a class="existingWikiWord" href="/nlab/show/spectrum">connective spectrum</a>.</p> <p>A symmetric monoidal category is a special case of the notion of <a class="existingWikiWord" href="/nlab/show/symmetric+pseudomonoid">symmetric pseudomonoid</a> in a <a class="existingWikiWord" href="/nlab/show/sylleptic+monoidal+2-category">sylleptic monoidal 2-category</a>.</p> <h2 id="definition">Definition</h2> <div class="num_defn"> <h6 id="definition_2">Definition</h6> <p>A <strong>symmetric monoidal category</strong> is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> for which the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>→</mo><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> B_{x,y} \colon x \otimes y \to y \otimes x </annotation></semantics></math></div> <p>satisfies the condition:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>∘</mo><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>=</mo><msub><mn>1</mn> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex"> B_{y,x} \circ B_{x,y} = 1_{x \otimes y} </annotation></semantics></math></div> <p>for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow><annotation encoding="application/x-tex">x, y</annotation></semantics></math></p> </div> <p>Intuitively this says that switching things twice <em>in the same direction</em> has no effect.</p> <p>Expanding this out a bit: a <strong>symmetric monoidal category</strong> is, to begin with a <a class="existingWikiWord" href="/nlab/show/category">category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> equipped with a <a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>⊗</mo><mo>:</mo><mi>M</mi><mo>×</mo><mi>M</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex"> \otimes : M \times M \to M </annotation></semantics></math></div> <p>called the <strong>tensor product</strong>, an object</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>M</mi></mrow><annotation encoding="application/x-tex"> 1 \in M </annotation></semantics></math></div> <p>called the <strong>unit object</strong>, a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub><mo>:</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi><mo>→</mo><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a_{x,y,z} : (x \otimes y) \otimes z \to x \otimes (y \otimes z) </annotation></semantics></math></div> <p>called the <strong>associator</strong>, a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>λ</mi> <mi>x</mi></msub><mo>:</mo><mn>1</mn><mo>⊗</mo><mi>x</mi><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> \lambda_x : 1 \otimes x \to x </annotation></semantics></math></div> <p>called the <strong>left unitor</strong>, a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>x</mi></msub><mo>:</mo><mi>x</mi><mo>⊗</mo><mn>1</mn><mo>→</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> \rho_x : x \otimes 1 \to x </annotation></semantics></math></div> <p>called the <strong>right unitor</strong>, and a natural isomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>:</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>→</mo><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> B_{x,y} : x \otimes y \to y \otimes x </annotation></semantics></math></div> <p>called the <strong>braiding</strong>. We then demand that the associator obey the <strong>pentagon identity</strong>, which says this diagram commutes:</p> <div style="text-align:center"><div> <svg xmlns="http://www.w3.org/2000/svg" width="480" height="320"> <defs> <marker id="se_marker_end_svg_39887_2" markerUnits="strokeWidth" orient="auto" viewBox="0 0 100 100" markerWidth="5" markerHeight="5" refX="50" refY="50"> <path id="svg_39887_3" d="m100,50l-100,40l30,-40l-30,-40l100,40z" fill="#000000" stroke="#000000" stroke-width="10"></path> </marker> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_39887_16"> <path stroke-width="10" stroke="#000000" fill="#000000" d="m100,50l-100,40l30,-40l-30,-40l100,40z" id="svg_39887_11"></path> </marker> <marker refY="50" refX="50" markerHeight="5" markerWidth="5" viewBox="0 0 100 100" orient="auto" markerUnits="strokeWidth" id="se_marker_end_svg_39887_15"> <path stroke-width="10" 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stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">(w\otimes x)\otimes(y\otimes z)</annotation> </semantics> </math> </foreignObject> <foreignObject height="20" width="136" font-size="16" id="svg_39887_13" y="120" x="1"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>w</mi> <mo>⊗</mo> <mi>x</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>z</mi> </mrow> <annotation encoding="application/x-tex">((w\otimes x)\otimes y)\otimes z</annotation> </semantics> </math> </foreignObject> <foreignObject x="343.666667" y="120" id="svg_39887_4" font-size="16" width="136" height="20"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>w</mi> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>y</mi> <mo>⊗</mo> <mi>z</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">w\otimes (x\otimes(y\otimes z))</annotation> </semantics> </math> </foreignObject> <foreignObject height="20" width="136" font-size="16" id="svg_39887_5" y="295" x="51.666667"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mo stretchy="false">(</mo> <mi>w</mi> <mo>⊗</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>z</mi> </mrow> <annotation encoding="application/x-tex">(w\otimes (x\otimes y))\otimes z</annotation> </semantics> </math> </foreignObject> <foreignObject x="317" y="295" id="svg_39887_6" font-size="16" width="136" height="20"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <mi>w</mi> <mo>⊗</mo> <mo stretchy="false">(</mo> <mo stretchy="false">(</mo> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo stretchy="false">)</mo> <mo>⊗</mo> <mi>z</mi> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">w\otimes ((x\otimes y)\otimes z)</annotation> </semantics> </math> </foreignObject> <foreignObject height="24" width="72" font-size="16" id="svg_39887_7" y="52" x="66.333333"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>w</mi> <mo>⊗</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">a_{w\otimes x,y,z}</annotation> </semantics> </math> </foreignObject> <foreignObject x="338" y="52" id="svg_39887_8" font-size="16" width="72" height="24"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>⊗</mo> <mi>z</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">a_{w,x,y\otimes z}</annotation> </semantics> </math> </foreignObject> <foreignObject height="24" width="92" font-size="16" id="svg_39887_9" y="215" x="-0.666667"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>,</mo> <mi>y</mi> </mrow> </msub> <mo>⊗</mo> <msub> <mn>1</mn> <mi>z</mi> </msub> </mrow> <annotation encoding="application/x-tex">a_{w,x,y}\otimes 1_{z}</annotation> </semantics> </math> </foreignObject> <foreignObject x="384.333333" y="215" id="svg_39887_10" font-size="16" width="92" height="24"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <msub> <mn>1</mn> <mi>w</mi> </msub> <mo>⊗</mo> <msub> <mi>a</mi> <mrow> <mi>x</mi> <mo>,</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">1_w\otimes a_{x,y,z}</annotation> </semantics> </math> </foreignObject> <foreignObject id="svg_39887_21" height="24" width="72" font-size="16" y="280" x="214.666667"> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline"> <semantics> <mrow> <msub> <mi>a</mi> <mrow> <mi>w</mi> <mo>,</mo> <mi>x</mi> <mo>⊗</mo> <mi>y</mi> <mo>,</mo> <mi>z</mi> </mrow> </msub> </mrow> <annotation encoding="application/x-tex">a_{w,x\otimes y,z}</annotation> </semantics> </math> </foreignObject> </g> </svg></div></div> <p>We demand that the associator and unitors obey the <strong>triangle identity</strong>, which says this diagram commutes:</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></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mn>1</mn><mo>,</mo><mi>y</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mrow><msub><mi>ρ</mi> <mi>x</mi></msub><mo>⊗</mo><msub><mn>1</mn> <mi>y</mi></msub></mrow></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mrow><msub><mn>1</mn> <mi>x</mi></msub><mo>⊗</mo><msub><mi>λ</mi> <mi>y</mi></msub></mrow></msub></mtd> <mtd></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>x</mi><mo>⊗</mo><mi>y</mi></mtd> <mtd></mtd> <mtd></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ & (x \otimes 1) \otimes y &\stackrel{a_{x,1,y}}{\longrightarrow} & x \otimes (1 \otimes y) \\ & {}_{\rho_x \otimes 1_y}\searrow && \swarrow_{1_x \otimes \lambda_y} & \\ && x \otimes y && } </annotation></semantics></math></div> <p>We demand that the braiding and associator obey the first <strong>hexagon identity</strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>⊗</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>x</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⊗</mo><msub><mn>1</mn> <mi>z</mi></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msub><mi>a</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></msup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>z</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>a</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>y</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msub><mn>1</mn> <mi>y</mi></msub><mo>⊗</mo><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>y</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (x \otimes y) \otimes z &\stackrel{a_{x,y,z}}{\to}& x \otimes (y \otimes z) &\stackrel{B_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{B_{x,y}\otimes 1_z} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{1_y \otimes B_{x,z}}{\to}& y \otimes (z \otimes x) } </annotation></semantics></math></div> <p>And lastly, we demand that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow></msub><msub><mi>B</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>=</mo><msub><mn>1</mn> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> B_{y,x} B_{x,y} = 1_{x \otimes y} . </annotation></semantics></math></div> <p>(The definition of <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> has <em>two</em> hexagon identities, but either one implies the other given this equation.)</p> <h2 id="properties">Properties</h2> <h3 id="the_2category_of_symmetric_monoidal_categories">The 2-category of symmetric monoidal categories</h3> <p>There is a <a class="existingWikiWord" href="/nlab/show/strict+2-category">strict 2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymmMonCat</mi></mrow><annotation encoding="application/x-tex">SymmMonCat</annotation></semantics></math> with:</p> <ul> <li>symmetric monoidal categories as objects,</li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a>s as morphisms,</li> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+natural+transformation">symmetric monoidal natural transformation</a>s as 2-morphisms.</li> </ul> <p>This 2-category has (weak) 2-<a class="existingWikiWord" href="/nlab/show/biproducts">biproducts</a> given by the cartesian product of underlying categories (analogously to how <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> has biproducts given by the cartesian product of underlying sets). For a proof, see <a href="#FongSpivak">Fong-Spivak, Theorem 2.3</a>, or for a more abstract version involving <a class="existingWikiWord" href="/nlab/show/pseudomonoids">pseudomonoids</a> <a href="#Schaeppi">Schaeppi, Appendix A</a>.</p> <h3 id="ModelsForConnectiveSpectra">As models for connective spectra</h3> <p>The <a class="existingWikiWord" href="/nlab/show/group+completion">group completion</a> of the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of a symmetric monoidal category is always an <a class="existingWikiWord" href="/nlab/show/infinite+loop+space">infinite loop space</a>, hence the degree-0-space of a <a class="existingWikiWord" href="/nlab/show/connective+spectrum">connective spectrum</a>. One calls this also the <a class="existingWikiWord" href="/nlab/show/K-theory">K-theory</a> spectrum of the symmetric monoidal category:</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></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Spectra</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>K</mi></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>SymmMonCat</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mi>Cat</mi></mtd> <mtd><munder><mo>→</mo><mi>N</mi></munder></mtd> <mtd><mi>sSet</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ &&&& Spectra \\ && {}^{\mathllap{K}}\nearrow && \downarrow \\ SymmMonCat &\to& Cat &\underset{N}{\to}& sSet } \,. </annotation></semantics></math></div> <p>This construction extended to an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>:</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>SymmMonCat</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mo>≃</mo></mover><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><msub><mo stretchy="false">)</mo> <mrow><mo>≥</mo><mn>0</mn></mrow></msub><mo>↪</mo><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> K : Ho(SymmMonCat) \stackrel{\simeq}{\to} Ho(Spectra)_{\geq 0} \hookrightarrow Ho(Spectra) </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> of the <a class="existingWikiWord" href="/nlab/show/stable+homotopy+category">stable homotopy category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ho</mi><mo stretchy="false">(</mo><mi>Spectra</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ho(Spectra)</annotation></semantics></math> on the <a class="existingWikiWord" href="/nlab/show/connective+spectra">connective spectra</a> and the <a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymmMonCat</mi></mrow><annotation encoding="application/x-tex">SymmMonCat</annotation></semantics></math>, regarded with the <a class="existingWikiWord" href="/nlab/show/transferred+model+structure">transferred structure</a> of a <a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a>.</p> <p>This is due to (<a href="#Thomason">Thomason, 95</a>). Further discussion is in (<a href="#Mandell">Mandell, 2010</a>).</p> <p>Notice that this is <em>almost</em> the complete analog in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> of the <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the <a class="existingWikiWord" href="/nlab/show/Thomason+model+structure">Thomason model structure</a> on <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a> and the standard <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure on simplicial sets</a>. Only that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymmMonCat</mi></mrow><annotation encoding="application/x-tex">SymmMonCat</annotation></semantics></math> cannot carry a <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> structure because it does not have all <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>s. In some sense the “colimit completion” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymmMonCat</mi></mrow><annotation encoding="application/x-tex">SymmMonCat</annotation></semantics></math> is the category of <a class="existingWikiWord" href="/nlab/show/multicategories">multicategories</a>. Once expects that this carries a model structure that refines the above equivalence of homotopy categories to a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a>.</p> <p>(This is currently being investigated by Elmendorf, Nikolaus and maybe others.)</p> <p>However, a subcategory of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymmMonCat</mi></mrow><annotation encoding="application/x-tex">SymmMonCat</annotation></semantics></math> whose objects are <a class="existingWikiWord" href="/nlab/show/permutative+categories">permutative categories</a> and maps are symmetric strict monoidal functors, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Perm</mi></mrow><annotation encoding="application/x-tex">Perm</annotation></semantics></math> has a model category structure which is transferred from the natural model category structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, see <a href="#Sharma">Sharma</a>. This model category structure is combinatorial, left-proper and a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>-model category structure. It is referred to as the natural model category structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Perm</mi></mrow><annotation encoding="application/x-tex">Perm</annotation></semantics></math>. The <a class="existingWikiWord" href="/nlab/show/coherence+theorem">coherence theorem</a> for symmetric monoidal categories states that each symmetric monoidal category is equivalent to a <a class="existingWikiWord" href="/nlab/show/permutative+category">permutative category</a>.</p> <h3 id="relation_to_categories">Relation to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-categories</h3> <p>The aforementioned natural model category of permutative categories is NOT a symmetric monoidal closed model category. This shortcoming was overcome in [Sharma] by constructing a Quillen equivalent <a class="existingWikiWord" href="/nlab/show/model+category">model category</a> which is symmetric monoidal closed. A (unnormalized) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-category is a functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Gamma^{op}</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Γ</mi> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">\Gamma^{op}</annotation></semantics></math> is a skeletal category of finite based sets and based maps. The category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-categories and natural transformations, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>, is a symmetric monoidal closed category under the <a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a> product. The aforementioned symmetric monoidal closed model category is constructed in <a href="#Sharma">Sharma</a> as a left-Bousfield localization of the projective model category structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Cat</mi></mrow><annotation encoding="application/x-tex">Cat</annotation></semantics></math>. A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-category is fibrant in this model category if it satisfies the Segal’s condition in which case it is referred to as a coherently commutative monoidal category. The main result of <a href="#Sharma">Sharma</a> is that an unnormalized version of the classical Segal’s nerve functor is the right <a class="existingWikiWord" href="/nlab/show/Quillen+functor">Quillen functor</a> of a <a class="existingWikiWord" href="/nlab/show/Quillen+equivalence">Quillen equivalence</a> between the natural model category of permutative categories and the symmetric monoidal closed model category of coherently commutative monoidal categories.</p> <h3 id="as_algebras_over_the_little_cubes_operad">As algebras over the little <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-cubes operad</h3> <p>A symmetric monoidal category is equivalently a category that is equipped with the structure of an <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over</a> the <a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little k-cubes operad</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>≥</mo><mn>3</mn></mrow><annotation encoding="application/x-tex">k \geq 3</annotation></semantics></math></p> <p>Details are in examples 1.2.3 and 1.2.4 of</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jacob+Lurie">Jacob Lurie</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔼</mi><mo stretchy="false">[</mo><mi>k</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{E}[k]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/Ek-Algebras">Algebras</a></li> </ul> <h3 id="tannaka_duality">Tannaka duality</h3> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h3 id="grothendieck_ring">Grothendieck ring</h3> <p>The <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> naturally has the structure of a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>, of an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian</a> monoidal category that of a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>, of an abelian <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> that of a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> and, finally, of an abelian symmetric monoidal category that of a <em><a class="existingWikiWord" href="/nlab/show/Lambda-ring">Lambda-ring</a></em>. See there for more.</p> <h3 id="internal_logic">Internal logic</h3> <p>The <a class="existingWikiWord" href="/nlab/show/internal+logic">internal logic</a> of (<a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a>) symmetric monoidal categories is called <em><a class="existingWikiWord" href="/nlab/show/linear+logic">linear logic</a></em>. This notably contains <a class="existingWikiWord" href="/nlab/show/quantum+logic">quantum logic</a>.</p> <h3 id="permutations">Permutations</h3> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \ge 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>𝔖</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \mathfrak{S}_{n}</annotation></semantics></math>, there is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow></msub><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>A</mi> <mrow><msup><mi>σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mi>A</mi> <mrow><msup><mi>σ</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma_{A_{1},...,A_{n}}: A_{1} \otimes ... \otimes A_{n} \rightarrow A_{\sigma^{-1}(1)} \otimes ... \otimes A_{\sigma^{-1}(n)}</annotation></semantics></math></div> <p>that is defined by the decomposition of a <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> in a product of adjacent <a class="existingWikiWord" href="/nlab/show/transpositions">transpositions</a> and generalizes the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> B:A_{1} \otimes A_{2} \rightarrow A_{2} \otimes A_{1} </annotation></semantics></math></div> <p>We recall that an adjacent transposition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[1,n] \rightarrow [1,n]</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \ge 2</annotation></semantics></math> is a permutation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>:</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo><mo>→</mo><mo stretchy="false">[</mo><mn>1</mn><mo>,</mo><mi>n</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">t:[1,n] \rightarrow [1,n]</annotation></semantics></math> such that there exists a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \le i \le n-1</annotation></semantics></math> such that:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mn>1</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi><mo>−</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>1</mn><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>i</mi><mo>+</mo><mn>2</mn><mo stretchy="false">)</mo><mo>=</mo><mi>i</mi><mo>+</mo><mn>2</mn><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><mi>t</mi><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo><mo>=</mo><mi>n</mi></mrow><annotation encoding="application/x-tex"> t(1)=1,..., t(i-1)=i-1, t(i) = i+1, t(i+1) = i, t(i+2)=i+2,.... ,t(n)=n </annotation></semantics></math></div> <p>This <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is then unique, and for every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \le i \le n-1</annotation></semantics></math>, we note <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t_{i}</annotation></semantics></math> the associated adjacent transposition.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \ge 2</annotation></semantics></math>, every permutation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>𝔖</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \mathfrak{S}_{n}</annotation></semantics></math> admits at least a decomposition under the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>=</mo><msub><mi>t</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mo>;</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>;</mo><msub><mi>t</mi> <mrow><msub><mi>i</mi> <mi>q</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">\sigma = t_{i_{1}}; ...;t_{i_{q}}</annotation></semantics></math> where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>≥</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">q \ge 0</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><msub><mi>i</mi> <mn>1</mn></msub><mo>,</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>,</mo><msub><mi>i</mi> <mi>q</mi></msub><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \le i_{1},....,i_{q} \le n-1</annotation></semantics></math>.</p> <p>In every symmetric monoidal category, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>≥</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">n \ge 2</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>≤</mo><mi>i</mi><mo>≤</mo><mi>n</mi><mo>−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1 \le i \le n-1</annotation></semantics></math>, we have a natural transformation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>i</mi></msub><mo>:</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mi>A</mi> <mi>n</mi></msub><mo>→</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mi>A</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>⊗</mo><msub><mi>A</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub><mo>⊗</mo><msub><mi>A</mi> <mi>i</mi></msub><mo>⊗</mo><msub><mi>A</mi> <mrow><mi>i</mi><mo>+</mo><mn>2</mn></mrow></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mi>A</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">t_{i}:A_{1} \otimes .... \otimes A_{n} \rightarrow A_{1} \otimes ... \otimes A_{i-1} \otimes A_{i+1} \otimes A_{i} \otimes A_{i+2} \otimes .... \otimes A_{n}</annotation></semantics></math></div> <p>defined by:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mi>i</mi></msub><mo>=</mo><msub><mn>1</mn> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mn>1</mn> <mrow><msub><mi>A</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow></msub><mo>⊗</mo><msub><mo lspace="0em" rspace="thinmathspace">B</mo> <mrow><msub><mi>A</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>A</mi> <mrow><mi>i</mi><mo>+</mo><mn>1</mn></mrow></msub></mrow></msub><mo>⊗</mo><msub><mn>1</mn> <mrow><msub><mi>A</mi> <mrow><mi>i</mi><mo>+</mo><mn>2</mn></mrow></msub></mrow></msub><mo>⊗</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>⊗</mo><msub><mn>1</mn> <mrow><msub><mi>A</mi> <mi>n</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">t_{i} = 1_{A_{1}} \otimes ... \otimes 1_{A_{i-1}} \otimes \B_{A_{i},A_{i+1}} \otimes 1_{A_{i+2}} \otimes ... \otimes 1_{A_{n}}</annotation></semantics></math></div> <p>Given a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo>∈</mo><msub><mi>𝔖</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\sigma \in \mathfrak{S}_{n}</annotation></semantics></math>, the associated natural transformation is then defined as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>t</mi> <mrow><msub><mi>i</mi> <mn>1</mn></msub></mrow></msub><mo>;</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>.</mo><mo>;</mo><msub><mi>t</mi> <mrow><msub><mi>i</mi> <mi>q</mi></msub></mrow></msub></mrow><annotation encoding="application/x-tex">t_{i_{1}};....;t_{i_{q}}</annotation></semantics></math> for any such decomposition of <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>. The fact that the result doesn’t depend on the particular decomposition is a consequence of the <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+symmetric+monoidal+categories">coherence theorem for symmetric monoidal categories</a>.</p> <h2 id="examples">Examples</h2> <ul> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> is necessarily symmetric monoidal, due to the essential uniqueness of the categorical <a class="existingWikiWord" href="/nlab/show/product">product</a>. This includes cases such as <a class="existingWikiWord" href="/nlab/show/Set">Set</a>, <a class="existingWikiWord" href="/nlab/show/Cat">Cat</a>.</p> </li> <li> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> some <a class="existingWikiWord" href="/nlab/show/field">field</a>, the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-vector spaces carries the standard structure of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> coming from the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>, over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, of vector spaces. The standard braiding that identifies <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>⊗</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">V \otimes W</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>⊗</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">W \otimes V</annotation></semantics></math> by mapping homogeneous elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>w</mi></mrow><annotation encoding="application/x-tex">v \otimes w</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi><mo>⊗</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">w \otimes v</annotation></semantics></math> obviously makes <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> into a symmetric monoidal category.</p> </li> <li> <p>The category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a>s, on the other hand, has two different symmetric monoidal extensions of the standard <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> monoidal structure. One is the trivial one from above, the other is the one that induces a a sign when two odd-graded vectors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi></mrow><annotation encoding="application/x-tex">v</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>w</mi></mrow><annotation encoding="application/x-tex">w</annotation></semantics></math> are passed past each other : <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>v</mi><mo>⊗</mo><mi>w</mi><mo>↦</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>w</mi><mo>⊗</mo><mi>v</mi></mrow><annotation encoding="application/x-tex">v \otimes w \mapsto - w \otimes v</annotation></semantics></math>. This non-trivial symmetric monoidal structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo stretchy="false">[</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">Vect[\mathbb{Z}_2]</annotation></semantics></math> defines the symmetric monoidal category of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a>.</p> </li> <li> <p>The monoidal category of <a class="existingWikiWord" href="/nlab/show/graded+modules">graded modules</a> over a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> (with the usual <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of graded modules</a>) can be made into a braided monoidal category with the braiding</p> </li> </ul> <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>V</mi><mo>⊗</mo><mi>W</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>W</mi><mo>⊗</mo><mi>V</mi></mtd></mtr> <mtr><mtd><mi>x</mi><mo>⊗</mo><mi>y</mi></mtd> <mtd><mo>↦</mo></mtd> <mtd><mi>y</mi><mo>⊗</mo><mi>x</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ V \otimes W &\longrightarrow& W \otimes V \\ x \otimes y &\mapsto& y \otimes x } \,. </annotation></semantics></math></div> <p>The braiding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>↦</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></msup><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x \otimes y \mapsto (-1)^{|x| |y|} y \otimes x</annotation></semantics></math> (where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|x|</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow><annotation encoding="application/x-tex">|y|</annotation></semantics></math> denote the degrees) is also commonly used.</p> <p>More generally, for any invertible element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>u</mi></mrow><annotation encoding="application/x-tex">u</annotation></semantics></math> of the base ring, there is the <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>↦</mo><msup><mi>u</mi> <mrow><mo stretchy="false">|</mo><mi>x</mi><mo stretchy="false">|</mo><mo stretchy="false">|</mo><mi>y</mi><mo stretchy="false">|</mo></mrow></msup><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">x \otimes y \mapsto u^{|x| |y|} y \otimes x</annotation></semantics></math>, and these braidings are the only possible. The resulting <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> is symmetric if and only if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>u</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">u^2 = 1</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> </li> <li> <p><strong>symmetric monoidal category</strong>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2Cn%29-category">symmetric monoidal (∞,n)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/permutative+category">permutative category</a>, <a class="existingWikiWord" href="/nlab/show/bipermutative+category">bipermutative category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger+category">symmetric monoidal dagger category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> , <a class="existingWikiWord" href="/nlab/show/closed+monoidal+%28%E2%88%9E%2C1%29-category">closed monoidal (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+closed+category">symmetric monoidal closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivariant+symmetric+monoidal+category">equivariant symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+bicategory">symmetric bicategory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+groupoid">symmetric monoidal groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+2-group">symmetric 2-group</a>, <a class="existingWikiWord" href="/nlab/show/abelian+%E2%88%9E-group">abelian ∞-group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supply+in+a+monoidal+category">supply in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoidal+category">commutative monoidal category</a></p> </li> </ul> <h2 id="references">References</h2> <p>Original references:</p> <ul> <li id="MacLane"> <p><a class="existingWikiWord" href="/nlab/show/Saunders+Mac+Lane">Saunders Mac Lane</a>, <em>Natural Associativity and Commutativity</em> , Rice University Studies <strong>49</strong> (1963) 28-46 [<a href="https://scholarship.rice.edu/handle/1911/62865">rice:1911/62865</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Jean+B%C3%A9nabou">Jean Bénabou</a>, <em>Algèbre élémentaire dans les catégories</em> (1964), C. R. Acad. Sci. Paris <strong>258</strong> (1964) 771-774 [<a href="https://gallica.bnf.fr/ark:/12148/bpt6k40102/f817">ark:12148/bpt6k40102/f817</a>]</p> </li> <li id="EilenbergKelly65"> <p><a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">Samuel Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/G.+Max+Kelly">G. Max Kelly</a>, Part III of: <em>Closed Categories</em>, <a class="existingWikiWord" href="/nlab/show/Samuel+Eilenberg">S. Eilenberg</a>, <a class="existingWikiWord" href="/nlab/show/D.+K.+Harrison">D. K. Harrison</a>, <a class="existingWikiWord" href="/nlab/show/S.+MacLane">S. MacLane</a>, <a class="existingWikiWord" href="/nlab/show/H.+R%C3%B6hrl">H. Röhrl</a> (eds.): <em><a class="existingWikiWord" href="/nlab/show/Proceedings+of+the+Conference+on+Categorical+Algebra+-+La+Jolla+1965">Proceedings of the Conference on Categorical Algebra - La Jolla 1965</a></em>, Springer (1966) 421-562 [<a href="https://doi.org/10.1007/978-3-642-99902-4">doi:10.1007/978-3-642-99902-4</a>]</p> </li> </ul> <p>Textbook accounts:</p> <ul> <li id="Borceux94"> <p><a class="existingWikiWord" href="/nlab/show/Francis+Borceux">Francis Borceux</a>, Section 6.1 of: <em>Handbook of Categorical Algebra</em> Vol. 2: <em>Categories and Structures</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://doi.org/10.1017/CBO9780511525865">doi:10.1017/CBO9780511525865</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math>, Encyclopedia of Mathematics and its Applications <strong>50</strong>, Cambridge University Press (1994)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Saunders+MacLane">Saunders MacLane</a>, Ch. XI of: <em><a class="existingWikiWord" href="/nlab/show/Categories+for+the+Working+Mathematician">Categories for the Working Mathematician</a></em>, Graduate Texts in Mathematics <strong>5</strong> Springer (second ed. 1997) [<a href="https://link.springer.com/book/10.1007/978-1-4757-4721-8">doi:10.1007/978-1-4757-4721-8</a>]</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>Tensor Categories</em>, AMS Mathematical Surveys and Monographs <strong>205</strong> (2015) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://bookstore.ams.org/surv-205">ISBN:978-1-4704-3441-0</a>, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></p> <blockquote> <p>(focused on <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a>)</p> </blockquote> </li> </ul> <p>Exposition of basics of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> and <a class="existingWikiWord" href="/nlab/show/categorical+algebra">categorical algebra</a>:</p> <ul> <li><em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+categories+and+toposes">geometry of physics – categories and toposes</a></em>, Section 2: <em><a href="geometry+of+physics+--+categories+and+toposes#BasicNotionsOfCategoricalAlgebra">Basic notions of categorical algebra</a></em></li> </ul> <p>A survey of definitions of symmetric monoidal categories, <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a>s and <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+natural+transformation">symmetric monoidal natural transformation</a>s, is also in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, <em><a href="http://math.ucr.edu/home/baez/qg-fall2004/definitions.pdf">Some definitions everyone should know</a></em></li> </ul> <p>For an elementary introduction to symmetric monoidal categories using string diagrams, see:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a> and <a class="existingWikiWord" href="/nlab/show/Mike+Stay">Mike Stay</a>, <a href="http://math.ucr.edu/home/baez/rosetta.pdf">Physics, topology, logic and computation: a Rosetta Stone</a>.</li> </ul> <p>The theorem that symmetric monoidal categories model all connective spectra is due to</p> <ul id="Thomason"> <li>R. Thomason, <em>Symmetric monoidal categories model all connective spectra</em> , Theory and applications of Categories, Vol. 1, No. 5, (1995) pp. 78-118</li> </ul> <p>More discussion is in</p> <ul id="Mandell"> <li><a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <em>An Inverse K-Theory Functor</em> (<a href="http://arxiv.org/abs/1002.3622">arXiv:1002.3622</a>)</li> </ul> <ul id="Sharma"> <li id="FongSpivak"> <p>Brendan Fong and David I, Spivak, <em>Supplying bells and whistles in symmetric monoidal categories</em>, <a href="https://arxiv.org/abs/1908.02633">arxiv</a>, 2019</p> </li> <li id="Schaeppi"> <p>Daniel Schäppi, <em>Ind-abelian categories and quasi-coherent sheaves</em>, <a href="https://arxiv.org/abs/1211.3678">arxiv</a>, 2014</p> </li> <li> <p>Amit Sharma, <em>Symmetric monoidal categories and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-categories</em> Theory and applications of Categories, Vol. 35, No. 14, (2020) pp. 417-512</p> </li> </ul> <ul> <li> <p>Stefano Kasangian and Fabio Rossi. <em>Some remarks on symmetry for a monoidal category</em>. Bulletin of the Australian Mathematical Society 23.2 (1981): 209-214.</p> </li> <li> <p>J. M. Egger. <em>On involutive monoidal categories</em>. Theory and Applications of Categories 25.14 (2011): 368-393.</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 12, 2024 at 22:41:42. 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