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1.1.4-5.2-10-8.2-13.5-10-11.1-5.2-30-15.3-35-27.3-2.5-6 2.8-13.8 9.4-13.6 6.9.2 13.4 7 17.5 12C70.9 34 75 43.8 86.3 47.4z"/> </svg> </span> <span class="webName">nLab</span> super vector space </h1> <div class="navigation"> <span class="skipNav"><a href='#navEnd'>Skip the Navigation Links</a> | </span> <span style="display:inline-block; width: 0.3em;"></span> <a href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/7503/#Item_2" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" 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<ul> <li> <p><a class="existingWikiWord" href="/nlab/show/algebra">algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry">geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/graded+object">graded object</a></p> </li> </ul> <h2 id="introductions">Introductions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="superalgebra">Superalgebra</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+commutative+monoid">super commutative monoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+abelian+group">super abelian group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+ring">super ring</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+ring">supercommutative ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+ring">exterior ring</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+ring">Clifford ring</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>, <a class="existingWikiWord" href="/nlab/show/SVect">SVect</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+algebra">super algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exterior+algebra">exterior algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Clifford+algebra">Clifford algebra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a> (<a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-infinity+algebra">super L-infinity algebra</a></p> </li> </ul> <h2 id="supergeometry">Supergeometry</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Cartesian+space">super Cartesian space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/SDiff">SDiff</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/NQ-supermanifold">NQ-supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/complex+supermanifold">complex supermanifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Euclidean+supermanifold">Euclidean supermanifold</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+spacetime">super spacetime</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Minkowski+spacetime">super Minkowski spacetime</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/integration+over+supermanifolds">integration over supermanifolds</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Berezin+integral">Berezin integral</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Lie+group">super translation group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Euclidean+group">super Euclidean group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+formal+smooth+%E2%88%9E-groupoid">super formal smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+line+2-bundle">super line 2-bundle</a></p> </li> </ul> <h2 id="supersymmetry">Supersymmetry</h2> <p><a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/division+algebra+and+supersymmetry">division algebra and supersymmetry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Poincare+Lie+algebra">super Poincare Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supermultiplet">supermultiplet</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BPS+state">BPS state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/M-theory+super+Lie+algebra">M-theory super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/type+II+super+Lie+algebra">type II super Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a>, <a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> </ul> <h2 id="supersymmetric_field_theory">Supersymmetric field theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superfield">superfield</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supersymmetric+quantum+mechanics">supersymmetric quantum mechanics</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/superparticle">superparticle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adinkra">adinkra</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+Yang-Mills+theory">super Yang-Mills theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity">supergravity</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/gauged+supergravity">gauged supergravity</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superstring+theory">superstring theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/type+II+string+theory">type II string theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/heterotic+string+theory">heterotic string theory</a></p> </li> </ul> </li> </ul> <h2 id="applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+model+for+elliptic+cohomology">geometric model for elliptic cohomology</a></li> </ul> <div> <p> <a href="/nlab/edit/supergeometry+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="linear_algebra">Linear algebra</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>, <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong></p> <p>flavors: <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+homotopy+theory">equivariant</a>, <a class="existingWikiWord" href="/nlab/show/rational+homotopy+theory">rational</a>, <a class="existingWikiWord" href="/nlab/show/p-adic+homotopy+theory">p-adic</a>, <a class="existingWikiWord" href="/nlab/show/proper+homotopy+theory">proper</a>, <a class="existingWikiWord" href="/nlab/show/geometric+homotopy+theory">geometric</a>, <a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+theory">cohesive</a>, <a class="existingWikiWord" href="/nlab/show/directed+homotopy+theory">directed</a>…</p> <p>models: <a class="existingWikiWord" href="/nlab/show/topological+homotopy+theory">topological</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+homotopy+theory">simplicial</a>, <a class="existingWikiWord" href="/nlab/show/localic+homotopy+theory">localic</a>, …</p> <p>see also <strong><a class="existingWikiWord" href="/nlab/show/algebraic+topology">algebraic topology</a></strong></p> <p><strong>Introductions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Topology+--+2">Introduction to Basic Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Introduction+to+Homotopy+Theory">Introduction to Abstract Homotopy Theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+homotopy+types">geometry of physics – homotopy types</a></p> </li> </ul> <p><strong>Definitions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>, <a class="existingWikiWord" href="/nlab/show/higher+homotopy">higher homotopy</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+type">homotopy type</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pi-algebra">Pi-algebra</a>, <a class="existingWikiWord" href="/nlab/show/spherical+object+and+Pi%28A%29-algebra">spherical object and Pi(A)-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+category+theory">homotopy coherent category theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/category+of+fibrant+objects">category of fibrant objects</a>, <a class="existingWikiWord" href="/nlab/show/cofibration+category">cofibration category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Waldhausen+category">Waldhausen category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+category">homotopy category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Ho%28Top%29">Ho(Top)</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+category+of+an+%28%E2%88%9E%2C1%29-category">homotopy category of an (∞,1)-category</a></li> </ul> </li> </ul> <p><strong>Paths and cylinders</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/left+homotopy">left homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cylinder+object">cylinder object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cone">mapping cone</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/right+homotopy">right homotopy</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+object">path object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/mapping+cocone">mapping cocone</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+universal+bundle">universal bundle</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/interval+object">interval object</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+localization">homotopy localization</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/infinitesimal+interval+object">infinitesimal interval object</a></p> </li> </ul> </li> </ul> <p><strong>Homotopy groups</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+group">homotopy group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+a+topos">fundamental group of a topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Brown-Grossman+homotopy+group">Brown-Grossman homotopy group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/categorical+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">categorical homotopy groups in an (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+homotopy+groups+in+an+%28%E2%88%9E%2C1%29-topos">geometric homotopy groups in an (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%28%E2%88%9E%2C1%29-category">fundamental (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+category">fundamental category</a></li> </ul> </li> </ul> <p><strong>Basic facts</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/fundamental+group+of+the+circle+is+the+integers">fundamental group of the circle is the integers</a></li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+theorem+of+covering+spaces">fundamental theorem of covering spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freudenthal+suspension+theorem">Freudenthal suspension theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Blakers-Massey+theorem">Blakers-Massey theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+homotopy+van+Kampen+theorem">higher homotopy van Kampen theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nerve+theorem">nerve theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Whitehead%27s+theorem">Whitehead's theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hurewicz+theorem">Hurewicz theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Galois+theory">Galois theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem</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='#details'>Details</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#StructuresInternalToSuperVectorSpaces'>Structures internal to super-vector spaces</a></li> <li><a href='#delignes_theorem_on_tensor_categories'>Deligne’s theorem on tensor categories</a></li> </ul> <li><a href='#related_entries'>Related entries</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <strong>super vector space</strong> is an <a class="existingWikiWord" href="/nlab/show/object">object</a> in the non-trivial <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> structure on the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a>: as an object it is just a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></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>, but the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of the underlying <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> is taken to be the non-trivial linear map which on elements of homogeneous degree is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>τ</mi> <mi>super</mi></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>v</mi><mo>⊗</mo><mi>w</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>w</mi><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><mi>w</mi><mo>⊗</mo><mi>v</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau^{super} \;\colon\; v \otimes w \;\mapsto\; (-1)^{deg(v) deg(w)} \, w \otimes v \,. </annotation></semantics></math></div> <p>We make this precise as definition <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a> below.</p> <p>Super vector spaces form the basis of <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> (over <a class="existingWikiWord" href="/nlab/show/ground+rings">ground rings</a> which are <a class="existingWikiWord" href="/nlab/show/fields">fields</a>) in direct analogy of how ordinary vector spaces form the basis of ordinary <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>. For more on this see <a href="#StructuresInternalToSuperVectorSpaces">below</a>, and for yet more see at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+superalgebra">geometry of physics – superalgebra</a></em>.</p> <h2 id="details">Details</h2> <div class="num_defn" id="VectorSpaces"> <h6 id="definition">Definition</h6> <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Vect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Vect_k</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> whose</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/object">objects</a> are <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>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/morphism">morphisms</a> are <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>-<a class="existingWikiWord" href="/nlab/show/linear+functions">linear functions</a> between these.</p> </li> </ul> <p>When the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <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> is understood or when its precise nature is irrelevant, we will often notationally suppress it and speak of just the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of vector spaces.</p> <p>This is the category inside which <a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a> takes place.</p> </div> <p>Of course the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> has some special properties. Not only are its objects “linear spaces”, but the whole category inherits linear structure of sorts. This is traditionally captured by the following terminology for <strong><a class="existingWikiWord" href="/nlab/show/additive+and+abelian+categories">additive and abelian categories</a></strong>. Notice that there are several different but equivalent ways to state the following properties (discussed behind the relevant links).</p> <div class="num_defn" id="AdditiveAndAbelianCategories"> <h6 id="definition_2">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/category">category</a>.</p> <ol> <li> <p>Say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> has <strong><a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a></strong> if it has <a class="existingWikiWord" href="/nlab/show/finite+products">finite products</a> and <a class="existingWikiWord" href="/nlab/show/finite+coproducts">finite coproducts</a> and if the canonical comparison morphism between these is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a>. We write <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 \oplus W</annotation></semantics></math> for the direct sum of two objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> <li> <p>Say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a></strong> if it has <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a> and in addition it is <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">enriched in abelian groups</a>, meaning that every <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> is equipped with the structure of an <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> such that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of morphisms is a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a>.</p> </li> <li> <p>Say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is an <strong><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></strong> if it is an <a class="existingWikiWord" href="/nlab/show/additive+category">additive category</a> and has property that its <a class="existingWikiWord" href="/nlab/show/monomorphisms">monomorphisms</a> are precisely the inclusions of <a class="existingWikiWord" href="/nlab/show/kernels">kernels</a> and its <a class="existingWikiWord" href="/nlab/show/epimorphisms">epimorphisms</a> are precisely the projections onto <a class="existingWikiWord" href="/nlab/show/cokernels">cokernels</a>.</p> </li> </ol> </div> <p>We also make the following definition 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>-linear category, but notice that conventions differ as to which extra properties beyond <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enrichment</a> to require on a linear category:</p> <div class="num_defn" id="LinearCategory"> <h6 id="definition_3">Definition</h6> <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a> (or more generally just a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a>), call 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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> a <strong><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>-<a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a></strong> if</p> <ol> <li> <p>it is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> (def. <a class="maruku-ref" href="#AdditiveAndAbelianCategories"></a>);</p> </li> <li> <p>its <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> have the structure 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>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (generally <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>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a>) such that <a class="existingWikiWord" href="/nlab/show/composition">composition</a> of morphisms in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a></p> </li> </ol> <p>and the underlying additive <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> structure of these <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> is that of the underlying <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>.</p> <p>In other words, a <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>-linear category is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> with the additional structure of a <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched category</a> (generally <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><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a>-enriched) such that the underlying <a class="existingWikiWord" href="/nlab/show/Ab-enriched+category">Ab-enrichment</a> according to def. <a class="maruku-ref" href="#AdditiveAndAbelianCategories"></a> is obtained from the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math>-enrichment under the <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo>→</mo><mi>Ab</mi></mrow><annotation encoding="application/x-tex">Vect \to Ab</annotation></semantics></math>.</p> <p>A <a class="existingWikiWord" href="/nlab/show/functor">functor</a> between <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>-linear categories is called a <strong><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>-<a class="existingWikiWord" href="/nlab/show/linear+functor">linear functor</a></strong> if its component functions on <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> are <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> with respect to the given <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>-linear structure, hence if it is a <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>-<a class="existingWikiWord" href="/nlab/show/enriched+functor">enriched functor</a>.</p> </div> <div class="num_example"> <h6 id="example">Example</h6> <p>The category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (def. <a class="maruku-ref" href="#VectorSpaces"></a>) is a <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>-<a class="existingWikiWord" href="/nlab/show/linear+category">linear category</a> according to def. <a class="maruku-ref" href="#LinearCategory"></a>.</p> <p>Here the abstract <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> is the usual direct sum of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, whence the name of the general concept.</p> <p>For <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,W</annotation></semantics></math> two <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, the vector space structure on the <a class="existingWikiWord" href="/nlab/show/hom-set">hom-set</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>Vect</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{Vect}(V,W)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>→</mo><mi>W</mi></mrow><annotation encoding="application/x-tex">\phi \colon V \to W</annotation></semantics></math> is given by “pointwise” multiplication and addition of functions:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>c</mi> <mn>2</mn></msub><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thinmathspace"></mspace><mi>v</mi><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><msub><mi>c</mi> <mn>1</mn></msub><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>c</mi> <mn>2</mn></msub><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (c_1 \phi_1 + c_2 \phi_2) \;\colon\, v \;\mapsto\; c_1 \phi_1(v) + c_2 \phi_2(v) </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">c_1, c_2 \in k</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>Hom</mi> <mi>Vect</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>W</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\phi_1, \phi_2 \in Hom_{Vect}(V,W)</annotation></semantics></math>.</p> </div> <p>Recall the basic construction of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>:</p> <div class="num_defn" id="TensorProductOfVectorSpaces"> <h6 id="definition_4">Definition</h6> <p>Given two <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> over some <a class="existingWikiWord" href="/nlab/show/field">field</a> <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>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>Vect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">V_1, V_2 \in Vect_k</annotation></semantics></math>, their <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> is the vector space denoted</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>V</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> V_1 \otimes_k V_2 \;\in\; Vect </annotation></semantics></math></div> <p>whose elements are <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> of elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1,v_2)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i \in V_i</annotation></semantics></math>, for the <a class="existingWikiWord" href="/nlab/show/equivalence+relation">equivalence relation</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>c</mi><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>c</mi><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (c v_1 , v_2) \;\sim\; (v_1, c v_2) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>+</mo><mi>v</mi><msub><mo>′</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><mi>v</mi><msub><mo>′</mo> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (v_1 + v'_1 , v_2) \; \sim \; (v_1,v_2) + (v'_1, v_2) </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>+</mo><mi>v</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∼</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mi>v</mi><msub><mo>′</mo> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (v_1 , v_2 + v'_2) \; \sim \; (v_1,v_2) + (v_1, v'_2) </annotation></semantics></math></div> <p>More abstractly this means that the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> is the vector space characterized by the fact that</p> <ol> <li> <p>it receives a <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> V_1 \times V_2 \longrightarrow V_1 \otimes V_2 </annotation></semantics></math></div> <p>(out of the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of the underlying sets)</p> </li> <li> <p>any other <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>⟶</mo><msub><mi>V</mi> <mn>3</mn></msub></mrow><annotation encoding="application/x-tex"> V_1 \times V_2 \longrightarrow V_3 </annotation></semantics></math></div> <p>factors through the above bilinear map via a unique <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mi>bilinear</mi></mover></mtd> <mtd><msub><mi>V</mi> <mn>3</mn></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↗</mo> <mpadded width="0"><mrow><mo>∃</mo><mo>!</mo><mspace width="thinmathspace"></mspace><mi>linear</mi></mrow></mpadded></msub></mtd></mtr> <mtr><mtd><msub><mi>V</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>V</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ V_1 \times V_2 &\overset{bilinear}{\longrightarrow}& V_3 \\ \downarrow & \nearrow_{\mathrlap{\exists ! \, linear}} \\ V_1 \otimes_k V_2 } </annotation></semantics></math></div></li> </ol> </div> <p>The existence of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>, def. <a class="maruku-ref" href="#TensorProductOfVectorSpaces"></a>, equips the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> of vector spaces with extra structure, which is a “<a class="existingWikiWord" href="/nlab/show/categorification">categorification</a>” of the familiar structure of a <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a>. One also says “<a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>” for <a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a> and therefore <a class="existingWikiWord" href="/nlab/show/categories">categories</a> equipped with a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> operation are also called <em><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></em>:</p> <div class="num_defn" id="MonoidalCategory"> <h6 id="definition_5">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></strong> is 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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> equipped with</p> <ol> <li> <p>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><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>×</mo><mi>𝒞</mi><mo>⟶</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex"> \otimes \;\colon\; \mathcal{C} \times \mathcal{C} \longrightarrow \mathcal{C} </annotation></semantics></math></div> <p>out of the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with itself, called the <strong><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></strong>,</p> </li> <li> <p>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>𝒞</mi></mrow><annotation encoding="application/x-tex"> 1 \in \mathcal{C} </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/unit+object">unit object</a></strong> or <strong><a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a></strong>,</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> a \;\colon\; ((-)\otimes (-)) \otimes (-) \overset{\simeq}{\longrightarrow} (-) \otimes ((-)\otimes(-)) </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/associator">associator</a></strong>,</p> </li> <li id="MonoidalCategoryUnitors"> <p>a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \ell \;\colon\; (1 \otimes (-)) \overset{\simeq}{\longrightarrow} (-) </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/left+unitor">left unitor</a></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><mi>r</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mn>1</mn><mover><mo>⟶</mo><mo>≃</mo></mover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> r \;\colon\; (-) \otimes 1 \overset{\simeq}{\longrightarrow} (-) </annotation></semantics></math></div> <p>called the <strong><a class="existingWikiWord" href="/nlab/show/right+unitor">right unitor</a></strong>,</p> </li> </ol> <p>such that the following two kinds of <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a>, for all objects involved:</p> <ol> <li> <p><strong>triangle 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></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></li> <li> <p>the <strong><a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon identity</a></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></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mi>w</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>α</mi> <mrow><mi>w</mi><mo>⊗</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup><mo>↗</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>α</mi> <mrow><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo>⊗</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><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><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mrow><mpadded width="0" lspace="-100%width"><mrow><msub><mi>α</mi> <mrow><mi>w</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow></mpadded><mo>⊗</mo><msub><mi>id</mi> <mi>z</mi></msub></mrow></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msub><mi>id</mi> <mi>w</mi></msub><mo>⊗</mo><msub><mi>α</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><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></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>α</mi> <mrow><mi>w</mi><mo>,</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></munder></mtd> <mtd></mtd> <mtd><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></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ && (w \otimes x) \otimes (y \otimes z) \\ & {}^{\mathllap{\alpha_{w \otimes x, y, z}}}\nearrow && \searrow^{\mathrlap{\alpha_{w,x,y \otimes z}}} \\ ((w \otimes x ) \otimes y) \otimes z && && (w \otimes (x \otimes (y \otimes z))) \\ {}^{\mathllap{\alpha_{w,x,y}} \otimes id_z }\downarrow && && \uparrow^{\mathrlap{ id_w \otimes \alpha_{x,y,z} }} \\ (w \otimes (x \otimes y)) \otimes z && \underset{\alpha_{w,x \otimes y, z}}{\longrightarrow} && w \otimes ( (x \otimes y) \otimes z ) } </annotation></semantics></math></div></li> </ol> </div> <p>As expected, we have the following basic example:</p> <div class="num_example" id="VectAsAMonoidalCategory"> <h6 id="example_2">Example</h6> <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> a <a class="existingWikiWord" href="/nlab/show/field">field</a>, the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mrow></mrow> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">{}_k</annotation></semantics></math> 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>-<a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> becomes a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) as follows</p> <ul> <li> <p>the abstract <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> is the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_k</annotation></semantics></math> from def. <a class="maruku-ref" href="#TensorProductOfVectorSpaces"></a>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> is the <a class="existingWikiWord" href="/nlab/show/field">field</a> <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> itself, regarded as a 1-dimensional vector space over itself;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> is the map that on representing <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> acts as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>α</mi> <mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>3</mn></msub></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \alpha_{V_{1}, V_2, V_3} \;\colon\; ((v_1, v_2), v_3) \mapsto (v_1, (v_2,v_3)) </annotation></semantics></math></div></li> <li> <p>the left <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a> is the map that on representing <a class="existingWikiWord" href="/nlab/show/tuples">tuples</a> is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mi>V</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>k</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>k</mi><mi>v</mi></mrow><annotation encoding="application/x-tex"> \ell_{V} \colon (k,v) \mapsto k v </annotation></semantics></math></div> <p>and the right unitor is similarly given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>r</mi> <mi>V</mi></msub><mo lspace="verythinmathspace">:</mo><mo stretchy="false">(</mo><mi>v</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↦</mo><mi>k</mi><mi>v</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> r_V \colon (v,k) \mapsto k v \,. </annotation></semantics></math></div></li> </ul> <p>That this satisifes the <a class="existingWikiWord" href="/nlab/show/pentagon+identity">pentagon identity</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) and the left and right unit identities is immediate on representing tuples.</p> </div> <p>But the point of the abstract definition of <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> is that there are also more exotic examples. The followig one is just a minimal enrichment of example <a class="maruku-ref" href="#VectAsAMonoidalCategory"></a>, and yet it will be important.</p> <div class="num_example" id="GradedVectorSpacesAsAMonoidaCategory"> <h6 id="example_3">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/group">group</a> (or in fact just a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a>). A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+space">graded vector space</a></strong> <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> is a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of vector spaces labeled by the elements in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><munder><mo>⊕</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msub><mi>V</mi> <mi>g</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V = \underset{g \in G}{\oplus} V_g \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>V</mi><mo>⟶</mo><mi>W</mi></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; V \longrightarrow W </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-graded vector spaces is a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> that respects this direct sum structure, hence equivalently a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/linear+maps">linear maps</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>g</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mi>g</mi></msub><mo>⟶</mo><msub><mi>W</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex"> \phi_g \;\colon\; V_g \longrightarrow W_g </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</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>ϕ</mi><mo>=</mo><munder><mo>⊕</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msub><mi>ϕ</mi> <mi>g</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi = \underset{g \in G}{\oplus} \phi_g \,. </annotation></semantics></math></div> <p>This defines a <a class="existingWikiWord" href="/nlab/show/category">category</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Vect</mi> <mi>G</mi></msup></mrow><annotation encoding="application/x-tex">Vect^G</annotation></semantics></math>. Equip this category with a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> which on the underlying vector spaces is just the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> from def. <a class="maruku-ref" href="#TensorProductOfVectorSpaces"></a>, equipped with the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-grading which is obtained by multiplying degree labels in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>V</mi><mo>⊗</mo><mi>W</mi><msub><mo stretchy="false">)</mo> <mi>g</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><munder><mo>⊕</mo><mfrac linethickness="0"><mrow><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow></mrow><mrow><mrow><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub><mo>=</mo><mi>g</mi></mrow></mrow></mfrac></munder><msub><mi>V</mi> <mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>V</mi> <mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (V \otimes W)_g \;\coloneqq\; \underset{{g_1, g_2 \in G} \atop {g_1 g_2 = g}}{\oplus} V_{g_1} \otimes_k V_{g_2} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> for the tensor product is the ground field <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>, regarded as being in the degree of the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">e \in G</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mn>1</mn> <mi>g</mi></msub><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>k</mi></mtd> <mtd><mo stretchy="false">|</mo><mi>g</mi><mo>=</mo><mi>e</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mo stretchy="false">|</mo><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 1_g \;=\; \left\{ \array{ k & | g = e \\ 0 & | otherwise } \right. \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/associator">associator</a> and <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> are just those of the monoidal structure on plain vector spaces, from example <a class="maruku-ref" href="#VectAsAMonoidalCategory"></a>.</p> </div> <p>One advantage of abstracting the concept of a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is that it allows to prove general statements uniformly for all kinds of tensor products, familiar ones and more exotic ones. The following lemma <a class="maruku-ref" href="#kel1"></a> and remark <a class="maruku-ref" href="#CoherenceForMonoidalCategories"></a> are two important such statements.</p> <div class="num_lemma" id="kel1"> <h6 id="lemma">Lemma</h6> <p><strong>(<a href="#Kelly64">Kelly 64</a>)</strong></p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, def. <a class="maruku-ref" href="#MonoidalCategory"></a>. Then the left and right <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> satisfy the following conditions:</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℓ</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>r</mi> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⊗</mo><mn>1</mn><mover><mo>⟶</mo><mo>≃</mo></mover><mn>1</mn></mrow><annotation encoding="application/x-tex">\ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1</annotation></semantics></math>;</p> </li> <li> <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><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x,y \in \mathcal{C}</annotation></semantics></math> the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mn>1</mn><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>α</mi> <mrow><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ℓ</mi> <mi>x</mi></msub><mo>⊗</mo><msub><mi>id</mi> <mi>y</mi></msub></mrow></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mn>1</mn><mo>⊗</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ℓ</mi> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>x</mi><mo>⊗</mo><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \array{ (1 \otimes x) \otimes y & & \\ {}^\mathllap{\alpha_{1, x, y}} \downarrow & \searrow^\mathrlap{\ell_x \otimes id_y} & \\ 1 \otimes (x \otimes y) & \underset{\ell_{x \otimes y}}{\longrightarrow} & x \otimes y } \,; </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>y</mi><mo>⊗</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>α</mi> <mrow><mn>1</mn><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>id</mi> <mi>x</mi></msub><mo>⊗</mo><msub><mi>r</mi> <mi>y</mi></msub></mrow></mpadded></msup></mtd> <mtd></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mn>1</mn></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>r</mi> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi></mrow></msub></mrow></munder></mtd> <mtd><mi>x</mi><mo>⊗</mo><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>;</mo></mrow><annotation encoding="application/x-tex"> \array{ x \otimes (y \otimes 1) & & \\ {}^\mathllap{\alpha^{-1}_{1, x, y}} \downarrow & \searrow^\mathrlap{id_x \otimes r_y} & \\ (x \otimes y) \otimes 1 & \underset{r_{x \otimes y}}{\longrightarrow} & x \otimes y } \,; </annotation></semantics></math></div></li> </ol> </div> <p>For <strong>proof</strong> see at <em><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></em> <a href="monoidal+category#kel1">this lemma</a> and <a href="monoidal+category#kel2">this lemma</a>.</p> <div class="num_remark" id="CoherenceForMonoidalCategories"> <h6 id="remark">Remark</h6> <p>Just as for an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> it is sufficient to demand <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mi>a</mi><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">1 a = a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mn>1</mn><mo>=</mo><mi>a</mi></mrow><annotation encoding="application/x-tex">a 1 = a</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>a</mi><mi>b</mi><mo stretchy="false">)</mo><mi>c</mi><mo>=</mo><mi>a</mi><mo stretchy="false">(</mo><mi>b</mi><mi>c</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(a b) c = a (b c)</annotation></semantics></math> in order to have that expressions of arbitrary length may be re-bracketed at will, so there is a <em><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></em> which states that all ways of freely composing the <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> and <a class="existingWikiWord" href="/nlab/show/associators">associators</a> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) to go from one expression to another will coincide. Accordingly, much as one may drop the notation for the bracketing in an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> altogether, so one may, with due care, reason about monoidal categories without always making all unitors and associators explicit.</p> <p>(Here the qualifier “freely” means informally that we must not use any non-formal identification between objects, and formally it means that the diagram in question must be in the image of a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> from a <em>free</em> monoidal category. For example if in a particular monoidal category it so happens that the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">X \otimes (Y \otimes Z)</annotation></semantics></math> is actually <em>equal</em> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Z</mi></mrow><annotation encoding="application/x-tex">(X \otimes Y)\otimes Z</annotation></semantics></math>, then the various ways of going from one expression to another using only associators <em>and</em> this “accidental” equality no longer need to coincide.)</p> </div> <p>The above discussion makes it clear that a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is like a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/semi-group">semi-group</a>, but “<a class="existingWikiWord" href="/nlab/show/categorified">categorified</a>”. Accordingly we may consider additional properties of <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/semi-groups">semi-groups</a> and correspondingly lift them to monoidal categories. A key such property is <em><a class="existingWikiWord" href="/nlab/show/commutative+ring">commutativity</a></em>. But while for a monoid commutativity is just an extra <a class="existingWikiWord" href="/nlab/show/property">property</a>, for a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> it involves choices of commutativity-<a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a> and hence is <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure+and+property">extra structure</a>. We will see <a href="#SuperGroupsAsSuperHopfAlgebras">below</a> that this is the very source of <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/categorification">categorification</a> of “commutativity” comes in two stages: <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> and <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric braiding</a>.</p> <div class="num_defn" id="BraidedMonoidalCategory"> <h6 id="definition_6">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></strong>, is a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) equipped with a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>τ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>→</mo><mi>y</mi><mo>⊗</mo><mi>x</mi></mrow><annotation encoding="application/x-tex"> \tau_{x,y} \;\colon\; x \otimes y \to y \otimes x </annotation></semantics></math></div> <p>(for all <a class="existingWikiWord" href="/nlab/show/objects">objects</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><mi>in</mi><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x,y in \mathcal{C}</annotation></semantics></math>) called the <strong><a class="existingWikiWord" href="/nlab/show/braiding">braiding</a></strong>, such that the following two kinds of <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a> for all <a class="existingWikiWord" href="/nlab/show/objects">objects</a> involved (“hexagon identities”):</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>τ</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>τ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⊗</mo><mi>Id</mi></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><mi>Id</mi><mo>⊗</mo><msub><mi>τ</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{\tau_{x,y \otimes z}}{\to}& (y \otimes z) \otimes x \\ \downarrow^{\tau_{x,y}\otimes Id} &&&& \downarrow^{a_{y,z,x}} \\ (y \otimes x) \otimes z &\stackrel{a_{y,x,z}}{\to}& y \otimes (x \otimes z) &\stackrel{Id \otimes \tau_{x,z}}{\to}& y \otimes (z \otimes x) } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><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><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mtd> <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>τ</mi> <mrow><mi>x</mi><mo>⊗</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mover></mtd> <mtd><mi>z</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mrow><mi>Id</mi><mo>⊗</mo><msub><mi>τ</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></msup></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mrow><msubsup><mi>a</mi> <mrow><mi>z</mi><mo>,</mo><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></msup></mtd></mtr> <mtr><mtd><mi>x</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi><mo>,</mo><mi>y</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>x</mi><mo>⊗</mo><mi>z</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd> <mtd><mover><mo>→</mo><mrow><msub><mi>τ</mi> <mrow><mi>x</mi><mo>,</mo><mi>z</mi></mrow></msub><mo>⊗</mo><mi>Id</mi></mrow></mover></mtd> <mtd><mo stretchy="false">(</mo><mi>z</mi><mo>⊗</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>y</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ x \otimes (y \otimes z) &\stackrel{a^{-1}_{x,y,z}}{\to}& (x \otimes y) \otimes z &\stackrel{\tau_{x \otimes y, z}}{\to}& z \otimes (x \otimes y) \\ \downarrow^{Id \otimes \tau_{y,z}} &&&& \downarrow^{a^{-1}_{z,x,y}} \\ x \otimes (z \otimes y) &\stackrel{a^{-1}_{x,z,y}}{\to}& (x \otimes z) \otimes y &\stackrel{\tau_{x,z} \otimes Id}{\to}& (z \otimes x) \otimes y } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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 lspace="verythinmathspace">:</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} \colon (x \otimes y) \otimes z \to x \otimes (y \otimes z)</annotation></semantics></math> denotes the components of the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝒞</mi> <mo>⊗</mo></msup></mrow><annotation encoding="application/x-tex">\mathcal{C}^\otimes</annotation></semantics></math>.</p> </div> <div class="num_defn" id="SymmetricMonoidalCategory"> <h6 id="definition_7">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></strong> is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> (def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></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>τ</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"> \tau_{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>τ</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow></msub><mo>∘</mo><msub><mi>τ</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"> \tau_{y,x} \circ \tau_{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> <div class="num_remark" id="SymmetricMonoidalCategoriesCoherenceTheorem"> <h6 id="remark_2">Remark</h6> <p>In analogy to the <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a> (remark <a class="maruku-ref" href="#CoherenceForMonoidalCategories"></a>, roughly speaking “all diagrams commute”) there is a <a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+symmetric+monoidal+categories">coherence theorem for symmetric monoidal categories</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>), saying that two parallel morphisms, built freely (see remark <a class="maruku-ref" href="#CoherenceForMonoidalCategories"></a>) from <a class="existingWikiWord" href="/nlab/show/associators">associators</a>, <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> and <a class="existingWikiWord" href="/nlab/show/braidings">braidings</a>, are equal if and only if they correspond to the same <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> of objects.</p> </div> <p>Consider the simplest non-trivial special case of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a> from example <a class="maruku-ref" href="#GradedVectorSpacesAsAMonoidaCategory"></a>, the case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">G = \mathbb{Z}/2</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/cyclic+group+of+order+two">cyclic group of order two</a>.</p> <div class="num_example" id="Z2Zgradedvectorspaces"> <h6 id="example_4">Example</h6> <p>A <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></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></strong> is a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of two vector spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>=</mo><msub><mi>V</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>V</mi> <mi>odd</mi></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> V = V_{even} \oplus V_{odd} \,, </annotation></semantics></math></div> <p>where we think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">V_{even}</annotation></semantics></math> as the summand that is graded by the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>, and of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">V_{odd}</annotation></semantics></math> as being the summand that is graded by the single non-trivial element.</p> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-graded vector spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>V</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; V_1 \longrightarrow V_2 </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> of the underlying vector spaces that respects the grading, hence equivalently a pair of linear maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>even</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex"> f_{even} \;\colon\; (V_1)_{even} \longrightarrow (V_1)_{even} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>odd</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex"> f_{odd} \;\colon\; (V_1)_{odd} \longrightarrow (V_1)_{odd} </annotation></semantics></math></div> <p>between then summands in even degree and in odd degree, respectively:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>=</mo><msub><mi>f</mi> <mi>even</mi></msub><mo>⊕</mo><msub><mi>f</mi> <mi>odd</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f = f_{even} \oplus f_{odd} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-graded vector space is the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a> of the underlying vector spaces, but with the grading obtained from multiplying the original gradings in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>. Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mo>)</mo></mrow><mo>⊕</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mo>)</mo></mrow></mrow><annotation encoding="application/x-tex"> (V_1 \otimes V_2)_{even} \;\coloneqq\; \left((V_1)_{even} \otimes (V_2)_{even}\right) \oplus \left((V_1)_{odd} \otimes (V_2)_{odd}\right) </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mo>)</mo></mrow><mo>⊕</mo><mrow><mo>(</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mi>even</mi></msub><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (V_1 \otimes V_2)_{odd} \;\coloneqq\; \left((V_1)_{even} \otimes (V_2)_{odd}\right) \oplus \left((V_1)_{odd} \otimes (V_2)_{even}\right) \,. </annotation></semantics></math></div> <p>As in example <a class="maruku-ref" href="#GradedVectorSpacesAsAMonoidaCategory"></a>, this definition makes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> def. <a class="maruku-ref" href="#MonoidalCategory"></a>.</p> </div> <div class="num_prop" id="TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces"> <h6 id="proposition">Proposition</h6> <p>There are, up to <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal</a> <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a>, precisely two choices for a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>τ</mi> <mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow></msub></mrow></mover><msub><mi>V</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex"> V_1 \otimes V_2 \stackrel{\tau_{V_1,V_2}}{\longrightarrow} V_2 \otimes V_1 </annotation></semantics></math></div> <p>on the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>Vect</mi> <mi>k</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msubsup><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Vect_k^{\mathbb{Z}/2}, \otimes_k)</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a> from def. <a class="maruku-ref" href="#Z2Zgradedvectorspaces"></a>:</p> <ol> <li> <p>the <strong>trivial braiding</strong> which is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> given on tuples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1,v_2)</annotation></semantics></math> representing an element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">V_1 \otimes V_2</annotation></semantics></math> (according to def. <a class="maruku-ref" href="#TensorProductOfVectorSpaces"></a>) by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>τ</mi> <mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow> <mi>triv</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \tau^{triv}_{V_1, V_2} \;\colon\; (v_1,v_2) \mapsto (v_2, v_1) </annotation></semantics></math></div></li> <li> <p>the <strong>super-braiding</strong> which is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural</a> <a class="existingWikiWord" href="/nlab/show/linear+function">linear function</a> given on tuples <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(v_1,v_2)</annotation></semantics></math> of <em>homogeneous degree</em> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>v</mi> <mi>i</mi></msub><mo>∈</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>i</mi></msub><msub><mo stretchy="false">)</mo> <mrow><msub><mi>σ</mi> <mi>i</mi></msub></mrow></msub><mo>↪</mo><msub><mi>V</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">v_i \in (V_i)_{\sigma_i} \hookrightarrow V_i</annotation></semantics></math>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mi>i</mi></msub><mo>∈</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\sigma_i \in \mathbb{Z}/2</annotation></semantics></math>) by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>τ</mi> <mrow><msub><mi>V</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>V</mi> <mn>2</mn></msub></mrow> <mi>super</mi></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>↦</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>v</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>v</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau^{super}_{V_1, V_2} \;\colon\; (v_1, v_2) \mapsto (-1)^{deg(v_1) deg(v_2)} \, (v_2,v_1) \,. </annotation></semantics></math></div></li> </ol> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Line(\mathcal{C}), \otimes, 1) \hookrightarrow (\mathcal{C}, \otimes, 1) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> on those <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L \in \mathcal{C}</annotation></semantics></math> which are <a class="existingWikiWord" href="/nlab/show/invertible+objects">invertible objects</a> under the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>, i.e. such that there is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L^{-1} \in \mathcal{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>L</mi><mo>⊗</mo><msup><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≃</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">L \otimes L^{-1} \simeq 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>L</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>⊗</mo><mi>L</mi><mo>≃</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">L^{-1} \otimes L \simeq 1</annotation></semantics></math>. Since the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> is clearly in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><mi>L</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(L)</annotation></semantics></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≃</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">1^{-1} \simeq 1</annotation></semantics></math>) and since with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>L</mi> <mn>2</mn></msub><mo>∈</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>↪</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">L_1, L_2 \in Line(\mathcal{C}) \hookrightarrow \mathcal{C}</annotation></semantics></math> also <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>L</mi> <mn>2</mn></msub><mo>∈</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">L_1 \otimes L_2 \in Line(\mathcal{C})</annotation></semantics></math> (with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>L</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>L</mi> <mn>2</mn></msub><msup><mo stretchy="false">)</mo> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo>≃</mo><msubsup><mi>L</mi> <mn>2</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>⊗</mo><msubsup><mi>L</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">(L_1 \otimes L_2)^{-1} \simeq L_2^{-1} \otimes L_1^{-1}</annotation></semantics></math>) the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> restricts to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(\mathcal{C})</annotation></semantics></math>.</p> <p>Accordingly any <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes,1)</annotation></semantics></math> restricts to a braiding on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Line(\mathcal{C}), \otimes, 1)</annotation></semantics></math>. Hence it is sufficient to show that there is an essentially unique non-trivial symmetric braiding on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Line(\mathcal{C}), \otimes, 1)</annotation></semantics></math>, and that this is the restriction of a braiding on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>.</p> <p>Now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Line(\mathcal{C}, \otimes , 1))</annotation></semantics></math> is necessarily a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> (the “<a class="existingWikiWord" href="/nlab/show/Picard+groupoid+of+a+monoidal+category">Picard groupoid</a>” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>) and in fact is what is called a <em><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></em>. As such we may regard it equivalently as a <a class="existingWikiWord" href="/nlab/show/homotopy+1-type">homotopy 1-type</a> with group structure, and as such it it is equivalent to its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</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> <mo>⊗</mo></msub><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> B_\otimes Line(\mathcal{C}) </annotation></semantics></math></div> <p>regarded as a <a class="existingWikiWord" href="/nlab/show/pointed+homotopy+type">pointed homotopy type</a>. (See at <em><a class="existingWikiWord" href="/nlab/show/looping+and+delooping">looping and delooping</a></em>).</p> <p>The <a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>B</mi><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0(Line(\mathcal{C})) \simeq \pi_1(B Line(\mathcal{C})) </annotation></semantics></math></div> <p>the <a class="existingWikiWord" href="/nlab/show/fundamental+group">fundamental group</a> of the delooping space.</p> <p>Now a symmetric braiding on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(\mathcal{C})</annotation></semantics></math> is precisely the structure that makes it a <a class="existingWikiWord" href="/nlab/show/symmetric+2-group">symmetric 2-group</a> which is equivalently the structure of a second <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mn>2</mn></msup><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B^2 Line(\mathcal{C})</annotation></semantics></math> (for the braiding) and then a third delooping <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mn>3</mn></msup><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B^3 Line(\mathcal{C})</annotation></semantics></math> (for the symmetry), regarded as a <a class="existingWikiWord" href="/nlab/show/pointed+homotopy+type">pointed homotopy type</a>.</p> <p>This way we have rephrased the question equivalently as a question about the possible <a class="existingWikiWord" href="/nlab/show/k-invariants">k-invariants</a> of spaces of this form.</p> <p>Now in the case at hand, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(Vect^{\mathbb{Z}/2})</annotation></semantics></math> has precisely two <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of objects, namely the <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a> <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> itself, regarded as being in even degree and regarded as being in odd degree. We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k^{1\vert 0}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k^{0 \vert 1}</annotation></semantics></math> for these, respectively. By the rules of the tensor product of <a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a> we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> k^{1\vert 0} \otimes_k k^{1\vert 0} \simeq k^{1\vert 0} </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex"> k^{1\vert 0} \otimes_k k^{0\vert 1} \simeq k^{0\vert 1} </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> k^{0 \vert 1} \otimes_k k^{0 \vert 1} \simeq k^{1 \vert 0} \,. </annotation></semantics></math></div> <p>In other words</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \pi_0(Line(Vect^{\mathbb{Z}/2})) \simeq \mathbb{Z}/2 \,. </annotation></semantics></math></div> <p>Now under the above homotopical identification the non-trivial braiding is identified with the elements</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><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mover><mo>⟶</mo><mrow><msubsup><mi>τ</mi> <mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow> <mi>super</mi></msubsup></mrow></mover><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>≃</mo><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> 1 = k^{1 \vert 0} \simeq k^{0\vert 1} \otimes_k k^{0 \vert 1} \stackrel{\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}}}{\longrightarrow} k^{0\vert 1} \otimes_k k^{0\vert 1} \simeq k^{1 \vert 0} = 1 </annotation></semantics></math></div> <p>Due to the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetry</a> condition (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msubsup><mi>τ</mi> <mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow> <mi>super</mi></msubsup><msup><mo stretchy="false">)</mo> <mn>2</mn></msup><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex"> (\tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}})^2 = id </annotation></semantics></math></div> <p>which implies that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>τ</mi> <mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup><mo>,</mo><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow> <mi>super</mi></msubsup><mo>∈</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mi>id</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>id</mi><mo stretchy="false">}</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tau^{super}_{k^{0\vert 1}, k^{0 \vert 1}} \in \{+ id, -id\} \,. </annotation></semantics></math></div> <p>Therefore for classifying just the symmetric braidings, it is sufficient to restrict the <a class="existingWikiWord" href="/nlab/show/hom-spaces">hom-spaces</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(Vect^{\mathbb{Z}/2})</annotation></semantics></math> from being either <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> or empty, to <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> being <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>=</mo><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo>−</mo><mn>1</mn><mo stretchy="false">}</mo><mo>↪</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2 = \{+1-1\} \hookrightarrow k</annotation></semantics></math> or empty. Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>Line</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\widetilde{Line}(sVect)</annotation></semantics></math> for the resulting <a class="existingWikiWord" href="/nlab/show/2-group">2-group</a>.</p> <p>In conclusion then the <a class="existingWikiWord" href="/nlab/show/equivalence+classes">equivalence classes</a> of possible <a class="existingWikiWord" href="/nlab/show/k-invariants">k-invariants</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>B</mi> <mn>3</mn></msup><mi>Line</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B^3 Line(sVect)</annotation></semantics></math>, hence the possible symmetric braiding on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Line(Vect^{\mathbb{Z}/2})</annotation></semantics></math> are in the degree-4 <a class="existingWikiWord" href="/nlab/show/ordinary+cohomology">ordinary cohomology</a> of the <a class="existingWikiWord" href="/nlab/show/Eilenberg-MacLane+space">Eilenberg-MacLane space</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K(\mathbb{Z}/2,3)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/coefficients">coefficients</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>. One finds (…)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>H</mi> <mn>4</mn></msup><mo stretchy="false">(</mo><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo>,</mo><mn>3</mn><mo stretchy="false">)</mo><mo>,</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> H^4(K(\mathbb{Z}/2, 3), \mathbb{Z}/2) \;\simeq\; \mathbb{Z}/2 \,. </annotation></semantics></math></div></div> <div class="num_defn" id="CategoryOfSuperVectorSpaces"> <h6 id="definition_8">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>)</p> <ul> <li> <p>whose underlying <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> is that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">\mathbb{Z}/2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/graded+vector+spaces">graded vector spaces</a> (example <a class="maruku-ref" href="#Z2Zgradedvectorspaces"></a>) and</p> </li> <li> <p>whose <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> (def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a>) is the unique non-trivial symmetric grading <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>τ</mi> <mi>super</mi></msup></mrow><annotation encoding="application/x-tex">\tau^{super}</annotation></semantics></math> from prop. <a class="maruku-ref" href="#TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces"></a></p> </li> </ul> <p>is called the <strong><a class="existingWikiWord" href="/nlab/show/category+of+super+vector+spaces">category of super vector spaces</a></strong>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sVect</mi> <mi>k</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msubsup><mi>Vect</mi> <mi>k</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msubsup><mo>,</mo><mo>⊗</mo><mo>=</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mn>1</mn><mo>=</mo><mi>k</mi><mo>,</mo><mi>τ</mi><mo>=</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sVect_k \;\coloneqq\; (Vect_k^{\mathbb{Z}/2}, \otimes = \otimes_k, 1 = k, \tau = \tau^{super} ) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_3">Remark</h6> <p>The non-full symmetric monoidal subcategory</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>Line</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mi>k</mi><mo>,</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\widetilde{Line}(sVect), \otimes_k, k, \tau^{super}) </annotation></semantics></math></div> <p>of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Line</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mi>k</mi><mo>,</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><mi>sVect</mi><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mi>k</mi><mo>,</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (Line(sVect) , \otimes_k, k, \tau^{super}) \hookrightarrow (sVect, \otimes_k, k, \tau^{super}) </annotation></semantics></math></div> <p>(on the two objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>1</mn><mo stretchy="false">|</mo><mn>0</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k^{1\vert 0} </annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>k</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">k^{0\vert 1}</annotation></semantics></math> and with <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a> restricted to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><mo stretchy="false">}</mo><mo>⊂</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\{+1,-1\} \subset k</annotation></semantics></math>, as in the proof of prop. <a class="maruku-ref" href="#TheTwoNontrivialBraidingsOnZ2GradedVectorSpaces"></a>) happens to be the <a class="existingWikiWord" href="/nlab/show/truncated+object+of+an+%28infinity%2C1%29-category">1-truncation</a> of the <a class="existingWikiWord" href="/nlab/show/looping">looping</a> of the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝕊</mi></mrow><annotation encoding="application/x-tex">\mathbb{S}</annotation></semantics></math>, regarded as a group-like <a class="existingWikiWord" href="/nlab/show/E-infinity+space">E-infinity space</a> (“<a class="existingWikiWord" href="/nlab/show/abelian+infinity-group">abelian infinity-group</a>”)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mover><mi>Line</mi><mo>˜</mo></mover><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>,</mo><mo>⊗</mo><mo>,</mo><mi>k</mi><mo>,</mo><msup><mi>τ</mi> <mi>super</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mo lspace="0em" rspace="thinmathspace">trunc</mo> <mn>1</mn></msub><mi>Ω</mi><mi>𝕊</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\widetilde{Line}(sVect), \otimes, k, \tau^{super}) \;\simeq\; \trunc_1 \Omega \mathbb{S} \,. </annotation></semantics></math></div> <p>It has been suggested (in <a href="super+algebra#Kapranov15">Kapranov 15</a>) that this and other phenomena are evidence that in the wider context of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>/<a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a> super-grading (and hence <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>) is to be regarded as but a shadow of grading in <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a> over the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a>. Notice that the <a class="existingWikiWord" href="/nlab/show/sphere+spectrum">sphere spectrum</a> is just the analog of the group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="StructuresInternalToSuperVectorSpaces">Structures internal to super-vector spaces</h3> <p>By <a class="existingWikiWord" href="/nlab/show/internalization">internalizing</a> <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a> and <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a> in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> of super vector spaces, one obtains the corresponding <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> and <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>.</p> <p>For example</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoids">monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/super+algebras">super algebras</a> and, more interestingly, <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a>;</p> </li> <li> <p>hence for example <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebras">commutative Hopf algebras</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebras">supercommutative Hopf algebras</a>, which are the <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of affine algebraic <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/super+Lie+algebras">super Lie algebras</a>;</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/manifolds">manifolds</a> modeled on objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a></p> </li> <li> <p>etc.</p> </li> </ul> <p>By the above definition, any structure in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> works just like the corresponding structure in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>, but with a sign inserted whenever two odd-graded symbols are interchanged. For more on this see also at <em><a class="existingWikiWord" href="/nlab/show/signs+in+supergeometry">signs in supergeometry</a></em>.</p> <h3 id="delignes_theorem_on_tensor_categories">Deligne’s theorem on tensor categories</h3> <p><a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a> says that all suitable <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> of subexponential growth have a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi></mrow><annotation encoding="application/x-tex">sVect</annotation></semantics></math> and are equivalent to <a class="existingWikiWord" href="/nlab/show/categories+of+representations">categories of representations</a> of affine algebraic <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a>.</p> <h2 id="related_entries">Related entries</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/supertrace">supertrace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superdeterminant">superdeterminant</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+vector+bundle">super vector bundle</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/chain+complex+of+super+vector+spaces">chain complex of super vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes+of+super+vector+spaces">model structure on chain complexes of super vector spaces</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+module">super module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/differential+graded-commutative+superalgebra">differential graded-commutative superalgebra</a>,</p> <p><a class="existingWikiWord" href="/nlab/show/model+structure+on+differential+graded-commutative+superalgebras">model structure on differential graded-commutative superalgebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Felix+A.+Berezin">Felix A. Berezin</a> (edited by <a class="existingWikiWord" href="/nlab/show/Alexandre+A.+Kirillov">Alexandre A. Kirillov</a>): <em>Linear Algebra in <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>-graded spaces</em>, Ch. 3 in: <em>Introduction to Superanalysis</em>, Mathematical Physics and Applied Mathematics <strong>9</strong>, Springer (1987) [<a href="https://doi.org/10.1007/978-94-017-1963-6_4">doi:10.1007/978-94-017-1963-6_4</a>]</p> </li> <li id="Varadarajan04"> <p><a class="existingWikiWord" href="/nlab/show/Veeravalli+Varadarajan">Veeravalli Varadarajan</a>, section 3.1 of: <em><a class="existingWikiWord" href="/nlab/show/Supersymmetry+for+mathematicians">Supersymmetry for mathematicians</a>: An introduction</em>, Courant Lecture Notes <strong>11</strong>, American Mathematical Society Providence, R.I (2004) [<a href="http://dx.doi.org/10.1090/cln/011">doi:10.1090/cln/011</a>]</p> </li> <li id="Westra09"> <p><a class="existingWikiWord" href="/nlab/show/Dennis+Westra">Dennis Westra</a>, section 3 of <em>Superrings and supergroups</em>, 2009 (<a href="http://www.mat.univie.ac.at/~michor/westra_diss.pdf">pdf</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on November 3, 2024 at 05:42:46. 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