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class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><span class="newWikiWord">Be?linson-Bernstein localization<a href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="monoidal_categories">Monoidal categories</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a>, <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+diagram">string diagram</a>, <a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/balanced+monoidal+category">balanced monoidal category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/twist">twist</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a></p> </li> </ul> <p><strong>With duals for objects</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+duals">category with duals</a> (list of them)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable object</a> (what they have)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/autonomous+category">autonomous category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/pivotal+category">pivotal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spherical+category">spherical category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/ribbon+category">ribbon category</a>, a.k.a. <a class="existingWikiWord" href="/nlab/show/tortile+category">tortile category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+closed+category">compact closed category</a></p> </li> </ul> <p><strong>With duals for morphisms</strong></p> <ul> <li> <p><span class="newWikiWord">monoidal dagger-category<a href="/nlab/new/monoidal+dagger-category">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+dagger-category">symmetric monoidal dagger-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dagger+compact+category">dagger compact category</a></p> </li> </ul> <p><strong>With traces</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/trace">trace</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/traced+monoidal+category">traced monoidal category</a></p> </li> </ul> <p><strong>Closed structure</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+closed+category">cartesian closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+category">closed category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/star-autonomous+category">star-autonomous category</a></p> </li> </ul> <p><strong>Special sorts of products</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/semicartesian+monoidal+category">semicartesian monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category+with+diagonals">monoidal category with diagonals</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/multicategory">multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/semisimple+category">semisimple category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/modular+tensor+category">modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+functor">monoidal functor</a></p> <p>(<a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax</a>, <a class="existingWikiWord" href="/nlab/show/oplax+monoidal+functor">oplax</a>, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong</a> <a class="existingWikiWord" href="/nlab/show/bilax+monoidal+functor">bilax</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+monoidal+functor">Frobenius</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></p> </li> </ul> <p><strong>Internal monoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in a symmetric monoidal category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+over+a+monoid">module over a monoid</a></p> </li> </ul> <p><strong id="_examples">Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/closed+monoidal+structure+on+presheaves">closed monoidal structure on presheaves</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Day+convolution">Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/coherence+theorem+for+monoidal+categories">coherence theorem for monoidal categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+Dold-Kan+correspondence">monoidal Dold-Kan correspondence</a></p> </li> </ul> <p><strong>In higher category theory</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/braided+monoidal+2-category">braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+bicategory">monoidal bicategory</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cartesian+bicategory">cartesian bicategory</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/k-tuply+monoidal+n-category">k-tuply monoidal n-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/little+cubes+operad">little cubes operad</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category">monoidal (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category">symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/compact+double+category">compact double category</a></p> </li> </ul> </div></div> <h4 id="superalgebra_and_supergeometry">Super-Algebra and Super-Geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a></strong> and (<a class="existingWikiWord" href="/nlab/show/synthetic+differential+supergeometry">synthetic</a> ) <strong><a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a></strong></p> <h2 id="background">Background</h2> <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> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#relevance'>Relevance</a></li> <ul> <li><a href='#RelevanceForSupersymmetry'>For supersymmetry</a></li> <li><a href='#RelevanceForHopfAlgebras'>For Tannaka duality of Hopf algebras</a></li> </ul> <li><a href='#background'>Background</a></li> <ul> <li><a href='#TensorProductsAndMonoidalCategories'>Tensor products and Super vector spaces</a></li> <li><a href='#CommutativeAlgebraInTensorCategories'>Commutative algebra in tensor categories and Affine super-spaces</a></li> <li><a href='#ModulesInTensorCategories'>Modules in tensor categories and Super vector bundles</a></li> <ul> <li><a href='#propposition'>Propposition</a></li> </ul> <li><a href='#GroupsAsHopfAlgebras'>Super-Groups as super-commutative Hopf algebras</a></li> <li><a href='#LinearRepresentationsAsComodules'>Linear super-representations as Comodules</a></li> <li><a href='#FiberFunctors'>Super Fiber functors and their automorphism supergroups</a></li> <li><a href='#superexterior_powers_and_schur_functors'>Super-exterior powers and Schur functors</a></li> </ul> <li><a href='#Statement'>Statement of the theorem</a></li> <li><a href='#proof_13'>Proof</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p><em>Deligne’s theorem on tensor categories</em> (<a href="#Deligne02">Deligne 02</a>, recalled as theorem <a class="maruku-ref" href="#TheTheorem"></a> below) establishes <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> between</p> <ol> <li> <p>linear <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> subject to just a mild size constraint (subexponential growth, def. <a class="maruku-ref" href="#SubexponentialGrowth"></a> below),</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a> (“<a class="existingWikiWord" href="/nlab/show/supersymmetries">supersymmetries</a>”), realizing these tensor categories as <a class="existingWikiWord" href="/nlab/show/categories+of+representations">categories of representations</a> of these supergroups.</p> </li> </ol> <p>As explained <span class="newWikiWord">there<a href="/nlab/new/%5Btensor+categories">?</a></span>, by tensor category we assume a symmetric braiding.</p> <h2 id="relevance">Relevance</h2> <h3 id="RelevanceForSupersymmetry">For supersymmetry</h3> <p>Since the concept of linear <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> arises very naturally in <a class="existingWikiWord" href="/nlab/show/mathematics">mathematics</a>, the theorem gives a purely mathematical “reason” for the relevance of <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> and <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>. It is reasonable to wonder why of all possible generalizations of <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a>, it is <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> that are singled out (from alternatives such as plain <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+algebras">graded algebras</a>, or in fact <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mi>n</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}/n</annotation></semantics></math>-graded algebras or general <a class="existingWikiWord" href="/nlab/show/noncommutative+algebras">noncommutative algebras</a> or the like), as they are notably in theoretical <a class="existingWikiWord" href="/nlab/show/physics">physics</a> (“<a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a>”), but also in mathematical fields such as <a class="existingWikiWord" href="/nlab/show/spin+geometry">spin geometry</a> (e.g. via the relation between <a class="existingWikiWord" href="/nlab/show/Majorana+spinors">Majorana spinors</a> and supersymmetry, <a href="Majorana+spinor#Supersymmetry">here</a>) and <a class="existingWikiWord" href="/nlab/show/topological+K-theory">topological K-theory</a> (for instance via its incarnation as <em><a class="existingWikiWord" href="/nlab/show/Karoubi+K-theory">Karoubi K-theory</a></em>, or via the descriptioon of <a class="existingWikiWord" href="/nlab/show/twisted+K-theory">twisted K-theory</a> by <em><a class="existingWikiWord" href="/nlab/show/super+line+2-bundles">super line 2-bundles</a></em>).</p> <p>But with <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 <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> appearing on general abstract grounds as the canonical structure to consider in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a>, Deligne’s theorem serves to base supercommutative superalgebra on this same general abstract foundation, showing that this is precisely the context in which full <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 tensor categories exhibit full <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a>.</p> <p>More concretely, in <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>, under the <a class="existingWikiWord" href="/nlab/show/Wigner+classification">Wigner classification</a>, <a class="existingWikiWord" href="/nlab/show/fundamental+particles">fundamental particles</a> are identified with <a class="existingWikiWord" href="/nlab/show/irreducible+representations">irreducible representations</a> of the <a class="existingWikiWord" href="/nlab/show/isometry+group">isometry group</a> of the local model of <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> (which are induced from <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional</a> representations of the “<a class="existingWikiWord" href="/nlab/show/Wigner+classification">Wigner's</a> <a class="existingWikiWord" href="/nlab/show/little+group">little group</a>” (<a href="Wigner+classification#Mackey68">Mackey 68</a>) ). Forming the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of two such representations corresponds to combining them as two <a class="existingWikiWord" href="/nlab/show/subsystems">subsystems</a> of a joint system. Therefore it is natural to demand that physical particle species should form complex-linear <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a>. Deligne’s theorem then gives that <a class="existingWikiWord" href="/nlab/show/supersymmetry">supersymmetry</a> is the most general context in which this works out. (In physics the irreducible representation in this context here are called the <em><a class="existingWikiWord" href="/nlab/show/supermultiplets">supermultiplets</a></em>.)</p> <p>More exposition of this point is at:</p> <ul> <li>PhysicsForums Insights, <em><a href="https://www.physicsforums.com/insights/supersymmetry-delignes-theorem/">Supersymmetry and Deligne’s theorem</a></em></li> </ul> <h3 id="RelevanceForHopfAlgebras">For Tannaka duality of Hopf algebras</h3> <p>By <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a>, <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a> in general are <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> of <a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebras">triangular Hopf algebras</a>. Hence Deligne’s theorem here implies that those triangular Hopf algebras over <a class="existingWikiWord" href="/nlab/show/algebraically+closed+fields">algebraically closed fields</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> whose category of representation has subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a> below) are equivalent to <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebras">supercommutative Hopf algebras</a>. See (<a href="#EtingofGelaki02">Etingof-Gelaki 02</a>) for more.</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> for <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> over <a class="existingWikiWord" href="/nlab/show/monoids">monoids</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoid">monoid</a>/<a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></th><th><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-module">2-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/sesquialgebra">sesquialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a> = <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/presentable+category">presentable category</a> with <a class="existingWikiWord" href="/nlab/show/colimit">colimit</a>-preserving <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/bialgebra">bialgebra</a></td><td style="text-align: left;">strict <a class="existingWikiWord" href="/nlab/show/2-ring">2-ring</a>: <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/hopfish+algebra">hopfish algebra</a> (correct version)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (without fiber functor)</td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/weak+Hopf+algebra">weak Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a> with generalized <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+bialgebra">quasitriangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+bialgebra">triangular bialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quasitriangular+Hopf+algebra">quasitriangular Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/quantum+group">quantum group</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebra">triangular Hopf algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/superalgebra">supercommutative</a> <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a> (<a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a>)</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> with <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> and Schur smallness</td></tr> <tr><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+double">Drinfeld double</a></td><td style="text-align: left;">form <a class="existingWikiWord" href="/nlab/show/Drinfeld+center">Drinfeld center</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trialgebra">trialgebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td></tr> </tbody></table> <p><strong>2-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+categories">module categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a></th><th><a class="existingWikiWord" href="/nlab/show/2-category+of+module+categories">2-category of module categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/2-algebra">2-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-module">3-module</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hopf+monoidal+category">Hopf monoidal category</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a> (with some duality and strictness structure)</td></tr> </tbody></table> <p><strong>3-Tannaka duality for <a class="existingWikiWord" href="/nlab/show/module+2-categories">module 2-categories</a> over <a class="existingWikiWord" href="/nlab/show/monoidal+2-categories">monoidal 2-categories</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/monoidal+2-category">monoidal 2-category</a></th><th><a class="existingWikiWord" href="/nlab/show/3-category+of+module+2-categories">3-category of module 2-categories</a></th></tr></thead><tbody><tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">Mod_A</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/3-algebra">3-algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mod</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">Mod_R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/4-module">4-module</a></td></tr> </tbody></table> </div> <h2 id="background">Background</h2> <p>This section provides exposition of the necessary background for the statement of Deligne’s theorem (theorem <a class="maruku-ref" href="#TheTheorem"></a> below).</p> <p>We start by introducing the basic concepts of <a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> along with the basic examples of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> and <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a>:</p> <ul> <li><em><a href="#TensorProductsAndMonoidalCategories">Tensor categories</a></em></li> </ul> <p>This allows to speak of <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a> <a class="existingWikiWord" href="/nlab/show/internalization">internal</a> to tensor categories. Specializing this to the tensor category of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> yields <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>. The <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of these are the <a class="existingWikiWord" href="/nlab/show/affine+variety">affine</a> <a class="existingWikiWord" href="/nlab/show/super+schemes">super schemes</a>. This we discuss in</p> <ul> <li><em><a href="#CommutativeAlgebraInTensorCategories">Commutative algebra in tensor categories and Affine super-spaces</a></em></li> </ul> <p>Next we introduce the concept of <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebras">commutative Hopf algebras</a> and explain how these are <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> to <a class="existingWikiWord" href="/nlab/show/group+objects">groups</a>. Then we use this to motivate and explain the concept of (affine algebraic) <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a> as <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> to <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebras">commutative Hopf algebras</a> internal to the tensor category of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a>, namely <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebras">supercommutative Hopf algebras</a>:</p> <ul> <li><em><a href="#GroupsAsHopfAlgebras">(Super-)Groups as (Super-)commutative Hopf algebras</a></em></li> </ul> <p>Finally we discuss how under this relation <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> of groups correspond to <a class="existingWikiWord" href="/nlab/show/comodules">comodules</a> over their formally dual <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebras">commutative Hopf algebras</a>, and we introduce the key class of categories of interest here: tensor-<a class="existingWikiWord" href="/nlab/show/categories+of+representations">categories of representations</a> of groups and of super-representations of super-groups:</p> <ul> <li><em><a href="#LinearRepresentationsAsComodules">Linear representations as comodules</a></em></li> </ul> <h3 id="TensorProductsAndMonoidalCategories">Tensor products and Super vector spaces</h3> <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/objects">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/morphisms">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, familar 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>) 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 every diagram built freely (see remark <a class="maruku-ref" href="#SymmetricMonoidalCategoriesCoherenceTheorem"></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> such that both sides of the diagram correspond to the same <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> of objects, coincide.</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>Consider furthermore the <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <a class="existingWikiWord" href="/nlab/show/core">core</a> (non-<a class="existingWikiWord" href="/nlab/show/full+subcategory">full</a> <a class="existingWikiWord" href="/nlab/show/subcategory">subcategory</a> including all the <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Line</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><mn>2</mn><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> Line(\mathcal{C}, \otimes , 1)_{iso} \;\in\; 2 Grp </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> now makes this a <em><a class="existingWikiWord" href="/nlab/show/2-group">2-group</a></em>, known as 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>. 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><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub></mrow><annotation encoding="application/x-tex"> B_\otimes Line(\mathcal{C})_{iso} </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><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub><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><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi_0(Line(\mathcal{C})_{iso}) \simeq \pi_1(B Line(\mathcal{C})_{iso}) </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><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub></mrow><annotation encoding="application/x-tex">Line(\mathcal{C})_{iso}</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><msub><mo stretchy="false">)</mo> <mi>iso</mi></msub><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})_{iso}) \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><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">\widetilde{Line}(Vect^{\mathbb{Z}/2})</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><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">B^3 Line(Vect^{\mathbb{Z}/2})</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>);</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 symmtric 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> is called the <strong><a class="existingWikiWord" href="/nlab/show/category+of+super+vector+spaces">category of super vector spaces</a></strong></p> </li> </ul> <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> <p>The following is evident but important</p> <div class="num_defn" id="InclusionOfVectorSpacesIntoSupervectorSpaces"> <h6 id="proposition_2">Proposition</h6> <p>The canonical inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Vect</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>sVect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> Vect_k \hookrightarrow sVect_k </annotation></semantics></math></div> <p>of the category of vector spaces (def. <a class="maruku-ref" href="#VectorSpaces"></a>) into that of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>) given by regarding a vector space as a super-vector space concentrated in even degree, extends to a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>).</p> </div> <div class="num_defn" id="ClosedMonoidalCategory"> <h6 id="definition_9">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) it is called a <strong><a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a></strong> if for each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">Y \in \mathcal{C}</annotation></semantics></math> the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi><mo>⊗</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>⊗</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y \otimes(-)\simeq (-)\otimes Y</annotation></semantics></math> has a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(Y,-)</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>𝒞</mi><munderover><mo>⊥</mo><munder><mo>⟶</mo><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⊗</mo><mi>Y</mi></mrow></mover></munderover><mi>𝒞</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \underoverset {\underset{hom(Y,-)}{\longrightarrow}} {\overset{(-) \otimes Y}{\longleftarrow}} {\bot} \mathcal{C} \,, </annotation></semantics></math></div> <p>hence if there are <a class="existingWikiWord" href="/nlab/show/natural+bijections">natural bijections</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mi>C</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(X \otimes Y, Z) \;\simeq\; Hom_{\mathcal{C}}{C}(X, hom(Y,Z)) </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>Z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X,Z \in \mathcal{C}</annotation></semantics></math>.</p> <p>Since for the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">X = 1</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</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> this means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{\mathcal{C}}(1,hom(Y,Z)) \simeq Hom_{\mathcal{C}}(Y,Z) \,, </annotation></semantics></math></div> <p>the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">hom(Y,Z) \in \mathcal{C}</annotation></semantics></math> is an enhancement of the ordinary <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>𝒞</mi></msub><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{C}}(Y,Z)</annotation></semantics></math> to an object 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>. Accordingly, it is also called the <strong><a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a></strong> between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Z</mi></mrow><annotation encoding="application/x-tex">Z</annotation></semantics></math>.</p> </div> <p>In a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a>, the adjunction isomorphism between <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> and <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> even holds internally:</p> <div class="num_prop" id="TensorHomAdjunctionIsoInternally"> <h6 id="proposition_3">Proposition</h6> <p>In a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>) there are <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> hom(X \otimes Y, Z) \;\simeq\; hom(X, hom(Y,Z)) </annotation></semantics></math></div> <p>whose image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hom_{\mathcal{C}}(1,-)</annotation></semantics></math> are the defining <a class="existingWikiWord" href="/nlab/show/natural+bijections">natural bijections</a> of def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math> be any object. By applying the defining natural bijections twice, there are composite natural bijections</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</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> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><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></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \begin{aligned} Hom_{\mathcal{C}}(A , hom(X \otimes Y, Z)) & \simeq Hom_{\mathcal{C}}(A \otimes (X \otimes Y), Z) \\ & \simeq Hom_{\mathcal{C}}((A \otimes X)\otimes Y, Z) \\ & \simeq Hom_{\mathcal{C}}(A \otimes X, hom(Y,Z)) \\ & \simeq Hom_{\mathcal{C}}(A, hom(X,hom(Y,Z))) \end{aligned} \,. </annotation></semantics></math></div> <p>Since this holds for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a> (the <a class="existingWikiWord" href="/nlab/show/fully+faithful+functor">fully faithfulness</a> of the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a>) says that there is an isomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom(X\otimes Y, Z) \simeq hom(X,hom(Y,Z))</annotation></semantics></math>. Moreover, by taking <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">A = 1</annotation></semantics></math> in the above and using the left <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a> isomorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">A \otimes (X \otimes Y) \simeq X \otimes Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>X</mi><mo>≃</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">A\otimes X \simeq X</annotation></semantics></math> we get a <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</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>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><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> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>⊗</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mo>≃</mo></mover></mtd> <mtd><msub><mi>Hom</mi> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>Y</mi><mo>,</mo><mi>Z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ Hom_{\mathcal{C}}(1,hom(X\otimes Y, )) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(1,hom(X,hom(Y,Z))) \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\simeq}} \\ Hom_{\mathcal{C}}(X \otimes Y, Z) &\overset{\simeq}{\longrightarrow}& Hom_{\mathcal{C}}(X, hom(Y,Z)) } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="LaxMonoidalFunctor"> <h6 id="definition_10">Definition</h6> <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><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})</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>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )</annotation></semantics></math> be two (pointed) <a class="existingWikiWord" href="/nlab/show/topologically+enriched+category">topologically enriched</a> <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>). A <strong>lax monoidal functor</strong> between them is</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><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D} \,, </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mn>1</mn> <mi>𝒟</mi></msub><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \epsilon \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) </annotation></semantics></math></div></li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</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>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu_{x,y} \;\colon\; F(x) \otimes_{\mathcal{D}} F(y) \longrightarrow F(x \otimes_{\mathcal{C}} y) </annotation></semantics></math></div> <p>for all <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></p> </li> </ol> <p>satisfying the following conditions:</p> <ol> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>)</strong> 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>z</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x,y,z \in \mathcal{C}</annotation></semantics></math> the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>a</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></munderover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>id</mi><mo>⊗</mo><msub><mi>μ</mi> <mrow><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo>,</mo><mi>z</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>a</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi><mo>,</mo><mi>z</mi></mrow> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo stretchy="false">(</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>z</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (F(x) \otimes_{\mathcal{D}} F(y)) \otimes_{\mathcal{D}} F(z) &\underoverset{\simeq}{a^{\mathcal{D}}_{F(x),F(y),F(z)}}{\longrightarrow}& F(x) \otimes_{\mathcal{D}}( F(y)\otimes_{\mathcal{D}} F(z) ) \\ {}^{\mathllap{\mu_{x,y} \otimes id}}\downarrow && \downarrow^{\mathrlap{id\otimes \mu_{y,z}}} \\ F(x \otimes_{\mathcal{C}} y) \otimes_{\mathcal{D}} F(z) && F(x) \otimes_{\mathcal{D}} ( F(x \otimes_{\mathcal{C}} y) ) \\ {}^{\mathllap{\mu_{x \otimes_{\mathcal{C}} y , z} } }\downarrow && \downarrow^{\mathrlap{\mu_{ x, y \otimes_{\mathcal{C}} z }}} \\ F( ( x \otimes_{\mathcal{C}} y ) \otimes_{\mathcal{C}} z ) &\underset{F(a^{\mathcal{C}}_{x,y,z})}{\longrightarrow}& F( x \otimes_{\mathcal{C}} ( y \otimes_{\mathcal{C}} z ) ) } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">a^{\mathcal{C}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>a</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">a^{\mathcal{D}}</annotation></semantics></math> denote the <a class="existingWikiWord" href="/nlab/show/associators">associators</a> of the monoidal categories;</p> </li> <li> <p><strong>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>)</strong> For all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">x \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><msub><mn>1</mn> <mi>𝒟</mi></msub><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>ϵ</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>ℓ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><msub><mn>1</mn> <mi>𝒞</mi></msub><mo>,</mo><mi>x</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>ℓ</mi> <mi>x</mi> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ 1_{\mathcal{D}} \otimes_{\mathcal{D}} F(x) &\overset{\epsilon \otimes id}{\longrightarrow}& F(1_{\mathcal{C}}) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\ell^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{1_{\mathcal{C}}, x }}} \\ F(x) &\overset{F(\ell^{\mathcal{C}}_x )}{\longleftarrow}& F(1 \otimes_{\mathcal{C}} 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>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><msub><mn>1</mn> <mi>𝒟</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>ϵ</mi></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>r</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>r</mi> <mi>x</mi> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ F(x) \otimes_{\mathcal{D}} 1_{\mathcal{D}} &\overset{id \otimes \epsilon }{\longrightarrow}& F(x) \otimes_{\mathcal{D}} F(1_{\mathcal{C}}) \\ {}^{\mathllap{r^{\mathcal{D}}_{F(x)}}}\downarrow && \downarrow^{\mathrlap{\mu_{x, 1_{\mathcal{C}} }}} \\ F(x) &\overset{F(r^{\mathcal{C}}_x )}{\longleftarrow}& F(x \otimes_{\mathcal{C}} 1 ) } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℓ</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">\ell^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℓ</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">\ell^{\mathcal{D}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">r^{\mathcal{C}}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>r</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">r^{\mathcal{D}}</annotation></semantics></math> denote the left and right <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a> of the two monoidal categories, respectively.</p> </li> </ol> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> and all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mu_{x,y}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a></strong>.</p> <p>If moreover <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><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})</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>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )</annotation></semantics></math> are equipped with the structure of <a class="existingWikiWord" href="/nlab/show/braided+monoidal+categories">braided monoidal categories</a> (def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a>) with <a class="existingWikiWord" href="/nlab/show/braidings">braidings</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>τ</mi> <mi>𝒞</mi></msup></mrow><annotation encoding="application/x-tex">\tau^{\mathcal{C}}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>τ</mi> <mi>𝒟</mi></msup></mrow><annotation encoding="application/x-tex">\tau^{\mathcal{D}}</annotation></semantics></math>, respectively, then the lax monoidal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is called a <strong><a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a></strong> if in addition the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a> 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></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><msubsup><mi>τ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow> <mi>𝒟</mi></msubsup></mrow></mover></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>y</mi><mo>,</mo><mi>x</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>F</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msubsup><mi>τ</mi> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow> <mi>𝒞</mi></msubsup><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><mi>F</mi><mo stretchy="false">(</mo><mi>y</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>x</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ F(x) \otimes_{\mathcal{C}} F(y) &\overset{\tau^{\mathcal{D}}_{F(x), F(y)}}{\longrightarrow}& F(y) \otimes_{\mathcal{D}} F(x) \\ {}^{\mathllap{\mu_{x,y}}}\downarrow && \downarrow^{\mathrlap{\mu_{y,x}}} \\ F(x \otimes_{\mathcal{C}} y ) &\underset{F(\tau^{\mathcal{C}}_{x,y} )}{\longrightarrow}& F( y \otimes_{\mathcal{C}} x ) } \,. </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϵ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>F</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ϵ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\;\colon\; (F_1,\mu_1, \epsilon_1) \longrightarrow (F_2, \mu_2, \epsilon_2)</annotation></semantics></math> between two (braided) lax monoidal functors is a <strong><a class="existingWikiWord" href="/nlab/show/monoidal+natural+transformation">monoidal natural transformation</a></strong>, in that it is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>f</mi> <mi>x</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>⟶</mo><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f_x \;\colon\; F_1(x) \longrightarrow F_2(x)</annotation></semantics></math> of the underlying functors</p> <p>compatible with the product and the unit in that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a> 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>:</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>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>f</mi><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mrow></mover></mtd> <mtd><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><msub><mi>μ</mi> <mn>1</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo stretchy="false">(</mo><msub><mi>μ</mi> <mn>2</mn></msub><msub><mo stretchy="false">)</mo> <mrow><mi>x</mi><mo>,</mo><mi>y</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>y</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ F_1(x) \otimes_{\mathcal{D}} F_1(y) &\overset{f(x)\otimes_{\mathcal{D}} f(y)}{\longrightarrow}& F_2(x) \otimes_{\mathcal{D}} F_2(y) \\ {}^{\mathllap{(\mu_1)_{x,y}}}\downarrow && \downarrow^{\mathrlap{(\mu_2)_{x,y}}} \\ F_1(x\otimes_{\mathcal{C}} y) &\underset{f(x \otimes_{\mathcal{C}} y ) }{\longrightarrow}& F_2(x \otimes_{\mathcal{C}} 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></mtd> <mtd></mtd> <mtd><msub><mn>1</mn> <mi>𝒟</mi></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ϵ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mrow><msub><mi>ϵ</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>F</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mtd> <mtd></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>f</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow></munder></mtd> <mtd></mtd> <mtd><msub><mi>F</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && 1_{\mathcal{D}} \\ & {}^{\mathllap{\epsilon_1}}\swarrow && \searrow^{\mathrlap{\epsilon_2}} \\ F_1(1_{\mathcal{C}}) &&\underset{f(1_{\mathcal{C}})}{\longrightarrow}&& F_2(1_{\mathcal{C}}) } \,. </annotation></semantics></math></div> <p>We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>MonFun</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">MonFun(\mathcal{C},\mathcal{D})</annotation></semantics></math> for the resulting <a class="existingWikiWord" href="/nlab/show/category">category</a> of lax monoidal functors between monoidal categories <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>, similarly <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BraidMonFun</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">BraidMonFun(\mathcal{C},\mathcal{D})</annotation></semantics></math> for the category of braided monoidal functors between <a class="existingWikiWord" href="/nlab/show/braided+monoidal+categories">braided monoidal categories</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>SymMonFun</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mi>𝒟</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">SymMonFun(\mathcal{C},\mathcal{D})</annotation></semantics></math> for the category of braided monoidal functors between <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a>.</p> </div> <div class="num_remark" id="SymmetricMonoidalFunctor"> <h6 id="remark_4">Remark</h6> <p>In the literature the term “monoidal functor” often refers by default to what in def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a> is called a <em>strong monoidal functor</em>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})</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>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}} )</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) then a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) between them is often called a <strong><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric monoidal functor</a></strong>.</p> </div> <div class="num_example"> <h6 id="example_5">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric 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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">Vect^{\mathbb{Z}/2}</annotation></semantics></math> (example <a class="maruku-ref" href="#Z2Zgradedvectorspaces"></a>) or of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> <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> (example <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>). Then there is an evident <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</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><mo>⟶</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> \mathcal{A} \longrightarrow Vect </annotation></semantics></math></div> <p>to the category of plain <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, which forgets the grading.</p> <p>In both cases this is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><msup><mi>Vect</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\mathcal{A} = Vect^{\mathbb{Z}/2}</annotation></semantics></math> it is also a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a>, but for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>=</mo><mi>sVect</mi></mrow><annotation encoding="application/x-tex">\mathcal{A} = sVect</annotation></semantics></math> it is not.</p> </div> <div class="num_prop" id="MonoidalFunctorComp"> <h6 id="proposition_4">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mover><mo>⟶</mo><mi>F</mi></mover><mi>𝒟</mi><mover><mo>⟶</mo><mi>G</mi></mover><mi>ℰ</mi></mrow><annotation encoding="application/x-tex">\mathcal{C} \overset{F}{\longrightarrow} \mathcal{D} \overset{G}{\longrightarrow} \mathcal{E}</annotation></semantics></math> two composable <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functors">lax monoidal functors</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) between <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a>, then their composite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo>∘</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">F \circ G</annotation></semantics></math> becomes a lax monoidal functor with structure morphisms</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mrow><mi>G</mi><mo>∘</mo><mi>F</mi></mrow></msup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mn>1</mn> <mi>ℰ</mi></msub><mover><mo>⟶</mo><mrow><msup><mi>ϵ</mi> <mi>G</mi></msup></mrow></mover><mi>G</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>G</mi><mo stretchy="false">(</mo><msup><mi>ϵ</mi> <mi>F</mi></msup><mo stretchy="false">)</mo></mrow></mover><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \epsilon^{G\circ F} \;\colon\; 1_{\mathcal{E}} \overset{\epsilon^G}{\longrightarrow} G(1_{\mathcal{D}}) \overset{G(\epsilon^F)}{\longrightarrow} G(F(1_{\mathcal{C}})) </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><msubsup><mi>μ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow> <mrow><mi>G</mi><mo>∘</mo><mi>F</mi></mrow></msubsup><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>ℰ</mi></msub><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msubsup><mi>μ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow> <mi>G</mi></msubsup></mrow></mover><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>G</mi><mo stretchy="false">(</mo><msubsup><mi>μ</mi> <mrow><msub><mi>c</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>c</mi> <mn>2</mn></msub></mrow> <mi>F</mi></msubsup><mo stretchy="false">)</mo></mrow></mover><mi>G</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mi>c</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>𝒞</mi></msub><msub><mi>c</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mu^{G \circ F}_{c_1,c_2} \;\colon\; G(F(c_1)) \otimes_{\mathcal{E}} G(F(c_2)) \overset{\mu^{G}_{F(c_1), F(c_2)}}{\longrightarrow} G( F(c_1) \otimes_{\mathcal{D}} F(c_2) ) \overset{G(\mu^F_{c_1,c_2})}{\longrightarrow} G(F( c_1 \otimes_{\mathcal{C}} c_2 )) \,. </annotation></semantics></math></div></div> <p>We now discuss one more extra property on monoidal categories</p> <div class="num_defn" id="DualizableObject"> <h6 id="definition_11">Definition</h6> <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>)</p> <p>Then right <strong><a class="existingWikiWord" href="/nlab/show/duality">duality</a></strong> between objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>,</mo><msup><mi>A</mi> <mo>*</mo></msup><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">A, A^\ast \in (\mathcal{C}, \otimes, 1)</annotation></semantics></math></p> <p>consists of</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</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>ev</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mi>A</mi><mo>⟶</mo><mn>1</mn></mrow><annotation encoding="application/x-tex"> ev_A\;\colon\;A^\ast \otimes A \longrightarrow 1 </annotation></semantics></math></div> <p>called the <em>counit</em> of the duality, or the <em><a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> map</em>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</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>i</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⟶</mo><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> i_A \;\colon\; 1 \longrightarrow A \otimes A^\ast </annotation></semantics></math></div> <p>called the <em>unit</em> or <em>coevaluation map</em></p> </li> </ol> <p>such that</p> <ul> <li> <p>(<a class="existingWikiWord" href="/nlab/show/triangle+identity">triangle identity</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟵</mo><mrow><msub><mi>id</mi> <mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow></msub><mo>⊗</mo><msub><mi>i</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msubsup><mi>α</mi> <mrow><msup><mi>A</mi> <mo>*</mo></msup><mo>,</mo><mi>A</mi><mo>,</mo><msup><mi>A</mi> <mo>*</mo></msup></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded> <mpadded width="0"><mrow><msubsup><mi>ℓ</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>r</mi> <mi>A</mi></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><msup><mi>A</mi> <mo>*</mo></msup></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>ev</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>id</mi> <mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow></msub></mrow></munder></mtd> <mtd><mn>1</mn><mo>⊗</mo><msup><mi>A</mi> <mo>*</mo></msup></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A^\ast \otimes (A \otimes A^\ast) &\overset{id_{A^\ast} \otimes i_A}{\longleftarrow}& A^\ast \otimes 1 \\ {}^{\mathllap{\alpha^{-1}_{A^\ast,A, A^\ast}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\ell^{-1}_A \circ r_A}}_{\mathrlap{\simeq}} \\ (A^\ast \otimes A) \otimes A^\ast &\underset{ev_A \otimes id_{A^\ast}}{\longrightarrow}& 1 \otimes A^\ast } </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><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><msub><mi>i</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow></mover></mtd> <mtd><mn>1</mn><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>α</mi> <mrow><mi>A</mi><mo>,</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>,</mo><mi>A</mi></mrow></msub></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded> <mpadded width="0"><mrow><msubsup><mi>r</mi> <mi>A</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>∘</mo><msub><mi>ℓ</mi> <mi>A</mi></msub></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>id</mi> <mi>A</mi></msub><mo>⊗</mo><msub><mi>ev</mi> <mi>A</mi></msub></mrow></munder></mtd> <mtd><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (A \otimes A^\ast) \otimes A &\overset{i_A \otimes id_A}{\longleftarrow}& 1 \otimes A \\ {}^{\mathllap{\alpha_{A,A^\ast, A}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r_A^{-1}\circ \ell_A}}_{\mathrlap{\simeq}} \\ A \otimes (A^\ast \otimes A) &\underset{id_A \otimes ev_A}{\longrightarrow}& A \otimes 1 } </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> of 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><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>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> denote the left and right <a class="existingWikiWord" href="/nlab/show/unitors">unitors</a>, respectively.</p> </li> </ul> <p>We say that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">A^\ast</annotation></semantics></math> is the right <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Similarly a left dual for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">A^\ast</annotation></semantics></math> and the structure of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as a right dual of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>A</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">A^\ast</annotation></semantics></math>. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+category">braided monoidal category</a>, then every right dual is canonically also a left dual.</p> <p>If in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> every object has a left and right dual, then it is called a <strong><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a></strong>.</p> </div> <div class="num_example" id="FiniteDimensionalVectorSpaces"> <h6 id="example_6">Example</h6> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>FinDimVect</mi> <mi>k</mi></msub><mo>↪</mo><msub><mi>Vect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> FinDimVect_k \hookrightarrow Vect_k </annotation></semantics></math></div> <p>be the <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> <a class="existingWikiWord" href="/nlab/show/FinDimVect">FinDimVect</a> of that of all <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (over the given <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>) on those which are <strong><a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a></strong>.</p> <p>Clearly 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>) restricts to those of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>, and so there is the induced <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> structure from example <a class="maruku-ref" href="#VectAsAMonoidalCategory"></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><msub><mi>FinDimVect</mi> <mi>k</mi></msub><mo>,</mo><mo>⊗</mo><mo>=</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mn>1</mn><mo>=</mo><mi>k</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo>,</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (FinDimVect_k, \otimes = \otimes_k, 1 = k ) \hookrightarrow (Vect_k, \otimes_k, k) \,. </annotation></semantics></math></div> <p>This is a a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a> (def. <a class="maruku-ref" href="#DualizableObject"></a>) in that for <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> any <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a>, its ordinary linear <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup><mo>≔</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> V^\ast \coloneqq hom(V,k) </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> in the abstract sense of def. <a class="maruku-ref" href="#DualizableObject"></a>.</p> <p>Here the evaluation map is literally the defining <a class="existingWikiWord" href="/nlab/show/evaluation">evaluation</a> map of linear duals (whence the name of the abstract concept)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ev</mi> <mi>V</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msup><mi>V</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>k</mi></msub><mi>V</mi><mo>≃</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>V</mi><mo>,</mo><mi>k</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>k</mi><mover><mo>⟶</mo><mrow></mrow></mover><mi>k</mi></mrow><annotation encoding="application/x-tex"> ev_{V} \;\colon\; V^\ast \otimes_k V \simeq hom(V,k) \otimes_k k \overset{}{\longrightarrow} k </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ev</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><mi>V</mi><mover><mo>→</mo><mi>ϕ</mi></mover><mi>k</mi><mo>,</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>↦</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>ϕ</mi><mo stretchy="false">(</mo><mi>v</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> ev \;\colon\; (V \stackrel{\phi}{\to} k, v) \;\;\mapsto \;\; \phi(v) \,. </annotation></semantics></math></div> <p>The co-evaluation map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>i</mi> <mi>V</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>k</mi><mo>⟶</mo><mi>V</mi><mo>⊗</mo><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex"> i_V \;\colon\; k \longrightarrow V \otimes V^\ast </annotation></semantics></math></div> <p>is the linear map that sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>∈</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">1 \in k</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>V</mi></msub><mo>∈</mo><mi>End</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>V</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">id_V \in End(V) \simeq V \otimes_k V^\ast</annotation></semantics></math> under the canonical identification of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V \otimes_k V^\ast</annotation></semantics></math> with the linear space of <a class="existingWikiWord" href="/nlab/show/linear+map">linear</a> <a class="existingWikiWord" href="/nlab/show/endomorphisms">endomorphisms</a> of <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>.</p> <p>If we choose a <a class="existingWikiWord" href="/nlab/show/linear+basis">linear basis</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e_i\}</annotation></semantics></math> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and a corresponding dual bases <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{e^i\}</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>V</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">V^\ast</annotation></semantics></math>, then the evaluation map is given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ev</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">(</mo><msup><mi>e</mi> <mi>i</mi></msup><mo>,</mo><msub><mi>e</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>↦</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>δ</mi> <mi>j</mi> <mi>i</mi></msubsup></mrow><annotation encoding="application/x-tex"> ev \;\colon\; (e^i, e_j) \mapsto e^i(e_j) = \delta^i_j </annotation></semantics></math></div> <p>(with the <a class="existingWikiWord" href="/nlab/show/Kronecker+delta">Kronecker delta</a> on the right) and the co-evaluation map is given by</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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>i</mi></munder><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo>,</mo><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> 1 \mapsto \underset{i}{\sum} (e_i, e^i) \,. </annotation></semantics></math></div> <p>In this perspective the <a class="existingWikiWord" href="/nlab/show/triangle+identities">triangle identities</a> are the statements that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><msup><mi>e</mi> <mi>i</mi></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><msup><mi>e</mi> <mi>j</mi></msup><mo>=</mo><msup><mi>e</mi> <mi>i</mi></msup></mrow><annotation encoding="application/x-tex"> \underset{j}{\sum} e^i(e_j) e^j = e^i </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><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mi>j</mi></munder><msub><mi>e</mi> <mi>j</mi></msub><msup><mi>e</mi> <mi>j</mi></msup><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>e</mi> <mi>i</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{j}{\sum} e_j e^j(e_i) = e_i \,. </annotation></semantics></math></div> <p>Physicists will recognize this as just the basic rules for <a class="existingWikiWord" href="/nlab/show/tensor">tensor</a> calculus in index-notation.</p> </div> <div class="num_example" id="FiniteDimensionalSuperVectorSpaces"> <h6 id="example_7">Example</h6> <p>Similarly, the full 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><mi>sFinDimVect</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><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"> (sFinDimVect, \otimes_k, k, \tau^{super}) \hookrightarrow (sVect, \otimes_k, k, \tau^{super}) </annotation></semantics></math></div> <p>of the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> from example <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>, on those of finite total dimension is a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>.</p> <p>Here we say that a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> <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> has <a class="existingWikiWord" href="/nlab/show/dimension">dimension</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><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>ℕ</mi><mo>×</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex"> (p\vert q) \in \mathbb{N} \times \mathbb{N} </annotation></semantics></math></div> <p>if its even part has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and its odd part has dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</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>dim</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇔</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mo>(</mo><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>even</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>p</mi><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mtext>and</mtext><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>dim</mi><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>odd</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mi>q</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim(V) = (p\vert q) \;\;\; \Leftrightarrow \;\;\; \left( dim(V_{even}) = p \;\;\;\;\text{and}\;\;\;\; dim(V_{odd}) = q \right) \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> of such a finite dimensional super vector space is just the linear <a class="existingWikiWord" href="/nlab/show/dual+vector+space">dual vector space</a> as in example <a class="maruku-ref" href="#FiniteDimensionalVectorSpaces"></a>, equipped with the evident grading:</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="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>⇒</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><msup><mi>V</mi> <mo>*</mo></msup><mo>=</mo><mo stretchy="false">(</mo><msubsup><mi>V</mi> <mi>even</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mo>⊕</mo><mo stretchy="false">(</mo><msubsup><mi>V</mi> <mi>odd</mi> <mo>*</mo></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> V = V_{even} \oplus V_{odd} \;\;\;\; \Rightarrow \;\;\;\; V^\ast = (V_{even}^\ast) \oplus (V_{odd}^\ast) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="CompactClosedMonoidalCategory"> <h6 id="proposition_5">Proposition</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) is a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>) with <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> between two objects given by the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of the <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> object with the <a class="existingWikiWord" href="/nlab/show/dual+object">dual object</a> of the domain object</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>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo>≃</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mi>B</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [A,B] \simeq A^* \otimes B \,. </annotation></semantics></math></div> <p>(The closed monoidal categories arising this way are called <strong><a class="existingWikiWord" href="/nlab/show/compact+closed+categories">compact closed categories</a></strong>).</p> </div> <div class="proof"> <h6 id="proof_3">Proof</h6> <p>The <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> that characterizes the <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[A,-]</annotation></semantics></math> as being <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> to the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \otimes (-)</annotation></semantics></math> is given by the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> natural isomorphism that characterizes <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><mo stretchy="false">[</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>C</mi><mo>,</mo><msup><mi>A</mi> <mo>*</mo></msup><mo>⊗</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>𝒞</mi><mo stretchy="false">(</mo><mi>C</mi><mo>⊗</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C}(C,[A,B]) \simeq \mathcal{C}(C, A^\ast \otimes B) \simeq \mathcal{C}(C \otimes A, B) \,. </annotation></semantics></math></div></div> <p>There are many <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> whose “<a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a>” operation is quite unlike the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces">tensor product of vector spaces</a>. Hence one says <em><a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></em> for monoidal categories that are also <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+categories">linear categories</a> and such that the tensor product functor suitably reflects that linear structure. There are slight variants of what people mean by a “<a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>”. Here we mean precisely the following:</p> <div class="num_defn" id="TensorCategory"> <h6 id="definition_12">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>, then 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/tensor+category">tensor category</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is an</p> <ol> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/linear+category">k-linear</a> (def. <a class="maruku-ref" href="#LinearCategory"></a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid</a> (def. <a class="maruku-ref" href="#DualizableObject"></a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>)</p> </li> </ol> <p>such that</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>×</mo><mi>𝒜</mi><mo>⟶</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\otimes \colon \mathcal{A} \times \mathcal{A} \longrightarrow \mathcal{A}</annotation></semantics></math> is in both arguments separately</p> <ol> <li> <p><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 (def. <a class="maruku-ref" href="#LinearCategory"></a>);</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+functor">exact</a>.</p> </li> </ol> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>End</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>≃</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">End(1) \simeq k</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/endomorphism+ring">endomorphism ring</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> coincides with <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>).</p> </li> </ol> </div> <p>In this form this is considered in (<a href="#Deligne02">Deligne 02, 0.1</a>).</p> <p>We consider now various types of size constraints on tensor categories. The Tannaka reconstruction theorem (theorem <a class="maruku-ref" href="#TheTheorem"></a> below) only assumes one of them (subexponential growth, def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>), but the others appear in the course of the proof of the theorem.</p> <ol> <li> <p>finiteness (def. <a class="maruku-ref" href="#FiniteTensorCategory"></a>)</p> </li> <li> <p>finite <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>-generation (def. <a class="maruku-ref" href="#FiniteTensorGeneration"></a>)</p> </li> <li> <p>subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>)</p> </li> </ol> <p>Recall the concept of <a class="existingWikiWord" href="/nlab/show/length+of+an+object">length of an object</a> in an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>, a generalization of the concept of <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of a <a class="existingWikiWord" href="/nlab/show/free+module">free module</a>/<a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>.</p> <div class="num_defn" id="FiniteLength"> <h6 id="definition_13">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 an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a>. Given an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math>, then a <em><a class="existingWikiWord" href="/nlab/show/Jordan-H%C3%B6lder+sequence">Jordan-Hölder sequence</a></em> or <em><a class="existingWikiWord" href="/nlab/show/composition+series">composition series</a></em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a finite <a class="existingWikiWord" href="/nlab/show/filtration">filtration</a>, i.e. a finite sequence of <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> unclusions into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, starting with the <a class="existingWikiWord" href="/nlab/show/zero+objects">zero objects</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>=</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>↪</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>↪</mo><mi>⋯</mi><mo>↪</mo><msub><mi>X</mi> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>X</mi> <mi>n</mi></msub><mo>=</mo><mi>X</mi></mrow><annotation encoding="application/x-tex"> 0 = X_0 \hookrightarrow X_1 \hookrightarrow \cdots \hookrightarrow X_{n-1} \hookrightarrow X_n = X </annotation></semantics></math></div> <p>such that at each stage <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">/</mo><msub><mi>X</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub></mrow><annotation encoding="application/x-tex">X_i/X_{i-1}</annotation></semantics></math> (i.e. the <a class="existingWikiWord" href="/nlab/show/coimage">coimage</a> of the <a class="existingWikiWord" href="/nlab/show/monomorphism">monomorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mrow><mi>i</mi><mo>−</mo><mn>1</mn></mrow></msub><mo>↪</mo><msub><mi>X</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_{i-1} \hookrightarrow X_i</annotation></semantics></math>) is a <a class="existingWikiWord" href="/nlab/show/simple+object">simple object</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>.</p> <p>If a Jordan-Hölder sequence for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> exists at all, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is said to be of <em>finite length</em>.</p> </div> <p>(e.g. <a href="#EGNO15">EGNO 15, def. 1.5.3</a>)</p> <div class="num_prop" id="JordanHolderSequenceHasDefiniteLength"> <h6 id="proposition_6">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/Jordan-H%C3%B6lder+theorem">Jordan-Hölder theorem</a>)</strong></p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> has finite length according to def. <a class="maruku-ref" href="#FiniteLength"></a>, then in fact all Jordan-Hölder sequences for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> have the same length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#EGNO15">EGNO 15, theorem 1.5.4</a>)</p> <div class="num_defn" id="LengthOfAnObject"> <h6 id="definition_14">Definition</h6> <p>If an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{C}</annotation></semantics></math> has finite length according to def. <a class="maruku-ref" href="#FiniteLength"></a>, then the length <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> of any of its Jordan-Hölder sequences, which is uniquely defined according to prop. <a class="maruku-ref" href="#JordanHolderSequenceHasDefiniteLength"></a>, is called the <em>length of the object</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#EGNO15">EGNO 15, def. 1.5.5</a>)</p> <div class="num_defn" id="FiniteTensorCategory"> <h6 id="definition_15">Definition</h6> <p>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/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>) is called <strong>finite</strong> (over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>) if</p> <ol> <li> <p>There are only <a class="existingWikiWord" href="/nlab/show/finite+number">finitely many</a> <a class="existingWikiWord" href="/nlab/show/simple+objects">simple objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> (hence it is a <a class="existingWikiWord" href="/nlab/show/finite+abelian+category">finite abelian category</a>), and each of them admits a <a class="existingWikiWord" href="/nlab/show/projective+presentation">projective presentation</a>.</p> </li> <li> <p>Each object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/object+of+finite+length">finite length</a>;</p> </li> <li> <p>For any two <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>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>b</mi></mrow><annotation encoding="application/x-tex">b</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>, the <a class="existingWikiWord" href="/nlab/show/hom-object">hom-object</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/vector+space">vector space</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>a</mi><mo>,</mo><mi>b</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\hom(a, b)</annotation></semantics></math> has <a class="existingWikiWord" href="/nlab/show/finite">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a>;</p> </li> </ol> </div> <div class="num_example"> <h6 id="example_8">Example</h6> <p>The category of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> is a finite tensor category according to def. <a class="maruku-ref" href="#FiniteTensorCategory"></a>. It has a single isomorphism class of <a class="existingWikiWord" href="/nlab/show/simple+objects">simple objects</a>, namely <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.</p> <p>Also category of finite dimensional <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> is a finite tensor category. This has two isomorphism classes of simple objects, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><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 = k^{1 \vert 0}</annotation></semantics></math> regarded in even degree, 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> regarded in odd degree.</p> </div> <p>The following finiteness condition is useful in the proof of the main theorem, but not necessary for its statement (according to <a href="#Deligne02">Deligne 02, bottom of p. 3</a>):</p> <div class="num_defn" id="FiniteTensorGeneration"> <h6 id="definition_16">Definition</h6> <p>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/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>) is called <em>finitely <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>-generated</em> if there exists an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">E \in \mathcal{A}</annotation></semantics></math> such that every other object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/subquotient">subquotient</a> of a <a class="existingWikiWord" href="/nlab/show/direct+sum">direct sum</a> of <a class="existingWikiWord" href="/nlab/show/tensor+products">tensor products</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>E</mi> <mrow><msup><mo>⊗</mo> <mi>n</mi></msup></mrow></msup></mrow><annotation encoding="application/x-tex">E^{\otimes^n}</annotation></semantics></math>, for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><munder><mo>⊕</mo><mi>i</mi></munder><msup><mi>E</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>↪</mo></mtd> <mtd><mo stretchy="false">(</mo><munder><mo>⊕</mo><mi>i</mi></munder><msup><mi>E</mi> <mrow><msup><mo>⊗</mo> <mrow><msub><mi>n</mi> <mi>i</mi></msub></mrow></msup></mrow></msup><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mi>Q</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && \underset{i}{\oplus} E^{\otimes^{n_i}} \\ && \downarrow \\ X &\hookrightarrow& (\underset{i}{\oplus} E^{\otimes^{n_i}})/Q } \,. </annotation></semantics></math></div> <p>Such <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> is called an <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>-generator</em> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>.</p> </div> <p>(<a href="#Deligne02">Deligne 02, 0.1</a>)</p> <p>The following is the main size constraint needed in the theorem. Notice that it is a “mild” constraint at least in the intuitive sense that it states just a minimum assumption on the expected behaviour of dimension (<a class="existingWikiWord" href="/nlab/show/length+of+an+object">length</a>) under tensor powers.</p> <div class="num_defn" id="SubexponentialGrowth"> <h6 id="definition_17">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) is said to have <strong>subexponential growth</strong>* if the <a class="existingWikiWord" href="/nlab/show/length">length</a> of tensor exponentials is no larger than the exponential of the length: for every <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there exists a <a class="existingWikiWord" href="/nlab/show/natural+number">natural number</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">N_X</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is of <a class="existingWikiWord" href="/nlab/show/length+of+an+object">length</a> at most <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">N_X</annotation></semantics></math>, and that also all <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> are of length bounded by the corresponding powers of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>X</mi></msub></mrow><annotation encoding="application/x-tex">N_X</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mo>∀</mo><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow></munder><mspace width="thinmathspace"></mspace><munder><mo>∃</mo><mrow><msub><mi>N</mi> <mi>X</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow></munder><mspace width="thinmathspace"></mspace><munder><mo>∀</mo><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow></munder><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mi>length</mi><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><msup><mo>⊗</mo> <mi>n</mi></msup></mrow></msup><mo stretchy="false">)</mo><mo>≤</mo><mo stretchy="false">(</mo><msub><mi>N</mi> <mi>X</mi></msub><msup><mo stretchy="false">)</mo> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underset{X \in \mathcal{A}}{\forall} \, \underset{N_X \in \mathbb{N}}{\exists} \, \underset{n \in \mathbb{N}}{\forall} \;\; length(X^{\otimes^n}) \leq (N_X)^n \,. </annotation></semantics></math></div></div> <p>(e.g. <a href="#EGNO15">EGNO 15, def. 9.11.1</a>)</p> <p>The evident example is the following:</p> <div class="num_example"> <h6 id="example_9">Example</h6> <p>The tensor category <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/FinDimVect">FinDimVect</a> of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> from example <a class="maruku-ref" href="#FiniteDimensionalVectorSpaces"></a> has subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>), for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mi>X</mi></msub><mo>=</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">N_X = dim(X)</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of a vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, we have</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>dim</mi><mrow><mo>(</mo><msup><mi>X</mi> <mrow><msup><mo>⊗</mo> <mi>n</mi></msup></mrow></msup><mo>)</mo></mrow><mo>=</mo><msup><mrow><mo>(</mo><mi>dim</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>)</mo></mrow> <mi>n</mi></msup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> dim\left( X^{\otimes^n} \right) = \left(dim(X)\right)^n \,. </annotation></semantics></math></div></div> <p>Categories that do not satisfy sub-exponential growth have come to be known as <em>Deligne categories</em>, see e.g. <a href="#Hu24">Hu 2024</a>.</p> <p>While many linear monoidal categories of interest do not satisfy finiteness or <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigidity</a> (def. <a class="maruku-ref" href="#DualizableObject"></a>), often they are such that all their objects are (formal) <a class="existingWikiWord" href="/nlab/show/inductive+limits">inductive limits</a> over “small” objects that do form a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>.</p> <div class="num_prop" id="IndObjectsInATensorCategory"> <h6 id="proposition_7">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>), such that</p> <ol> <li> <p>all <a class="existingWikiWord" href="/nlab/show/object+of+finite+length">objects have finite length</a>;</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/hom+spaces">hom spaces</a> are of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></p> </li> </ol> <p>then for its <a class="existingWikiWord" href="/nlab/show/category+of+ind-objects">category of ind-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ind(\mathcal{A})</annotation></semantics></math> the following holds</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ind(\mathcal{A})</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a></p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>↪</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{A} \hookrightarrow Ind(\mathcal{A})</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a></p> <ol> <li> <p>which stable under forming <a class="existingWikiWord" href="/nlab/show/subquotients">subquotients</a></p> </li> <li> <p>such that that every <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Ind(\mathcal{A})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/filtered+colimit">filtered colimit</a> of those of its <a class="existingWikiWord" href="/nlab/show/subobjects">subobjects</a> that are in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>;</p> </li> </ol> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">Ind</mo><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Ind(\mathcal{A})</annotation></semantics></math> inherits a <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><mi>X</mi><mo>⊗</mo><mi>Y</mi></mtd> <mtd><mo>≃</mo><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mo>⊗</mo><mo stretchy="false">(</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>i</mi></msub><msub><mi>Y</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>≃</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mrow><mi>i</mi><mo>,</mo><mi>j</mi></mrow></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>⊗</mo><msub><mi>X</mi> <mi>j</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} X \otimes Y & \simeq (\underset{\longrightarrow}{\lim}_i X_i) \otimes (\underset{\longrightarrow}{\lim}_i Y_i) \\ & \simeq \underset{\longrightarrow}{\lim}_{i,j} (X_i \otimes X_j) \end{aligned} </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>X</mi> <mi>j</mi></msub><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X_i,X_j \in \mathcal{A}</annotation></semantics></math>, by the above.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ind(\mathcal{A})</annotation></semantics></math> satisfies all the axioms of def. <a class="maruku-ref" href="#TensorCategory"></a> except that it fails to be <a class="existingWikiWord" href="/nlab/show/essentially+small+category">essentially small</a> and <a class="existingWikiWord" href="/nlab/show/rigid+category">rigid category</a>. In detail</p> <ul> <li>an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ind(\mathcal{A})</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/dualizable+object">dualizable</a> precisely if it is in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>.</li> </ul> </li> </ol> </div> <div class="num_example"> <h6 id="example_10">Example</h6> <p>The category of all vector spaces is the category of <a class="existingWikiWord" href="/nlab/show/ind-objects">ind-objects</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> of <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+spaces">finite dimensional vector spaces</a> (example <a class="maruku-ref" href="#FiniteDimensionalVectorSpaces"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Vect</mi><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>FinDimVect</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Vect \simeq Ind(FinDimVect) \,. </annotation></semantics></math></div> <p>Similarly the category of all <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>) is the category of ind-objects of that of finite-dimensional super vector spaces (example <a class="maruku-ref" href="#FiniteDimensionalSuperVectorSpaces"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>sVect</mi><mo>≃</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>sFinDimVect</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sVect \simeq Ind(sFinDimVect) \,. </annotation></semantics></math></div></div> <h3 id="CommutativeAlgebraInTensorCategories">Commutative algebra in tensor categories and Affine super-spaces</h3> <p>The key idea of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> is that it is nothing but plain <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a> but “<a class="existingWikiWord" href="/nlab/show/internalization">internalized</a>” not in ordinary <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a>, but in <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a>. This is made precise by def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a> and ef. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a> below.</p> <p>The key idea then of <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> is to define super-<a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> to be spaces whose <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> are <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>. This is not the case for any “ordinary” space such as a <a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a> or a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>. But these spaces may be <em>characterized dually</em> via their algebras of functions, and hence it makes sense to generalize the latter.</p> <p>For <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> the <a class="existingWikiWord" href="/nlab/show/duality">duality</a> statement is the following:</p> <div class="num_prop" id="EmbeddingOfSmoothManifoldsIntoRAlgebras"> <h6 id="proposition_8">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a>)</strong></p> <p>The <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><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SmoothMfd</mi><mo>⟶</mo><msubsup><mi>Alg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex"> C^\infty(-) \;\colon\; SmoothMfd \longrightarrow Alg_{\mathbb{R}}^{op} </annotation></semantics></math></div> <p>which sends a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> (<a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimensional</a>, <a class="existingWikiWord" href="/nlab/show/paracompact+topological+space">paracompact</a>, <a class="existingWikiWord" href="/nlab/show/second+countable+topological+space">second countable</a>) to (the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of) its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> is a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a>.</p> <p>In other words, for two <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</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></mrow><annotation encoding="application/x-tex">X,Y</annotation></semantics></math> there is a <a class="existingWikiWord" href="/nlab/show/natural+bijection">natural bijection</a> between the <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</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></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> and the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>←</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)\leftarrow C^\infty(Y)</annotation></semantics></math>.</p> </div> <p>A <strong>proof</strong> is for instance in (<a href="embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras#KolarSlovakMichor93">Kolar-Slovak-Michor 93, lemma 35.8, corollaries 35.9, 35.10</a>).</p> <p>This says that we may <em>identify</em> <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> as the “<a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a>” of certain <a class="existingWikiWord" href="/nlab/show/associative+algebras">associative algebras</a>, namely those in the image of the above full embedding. Accordingly then, any larger class of associative algebras than this may be thought of as the class of <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> to a generalized kind of manifold, defined thereby. Given any associative algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, then we may think of it as representing a space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math> which is such that it has <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> as its <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a>.</p> <p>This <a class="existingWikiWord" href="/nlab/show/duality">duality</a> between certain <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> and their <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> is profound. In <a class="existingWikiWord" href="/nlab/show/physics">physics</a> it has always been used implicitly, in fact it was so ingrained into theoretical physics that it took much effort to abstract away from <a class="existingWikiWord" href="/nlab/show/coordinate+system">coordinate functions</a> to discover global <a class="existingWikiWord" href="/nlab/show/Riemannian+geometry">Riemannian geometry</a> in the guise of“<a class="existingWikiWord" href="/nlab/show/general+relativity">general relativity</a>”. As mathematics, an early prominent duality theorem is <a class="existingWikiWord" href="/nlab/show/Gelfand+duality">Gelfand duality</a> (between <a class="existingWikiWord" href="/nlab/show/topological+spaces">topological spaces</a> and <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a>) which served as motivation for the very definition of <a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>, where <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> are nothing but the <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>/<a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a>. Passing to <a class="existingWikiWord" href="/nlab/show/non-commutative+algebras">non-commutative algebras</a> here yields <a class="existingWikiWord" href="/nlab/show/non-commutative+geometry">non-commutative geometry</a>, and so forth. In great generality this duality between spaces and their function algebras appears as “<a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a>” between <a class="existingWikiWord" href="/nlab/show/presheaves">presheaves</a> and <a class="existingWikiWord" href="/nlab/show/copresheaves">copresheaves</a>.</p> <p>In <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> we are concerned with spaces that are formally dual to associative algebras which are “very mildly” non-commutative, namely <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>. These are in fact <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> when viewed internal to <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a> below). The corresponding <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> spaces are, depending on some technical details, <em><a class="existingWikiWord" href="/nlab/show/super+schemes">super schemes</a></em> or <em><a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a></em>. In the <a class="existingWikiWord" href="/nlab/show/physics">physics</a> literature, such spaces are usually just called <em><a class="existingWikiWord" href="/nlab/show/superspaces">superspaces</a></em>.</p> <p>We now make this precise.</p> <div class="num_defn" id="MonoidsInMonoidalCategory"> <h6 id="definition_18">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def <a class="maruku-ref" href="#MonoidalCategory"></a>), then a <strong><a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid internal to</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \in \mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">e \;\colon\; 1 \longrightarrow A</annotation></semantics></math> (called the <em><a class="existingWikiWord" href="/nlab/show/unit">unit</a></em>)</p> </li> <li> <p>a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\mu \;\colon\; A \otimes A \longrightarrow A</annotation></semantics></math> (called the <em>product</em>);</p> </li> </ol> <p>such that</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>A</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>a</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>μ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>A</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>μ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (A\otimes A) \otimes A &\underoverset{\simeq}{a_{A,A,A}}{\longrightarrow}& A \otimes (A \otimes A) &\overset{A \otimes \mu}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{\mu \otimes A}}\downarrow && && \downarrow^{\mathrlap{\mu}} \\ A \otimes A &\longrightarrow& &\overset{\mu}{\longrightarrow}& A } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>;</p> </li> <li id="UnitalityMonoid"> <p>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>1</mn><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟵</mo><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ℓ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>μ</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>r</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ 1 \otimes A &\overset{e \otimes id}{\longrightarrow}& A \otimes A &\overset{id \otimes e}{\longleftarrow}& A \otimes 1 \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\mu}} & & \swarrow_{\mathrlap{r}} \\ && A } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>r</mi></mrow><annotation encoding="application/x-tex">r</annotation></semantics></math> are the left and right unitor isomorphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> </ol> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes , 1)</annotation></semantics></math> has the structure of a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>τ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1, \tau)</annotation></semantics></math> with symmetric <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math>, then a monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu, e)</annotation></semantics></math> as above is called a <strong><a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1, B)</annotation></semantics></math> if in addition</p> <ul> <li> <p>(commutativity) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>τ</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi></mrow></msub></mrow></munderover></mtd> <mtd></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>μ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>μ</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>A</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A && \underoverset{\simeq}{\tau_{A,A}}{\longrightarrow} && A \otimes A \\ & {}_{\mathllap{\mu}}\searrow && \swarrow_{\mathrlap{\mu}} \\ && A } \,. </annotation></semantics></math></div></li> </ul> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>e</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>μ</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A_1, \mu_1, e_1)\longrightarrow (A_2, \mu_2, f_2)</annotation></semantics></math> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>A</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f \;\colon\; A_1 \longrightarrow A_2 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, such that the following two <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagrams commute</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>f</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ A_1 \otimes A_1 &\overset{f \otimes f}{\longrightarrow}& A_2 \otimes A_2 \\ {}^{\mathllap{\mu_1}}\downarrow && \downarrow^{\mathrlap{\mu_2}} \\ A_1 &\underset{f}{\longrightarrow}& A_2 } </annotation></semantics></math></div> <p>and</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><msub><mn>1</mn> <mi>𝒸</mi></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msub><mi>A</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>e</mi> <mn>2</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mi>A</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ 1_{\mathcal{c}} &\overset{e_1}{\longrightarrow}& A_1 \\ & {}_{\mathllap{e_2}}\searrow & \downarrow^{\mathrlap{f}} \\ && A_2 } \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon(\mathcal{C}, \otimes,1)</annotation></semantics></math> for the <strong><a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathcal{C}, \otimes, 1)</annotation></semantics></math> for its subcategory of commutative monoids.</p> </div> <div class="num_example" id="MonoidsInVectAreAssociativeAlgebras"> <h6 id="example_11">Example</h6> <p>A <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> according to def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a> in the <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> from example <a class="maruku-ref" href="#TensorProductOfVectorSpaces"></a> is equivalently an ordinary <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> over the given <a class="existingWikiWord" href="/nlab/show/ground+field">ground field</a>. Similarly a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> in <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> is an ordinary <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a>. Moreover, in both cases the <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> of monoids agree with usual algebra homomorphisms. Hence there are <a class="existingWikiWord" href="/nlab/show/equivalences+of+categories">equivalences of categories</a>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Alg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> Mon(Vect_k) \simeq Alg_k </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>CAlg</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CMon(Vect_k) \simeq CAlg_k \,. </annotation></semantics></math></div></div> <div class="num_example" id="GradedAlgebras"> <h6 id="example_12">Example</h6> <p>For <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 <a class="existingWikiWord" href="/nlab/show/group">group</a>, then 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+algebra">graded</a> <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a></strong> is a <a class="existingWikiWord" href="/nlab/show/monoid+object">monoid object</a> according to def. <a class="maruku-ref" href="#MonoidsInVectAreAssociativeAlgebras"></a> in 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>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>.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>Alg</mi> <mi>k</mi> <mi>G</mi></msubsup><mo>≃</mo><mi>Mon</mi><mo stretchy="false">(</mo><msubsup><mi>Vect</mi> <mi>k</mi> <mi>G</mi></msubsup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Alg_k^G \simeq Mon(Vect_k^G) \,. </annotation></semantics></math></div> <p>This means that a <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 algebra is</p> <ol> <li> <p>a <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><munder><mo>⊕</mo><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow></munder><msub><mi>A</mi> <mi>g</mi></msub></mrow><annotation encoding="application/x-tex">A = \underset{g\in G}{\oplus} A_g</annotation></semantics></math></p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/associative+algebra">associative algebra</a> structure on the underlying vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math></p> </li> </ol> <p>such that for two elements of homogeneous degree, i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>∈</mo><msub><mi>A</mi> <mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></msub><mo>↪</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a_1 \in A_{g_1} \hookrightarrow A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>A</mi> <mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></msub><mo>↪</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a_2 \in A_{g_2} \hookrightarrow A</annotation></semantics></math> then their product is in degre <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">g_1 g_2</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><mi>a</mi> <mrow><msub><mi>g</mi> <mn>1</mn></msub></mrow></msub><msub><mi>a</mi> <mrow><msub><mi>g</mi> <mn>2</mn></msub></mrow></msub><mo>∈</mo><msub><mi>A</mi> <mrow><msub><mi>g</mi> <mn>1</mn></msub><msub><mi>g</mi> <mn>2</mn></msub></mrow></msub><mo>↪</mo><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a_{g_1} a_{g_2} \in A_{g_1 g_2} \hookrightarrow A \,. </annotation></semantics></math></div></div> <p>Example <a class="maruku-ref" href="#MonoidsInVectAreAssociativeAlgebras"></a> motivates the following definition:</p> <div class="num_defn" id="SupercommutativeSuperalgebra"> <h6 id="definition_19">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a></strong> is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) in the <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+super+vector+spaces">category of super vector spaces</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>). We write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sCAlg</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">sCAlg_k</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/category">category</a> of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> with the induced <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a> between them:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sCAlg</mi> <mi>k</mi></msub><mspace width="thickmathspace"></mspace><mo>≔</mo><mspace width="thickmathspace"></mspace><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sCAlg_k \;\coloneqq\; CMon(sVect_k) \,. </annotation></semantics></math></div> <p>Unwinding what this means, then a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is</p> <ol> <li> <p>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>-graded associative algebra according to example <a class="maruku-ref" href="#GradedAlgebras"></a>;</p> </li> <li> <p>such that for any two elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi></mrow><annotation encoding="application/x-tex">a, b</annotation></semantics></math> of homogeneous degree, their product satisfies</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><mi>b</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>a</mi><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo></mrow></msup><mspace width="thinmathspace"></mspace><mi>b</mi><mi>a</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a b \; = \; (-1)^{deg(a) deg(b)}\, b a \,. </annotation></semantics></math></div></li> </ol> </div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>In view of def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a> we might define a not-necessarily supercommutative superalgebra to be a <a class="existingWikiWord" href="/nlab/show/monoid">monoid</a> (not necessarily commutative) in <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a>, and write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sAlg</mi> <mi>k</mi></msub><mo>≔</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sAlg_k \coloneqq Mon(sVect) \,. </annotation></semantics></math></div> <p>However, since the definition of not-necessarily commutative monoids (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) does not invoke the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> of the ambient <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>, and since <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> differ from <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> only via their <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> (example <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>), this yields equivalently just the <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 algebras froom example <a class="maruku-ref" href="#GradedAlgebras"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>sAlg</mi> <mi>k</mi></msub><mo>≃</mo><msubsup><mi>Alg</mi> <mi>k</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> sAlg_k \simeq Alg_k^{\mathbb{Z}/2} \,. </annotation></semantics></math></div> <p>Hence the heart of <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> is <em><a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">super-commutativity</a></em>.</p> </div> <div class="num_example" id="GrassmannAlgebra"> <h6 id="example_13">Example</h6> <p>The <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> which is <a class="existingWikiWord" href="/nlab/show/free+construction">freely generated</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>θ</mi> <mi>i</mi></msub><msubsup><mo stretchy="false">}</mo> <mrow><mi>i</mi><mo>=</mo><mn>1</mn></mrow> <mi>n</mi></msubsup></mrow><annotation encoding="application/x-tex">\{\theta_i\}_{i = 1}^n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+algebra">tensor algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>•</mo></msup><msup><mi>ℝ</mi> <mi>n</mi></msup></mrow><annotation encoding="application/x-tex">T^\bullet \mathbb{R}^n</annotation></semantics></math>, with the generators <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">\theta_i</annotation></semantics></math> in odd degree, by the ideal generated by the relations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>θ</mi> <mi>i</mi></msub><msub><mi>θ</mi> <mi>j</mi></msub><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mi>θ</mi> <mi>j</mi></msub><msub><mi>θ</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex"> \theta_i \theta_j = - \theta_j \theta_i </annotation></semantics></math></div> <p>for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>,</mo><mi>j</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mi>n</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i,j \in \{1, \cdots, n\}</annotation></semantics></math>.</p> <p>This is also called a <em><a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a></em>, in honor of (<a href="#Grassmann1844">Grassmann 1844</a>), who introduced and studied the super-sign rule in def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a> a century ahead of his time.</p> <p>We also denote this algebra by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>n</mi></msup><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo>∈</mo><mspace width="thickmathspace"></mspace><msub><mi>sCAlg</mi> <mi>ℝ</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^n) \;\in\; sCAlg_{\mathbb{R}} \,. </annotation></semantics></math></div></div> <div class="num_example" id="HomotopyCommutativeRingSpectrum"> <h6 id="example_14">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/homotopy+commutative+ring+spectrum">homotopy commutative ring spectrum</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> (i.e., via the <a class="existingWikiWord" href="/nlab/show/Brown+representability+theorem">Brown representability theorem</a>, a <a class="existingWikiWord" href="/nlab/show/multiplicative+cohomology+theory">multiplicative</a> <a class="existingWikiWord" href="/nlab/show/generalized+%28Eilenberg-Steenrod%29+cohomology">generalized cohomology theory</a>), then its <a class="existingWikiWord" href="/nlab/show/stable+homotopy+groups">stable homotopy groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>E</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_\bullet(E)</annotation></semantics></math> inherit the structure of a super-commutative ring.</p> <p>See at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a></em> in the section <a href="Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra">1-2 Homotopy commutative ring spectra</a> <a href="Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyGroupsOfHomotopyCommutativeRingSpectrum">this proposition</a>.</p> </div> <p>The following is an elementary but fundamental fact about the relation between commutative algbra and supercommutative superalgebra. It is implicit in much of the literature, but maybe the only place where it has been made explicit before is (<a href="#CarchediRoytenberg12">Carchedi-Roytenberg 12, example 3.18</a>).</p> <div class="num_prop" id="InclusionOfCAlgIntosCAlg"> <h6 id="proposition_9">Proposition</h6> <p>There is a <a class="existingWikiWord" href="/nlab/show/full+subcategory">full subcategory</a> inclusion</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>CAlg</mi> <mi>k</mi></msub></mtd> <mtd><mo>↪</mo></mtd> <mtd><msub><mi>sCAlg</mi> <mi>k</mi></msub></mtd></mtr> <mtr><mtd><mo>=</mo></mtd> <mtd></mtd> <mtd><mo>=</mo></mtd></mtr> <mtr><mtd><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd> <mtd><mo>↪</mo></mtd> <mtd><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ CAlg_k &\hookrightarrow& sCAlg_k \\ = && = \\ CMon(Vect_k) &\hookrightarrow& CMon(sVect_k) } </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/commutative+algebras">commutative algebras</a> (example <a class="maruku-ref" href="#MonoidsInVectAreAssociativeAlgebras"></a>) into <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> (def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a>) induced via prop. <a class="maruku-ref" href="#MonoidsPreservedByLaxMonoidalFunctor"></a> from the full inclusion</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>i</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Vectk</mi><mo>↪</mo><msub><mi>sVect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> i \;\colon\; Vectk \hookrightarrow sVect_k </annotation></semantics></math></div> <p>of <a class="existingWikiWord" href="/nlab/show/vector+spaces">vector spaces</a> (def. <a class="maruku-ref" href="#VectorSpaces"></a>) into <a class="existingWikiWord" href="/nlab/show/super+vector+spaces">super vector spaces</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>), which is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> by prop. <a class="maruku-ref" href="#InclusionOfVectorSpacesIntoSupervectorSpaces"></a>. Hence this regards a commutative algebra as a superalgebra concentrated in even degree.</p> <p>This inclusion functor has both a <a class="existingWikiWord" href="/nlab/show/left+adjoint">left</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> and a <a class="existingWikiWord" href="/nlab/show/right+adjoint">right</a> <a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a> , (an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> exibiting a <a class="existingWikiWord" href="/nlab/show/reflective+subcategory">reflective subcategory</a> and <a class="existingWikiWord" href="/nlab/show/coreflective+subcategory">coreflective subcategory</a> inclusion, an “<a class="existingWikiWord" href="/nlab/show/adjoint+cylinder">adjoint cylinder</a>”):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>CAlg</mi> <mi>k</mi></msub><munderover><mo>↪</mo><munder><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>odd</mi></msub></mrow></mover></munderover><msub><mi>sCAlg</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CAlg_k \underoverset {\underset{(-)_{even}}{\longleftarrow}} {\overset{(-)/(-)_{odd}}{\longleftarrow}} {\hookrightarrow} sCAlg_k \,. </annotation></semantics></math></div> <p>Here</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a> <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><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">(-)_{even}</annotation></semantics></math> sends a supercommutative superalgebra to its even part <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><msub><mi>A</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">A \mapsto A_{even}</annotation></semantics></math>;</p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> <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><mo stretchy="false">)</mo><mo stretchy="false">/</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">(-)/(-)_{even}</annotation></semantics></math> sends a supercommutative superalgebra to the <a class="existingWikiWord" href="/nlab/show/quotient">quotient</a> by the ideal which is generated by its odd part <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>↦</mo><mi>A</mi><mo stretchy="false">/</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>odd</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \mapsto A/(A_{odd})</annotation></semantics></math> (hence it sets all elements to zero which may be written as a product such that at least one factor is odd-graded).</p> </li> </ol> </div> <div class="proof"> <h6 id="proof_4">Proof</h6> <p>The full inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math> is evident. To see the <a class="existingWikiWord" href="/nlab/show/adjunctions">adjunctions</a> observe their characteristic <a class="existingWikiWord" href="/nlab/show/natural+bijections">natural bijections</a> between <a class="existingWikiWord" href="/nlab/show/hom-sets">hom-sets</a>: If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>ordinary</mi></msub></mrow><annotation encoding="application/x-tex">A_{ordinary}</annotation></semantics></math> is an ordinary commutative algebra regarded as a superalgeba <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>ordinary</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">i(A_{ordinary})</annotation></semantics></math> concentrated in even degree, and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is any superalgebra,</p> <ol> <li> <p>then every super-algebra homomorphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mi>ordinary</mi></msub><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">A_{ordinary} \to B</annotation></semantics></math> must factor through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">B_{even}</annotation></semantics></math>, simply because super-algebra homomorpism by definition respect the <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>-grading. This gives a natual bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msub><mi>sCAlg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>i</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>ordinary</mi></msub><mo stretchy="false">)</mo><mo>,</mo><mi>B</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msub><mi>CAlg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><msub><mi>A</mi> <mrow><mi>ordinary</mi><mo>,</mo><msub><mi>B</mi> <mi>even</mi></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Hom_{sCAlg_k}(i(A_{ordinary}), B) \simeq Hom_{CAlg_k}(A_{ordinary,B_{even}}) \,, </annotation></semantics></math></div></li> <li> <p>every super-algebra homomorphism of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>i</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>ordinary</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">B \to i(A_{ordinary})</annotation></semantics></math> must send every odd element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> to 0, again because homomorphism have to respect the <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>-grading, and since homomorphism of course also preserve products, this means that the entire ideal generated by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>B</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">B_{odd}</annotation></semantics></math> must be sent to zero, hence the homomorphism must facto through the <a class="existingWikiWord" href="/nlab/show/projection">projection</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi><mo>→</mo><mi>B</mi><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>odd</mi></msub></mrow><annotation encoding="application/x-tex">B \to B/B_{odd}</annotation></semantics></math>, which gives a natural bijection</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>Hom</mi> <mrow><msub><mi>sCalg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><mi>i</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mi>ordinary</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><msub><mi>Alg</mi> <mi>k</mi></msub></mrow></msub><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">/</mo><msub><mi>B</mi> <mi>odd</mi></msub><mo>,</mo><msub><mi>A</mi> <mi>ordinary</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Hom_{sCalg_k}(B, i(A_{ordinary})) \simeq Hom_{Alg_k}(B/B_{odd}, A_{ordinary}) \,. </annotation></semantics></math></div></li> </ol> </div> <p>It is useful to make explicit the following <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a> perspective on <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a>:</p> <div class="num_defn" id="Affines"> <h6 id="definition_20">Definition</h6> <p>For <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 <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, then we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> Aff(\mathcal{C}) \coloneqq CMon(\mathcal{C})^{op} </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of the <a class="existingWikiWord" href="/nlab/show/category+of+commutative+monoids">category of commutative monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \in CMon(\mathcal{C})</annotation></semantics></math> we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Aff</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(A) \in Aff(\mathcal{C}) </annotation></semantics></math></div> <p>for the same object, regarded in the opposite category. We also call this the <strong><a class="existingWikiWord" href="/nlab/show/affine+scheme">affine scheme</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Conversely, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>Aff</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \in Aff(\mathcal{C})</annotation></semantics></math>, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{O}(X) \in CMon(\mathcal{C}) </annotation></semantics></math></div> <p>for the same object, regarded in the category of commutative monoids. We also call this the <strong><a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a></strong> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <div class="num_defn" id="AffineSuperSchemes"> <h6 id="definition_21">Definition</h6> <p>For the special case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">mathal</mo><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathal{C} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>) in def. <a class="maruku-ref" href="#Affines"></a>, then we say that the objects in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msubsup><mi>scAlg</mi> <mi>k</mi> <mi>op</mi></msubsup><mo>=</mo><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex"> Aff(sVect_k) = scAlg_k^{op} = CMon(sVect_k)^{op} </annotation></semantics></math></div> <p>are <strong>affine <a class="existingWikiWord" href="/nlab/show/super+schemes">super schemes</a></strong> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </div> <div class="num_example" id="OrdinaryCAlgAssCAlg"> <h6 id="example_15">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><msub><mi>CAlg</mi> <mi>ℝ</mi></msub></mrow><annotation encoding="application/x-tex">A \in CAlg_{\mathbb{R}}</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/commutative+algebra">commutative algebra</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>, then of course this becomes a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> by regarding it as being concentrated in even degrees. Accordingly, via def. <a class="maruku-ref" href="#Affines"></a>, ordinary <a class="existingWikiWord" href="/nlab/show/affine+schemes">affine schemes</a> <a class="existingWikiWord" href="/nlab/show/full+subcategory">fully embed</a> into affine <a class="existingWikiWord" href="/nlab/show/super+schemes">super schemes</a> (def. <a class="maruku-ref" href="#AffineSuperSchemes"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mo>↪</mo><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Aff(Vect_k) \hookrightarrow Aff(sVect_k) \,. </annotation></semantics></math></div> <p>In particular for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mi>p</mi></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^p</annotation></semantics></math> an ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+space">Cartesian space</a>, this becomes an affine <a class="existingWikiWord" href="/nlab/show/superscheme">superscheme</a> in even degree, under the above embedding. As such, it is usually written</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>∈</mo><mi>Aff</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{p \vert 0} \in Aff(sVect_k) \,. </annotation></semantics></math></div></div> <div class="num_example" id="Superpoint"> <h6 id="example_16">Example</h6> <p>The formal dual space, according to def. <a class="maruku-ref" href="#Affines"></a> (example <a class="maruku-ref" href="#Affines"></a>) to a <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)</annotation></semantics></math> (example <a class="maruku-ref" href="#GrassmannAlgebra"></a>) is to be thought of as a space which is “so tiny” that the coefficients of the <a class="existingWikiWord" href="/nlab/show/Taylor+expansion">Taylor expansion</a> of any real-valued function on it become “so very small” as to be actually equal to zero, at least after the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi></mrow><annotation encoding="application/x-tex">q</annotation></semantics></math>-th power.</p> <p>For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mn>2</mn></mrow><annotation encoding="application/x-tex">q = 2</annotation></semantics></math> then a general element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)</annotation></semantics></math> is of the form</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>a</mi> <mn>0</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>1</mn></msub><msub><mi>θ</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>2</mn></msub><msub><mi>θ</mi> <mn>2</mn></msub><mo>+</mo><msub><mi>a</mi> <mn>12</mn></msub><msub><mi>θ</mi> <mn>1</mn></msub><msub><mi>θ</mi> <mn>2</mn></msub><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>∈</mo><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f = a_0 + a_1 \theta_1 + a_2 \theta_2 + a_{12} \theta_1 \theta_2 \;\;\;\in \wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q) \,. </annotation></semantics></math></div> <p>for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>a</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>a</mi> <mn>12</mn></msub><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">a_1,a_2, a_{12} \in \mathbb{R}</annotation></semantics></math>, to be compared with the <a class="existingWikiWord" href="/nlab/show/Taylor+expansion">Taylor expansion</a> of a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ℝ</mi> <mn>2</mn></msup><mo>→</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">g \colon \mathbb{R}^2 \to \mathbb{R}</annotation></semantics></math>, which is of the form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>x</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>g</mi><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>g</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mn>1</mn></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>x</mi> <mn>1</mn></msub><mo>+</mo><mfrac><mrow><mo>∂</mo><mi>g</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>x</mi> <mn>2</mn></msub><mo>+</mo><mfrac><mrow><msup><mo>∂</mo> <mn>2</mn></msup><mi>g</mi></mrow><mrow><mo>∂</mo><msub><mi>x</mi> <mn>1</mn></msub><mo>∂</mo><msub><mi>x</mi> <mn>2</mn></msub></mrow></mfrac><mo stretchy="false">(</mo><mn>0</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><msub><mi>x</mi> <mn>1</mn></msub><msub><mi>x</mi> <mn>2</mn></msub><mo>+</mo><mi>⋯</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g(x_1, x_2) = g(0) + \frac{\partial g}{\partial x_1}(0)\, x_1 + \frac{\partial g}{\partial x_2}(0)\, x_2 + \frac{\partial^2 g}{\partial x_1 \partial x_2}(0) \, x_1 x_2 + \cdots \,. </annotation></semantics></math></div> <p>Therefore the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> <a class="existingWikiWord" href="/nlab/show/space">space</a> to a <a class="existingWikiWord" href="/nlab/show/Grassmann+algebra">Grassmann algebra</a> behaves like an infinitesimal neighbourhood of a point. Hence these are also called <strong><a class="existingWikiWord" href="/nlab/show/superpoints">superpoints</a></strong> and one writes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>≔</mo><mi>Spec</mi><mo stretchy="false">(</mo><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>q</mi></msup><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{0\vert q} \coloneqq Spec(\wedge^\bullet_{\mathbb{R}}(\mathbb{R}^q)) \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_17">Example</h6> <p>Combining example <a class="maruku-ref" href="#OrdinaryCAlgAssCAlg"></a> with example <a class="maruku-ref" href="#Superpoint"></a>, and using prop. <a class="maruku-ref" href="#ProductsInAff"></a>, we obtain the affine <a class="existingWikiWord" href="/nlab/show/super+schemes">super schemes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>≔</mo><msup><mi>ℝ</mi> <mrow><mi>p</mi><mo stretchy="false">|</mo><mn>0</mn></mrow></msup><mo>×</mo><msup><mi>ℝ</mi> <mrow><mn>0</mn><mo stretchy="false">|</mo><mi>q</mi></mrow></msup><mo>≃</mo><mi>Spec</mi><mrow><mo>(</mo><munder><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><msup><mi>ℝ</mi> <mi>p</mi></msup><mo stretchy="false">)</mo></mrow><mo>⏟</mo></munder><msub><mo>⊗</mo> <mi>ℝ</mi></msub><msubsup><mo>∧</mo> <mi>ℝ</mi> <mo>•</mo></msubsup><msup><mi>ℝ</mi> <mi>q</mi></msup><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbb{R}^{p \vert q} \coloneqq \mathbb{R}^{p\vert 0} \times \mathbb{R}^{0\vert q} \simeq Spec\left( \underbrace{C^\infty(\mathbb{R}^p)} \otimes_{\mathbb{R}} \wedge^\bullet_{\mathbb{R}} \mathbb{R}^q \right) \,. </annotation></semantics></math></div> <p>These may be called the <strong><a class="existingWikiWord" href="/nlab/show/super+Cartesian+spaces">super Cartesian spaces</a></strong>. The play the same role in the theory of <a class="existingWikiWord" href="/nlab/show/supermanifolds">supermanifolds</a> as the ordinary <a class="existingWikiWord" href="/nlab/show/Cartesian+spaces">Cartesian spaces</a> do for <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. See at <em><a class="existingWikiWord" href="/nlab/show/geometry+of+physics+--+supergeometry">geometry of physics – supergeometry</a></em> for more on this.</p> </div> <div class="num_defn" id="ParityAutomorphism"> <h6 id="definition_22">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a>), its <strong>parity involution</strong> is the algebra <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>par</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mover><mo>⟶</mo><mo>≃</mo></mover><mi>A</mi></mrow><annotation encoding="application/x-tex"> par \;\colon\; A \overset{\simeq}{\longrightarrow} A </annotation></semantics></math></div> <p>which on homogeneously graded elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi></mrow><annotation encoding="application/x-tex">a</annotation></semantics></math> of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mo>∈</mo><mo stretchy="false">{</mo><mi>even</mi><mo>,</mo><mi>odd</mi><mo stretchy="false">}</mo><mo>=</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">deg(a) \in \{even,odd\} = \mathbb{Z}/2\mathbb{Z}</annotation></semantics></math> is multiplication by the degree</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>a</mi><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><mi>a</mi><mo stretchy="false">)</mo></mrow></msup><mi>a</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> a \mapsto (-1)^{deg(a)}a \,. </annotation></semantics></math></div> <p>(e.g. <a href="http://arxiv.org/abs/1303.1916">arXiv:1303.1916, 7.5</a>)</p> <p>Dually, via def. <a class="maruku-ref" href="#Affines"></a>, this means that every affine <a class="existingWikiWord" href="/nlab/show/super+scheme">super scheme</a> has a canonical <a class="existingWikiWord" href="/nlab/show/involution">involution</a>.</p> </div> <p>Here are more general and more abstract examples of commutative monoids, which will be useful to make explicit:</p> <div class="num_example" id="MonoidGivenByTensorUnit"> <h6 id="example_18">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>), then the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) with product given by either the left or right <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a></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> <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><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \ell_1 = r_1 \;\colon\; 1 \otimes 1 \overset{\simeq}{\longrightarrow} 1 \,. </annotation></semantics></math></div> <p>By lemma <a class="maruku-ref" href="#kel1"></a>, these two morphisms coincide and define an <a class="existingWikiWord" href="/nlab/show/associativity">associative</a> product with unit the identity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo lspace="verythinmathspace">:</mo><mn>1</mn><mo>→</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">id \colon 1 \to 1</annotation></semantics></math>.</p> <p>If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes , 1)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>), then this monoid is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid</a>.</p> </div> <div class="num_example" id="TensorProductOfTwoCommutativeMonoids"> <h6 id="example_19">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>), and given two <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoids</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>μ</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_i, \mu_i, e_i)</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi><mo>∈</mo><mo stretchy="false">{</mo><mn>1</mn><mo>,</mo><mn>2</mn><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">i \in \{1,2\}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), then the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">E_1 \otimes E_2</annotation></semantics></math> becomes itself a commutative monoid with unit morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>e</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mover><mo>⟶</mo><mo>≃</mo></mover><mn>1</mn><mo>⊗</mo><mn>1</mn><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>e</mi> <mn>2</mn></msub></mrow></mover><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> e \;\colon\; 1 \overset{\simeq}{\longrightarrow} 1 \otimes 1 \overset{e_1 \otimes e_2}{\longrightarrow} E_1 \otimes E_2 </annotation></semantics></math></div> <p>(where the first isomorphism is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>ℓ</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup><mo>=</mo><msubsup><mi>r</mi> <mn>1</mn> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow><annotation encoding="application/x-tex">\ell_1^{-1} = r_1^{-1}</annotation></semantics></math> (lemma <a class="maruku-ref" href="#kel1"></a>)) and with product morphism 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>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><msub><mi>τ</mi> <mrow><msub><mi>E</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>E</mi> <mn>1</mn></msub></mrow></msub><mo>⊗</mo><mi>id</mi></mrow></mover><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>μ</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>μ</mi> <mn>2</mn></msub></mrow></mover><msub><mi>E</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>E</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> E_1 \otimes E_2 \otimes E_1 \otimes E_2 \overset{id \otimes \tau_{E_2, E_1} \otimes id}{\longrightarrow} E_1 \otimes E_1 \otimes E_2 \otimes E_2 \overset{\mu_1 \otimes \mu_2}{\longrightarrow} E_1 \otimes E_2 </annotation></semantics></math></div> <p>(where we are notationally suppressing the <a class="existingWikiWord" href="/nlab/show/associators">associators</a> and where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/braiding">braiding</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>).</p> <p>That this definition indeed satisfies associativity and commutativity follows from the corresponding properties of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>E</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>μ</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>i</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E_i,\mu_i, e_i)</annotation></semantics></math>, and from the hexagon identities for the braiding (def. <a class="maruku-ref" href="#BraidedMonoidalCategory"></a>) and from symmetry of the braiding.</p> <p>Similarly one checks that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>E</mi> <mn>1</mn></msub><mo>=</mo><msub><mi>E</mi> <mn>2</mn></msub><mo>=</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E_1 = E_2 = E</annotation></semantics></math> then the unit maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>≃</mo><mi>E</mi><mo>⊗</mo><mn>1</mn><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mover><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> E \simeq E \otimes 1 \overset{id \otimes e}{\longrightarrow} E \otimes E </annotation></semantics></math></div><div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>≃</mo><mn>1</mn><mo>⊗</mo><mi>E</mi><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mn>1</mn></mrow></mover><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> E \simeq 1 \otimes E \overset{e \otimes 1}{\longrightarrow} E \otimes E </annotation></semantics></math></div> <p>and the product map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>E</mi><mo>⊗</mo><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \mu \;\colon\; E \otimes E \longrightarrow E </annotation></semantics></math></div> <p>and the braiding</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>E</mi><mo>,</mo><mi>E</mi></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>E</mi><mo>⊗</mo><mi>E</mi><mo>⟶</mo><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex"> \tau_{E,E} \;\colon\; E \otimes E \longrightarrow E \otimes E </annotation></semantics></math></div> <p>are monoid homomorphisms, with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E \otimes E</annotation></semantics></math> equipped with the above monoid structure.</p> </div> <p>Monoids are preserved by <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functors">lax monoidal functors</a>:</p> <div class="num_prop" id="MonoidsPreservedByLaxMonoidalFunctor"> <h6 id="proposition_10">Proposition</h6> <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><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes_{\mathcal{C}}, 1_{\mathcal{C}})</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>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{D}, \otimes_{\mathcal{D}},1_{\mathcal{D}})</annotation></semantics></math> be two <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒞</mi><mo>⟶</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F \;\colon\; \mathcal{C} \longrightarrow \mathcal{D}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/lax+monoidal+functor">lax monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) between them.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu_A,e_A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">F(A) \in \mathcal{D}</annotation></semantics></math> becomes a monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>μ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo>,</mo><msub><mi>e</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(F(A), \mu_{F(A)}, e_{F(A)})</annotation></semantics></math> by setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mi>A</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mu_{F(A)} \;\colon\; F(A) \otimes_{\mathcal{C}} F(A) \overset{}{\longrightarrow} F(A \otimes_{\mathcal{C}} A) \overset{F(\mu_A)}{\longrightarrow} F(A) </annotation></semantics></math></div> <p>(where the first morphism is the structure morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>) and setting</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mn>1</mn> <mi>𝒟</mi></msub><mo>⟶</mo><mi>F</mi><mo stretchy="false">(</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mi>F</mi><mo stretchy="false">(</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow></mover><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> e_{F(A)} \;\colon\; 1_{\mathcal{D}} \longrightarrow F(1_{\mathcal{C}}) \overset{F(e_A)}{\longrightarrow} F(A) </annotation></semantics></math></div> <p>(where again the first morphism is the corresponding structure morphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>).</p> <p>This construction extends to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Mon</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Mon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Mon</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Mon(F) \;\colon\; Mon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow Mon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) </annotation></semantics></math></div> <p>from the <a class="existingWikiWord" href="/nlab/show/category+of+monoids">category of monoids</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> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) to that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math>.</p> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒟</mi></mrow><annotation encoding="application/x-tex">\mathcal{D}</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) then so is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(A)</annotation></semantics></math>, and this construction extends to a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>F</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒞</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒞</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒟</mi><mo>,</mo><msub><mo>⊗</mo> <mi>𝒟</mi></msub><mo>,</mo><msub><mn>1</mn> <mi>𝒟</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> CMon(F) \;\colon\; CMon(\mathcal{C}, \otimes_{\mathcal{C}}, 1_{\mathcal{C}}) \longrightarrow CMon(\mathcal{D},\otimes_{\mathcal{D}}, 1_{\mathcal{D}}) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_5">Proof</h6> <p>This follows immediately from combining the associativity and unitality (and symmetry) constraints of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> with those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </div> <h3 id="ModulesInTensorCategories">Modules in tensor categories and Super vector bundles</h3> <p>Above (in def. <a class="maruku-ref" href="#Affines"></a>) we considered spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> from a dual perspective, as determined by their <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(X)</annotation></semantics></math>. In the same spirit then we are to express various constructions on and with spaces in terms of dual algebraic constructions.</p> <p>A key such construction is that of <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Here we discuss the corresponding algebraic incarnation of these, namely as <em><a class="existingWikiWord" href="/nlab/show/modules">modules</a></em> over <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a>.</p> <p>Suppose that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mover><mo>→</mo><mi>p</mi></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">V \stackrel{p}{\to} X</annotation></semantics></math> is an ordinary smooth real <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. A <a class="existingWikiWord" href="/nlab/show/section">section</a> of this vector bundle is a smooth function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\sigma \colon X \to V</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∘</mo><mi>σ</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">p \circ \sigma = id</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>V</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>σ</mi></mpadded></msup><mo>↗</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>p</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && V \\ & {}^{\mathllap{\sigma}}\nearrow & \downarrow^{\mathrlap{p}} \\ X &=& X } \,. </annotation></semantics></math></div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(V)</annotation></semantics></math> for the set of all such sections. Observe that this set inherits various extra <a class="existingWikiWord" href="/nlab/show/stuff%2C+structure%2C+property">structure</a>.</p> <p>First of all, since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> is a vector bundle, we have <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a>-wise the vector space operations. This means that given two elements <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>ℝ</mi></mrow><annotation encoding="application/x-tex">c_1, c_2 \in \mathbb{R}</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a>, and given two sections <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\sigma_2</annotation></semantics></math>, we may form in each <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/linear+combination">linear combination</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo><mo>+</mo><msub><mi>c</mi> <mn>2</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>x</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">c_1 \sigma_1(x) + c_2 \sigma_2(x)</annotation></semantics></math>. This hence yields a new section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>c</mi> <mn>1</mn></msub><msub><mi>σ</mi> <mn>1</mn></msub><mo>+</mo><msub><mi>c</mi> <mn>2</mn></msub><msub><mi>σ</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">c_1 \sigma_1 + c_2 \sigma_2</annotation></semantics></math>. Hence the set of sections of a vector bundle naturally forms itself a vector space.</p> <p>But there is more structure. We need not multiply with the same element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∈</mo><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">c \in \mathbb{R}</annotation></semantics></math> in each fiber, but we may multiply the section in each fiber by a different element, as long as the choice of element varies smoothly with the fibers, so that the resulting section is still smooth.</p> <p>In other words, every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f \in C^\infty(X)</annotation></semantics></math> in the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-algebra of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, takes a smooth section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>σ</mi></mrow><annotation encoding="application/x-tex">\sigma</annotation></semantics></math> of <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> to a new smooth section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>σ</mi></mrow><annotation encoding="application/x-tex">f \cdot \sigma</annotation></semantics></math>. This operation enjoys some evident properties. It is <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear</a> in the real vector spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(V)</annotation></semantics></math>, and it satisfies the “<a class="existingWikiWord" href="/nlab/show/action">action property</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><mi>f</mi><mi>g</mi><mo stretchy="false">)</mo><mo>⋅</mo><mi>σ</mi><mo>=</mo><mi>f</mi><mo>⋅</mo><mo stretchy="false">(</mo><mi>g</mi><mo>⋅</mo><mi>σ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (f g) \cdot \sigma = f\cdot (g \cdot \sigma) </annotation></semantics></math></div> <p>for any two smooth functions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f,g \in C^\infty(X)</annotation></semantics></math>.</p> <p>One says that a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> such as <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(V)</annotation></semantics></math> equipped with an <a class="existingWikiWord" href="/nlab/show/action">action</a> of an algebra <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> this way is a <em><a class="existingWikiWord" href="/nlab/show/module">module</a></em> over <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>.</p> <p>In conclusion, any <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> gives rise to an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(V)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a>.</p> <p>The <em><a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a></em> states sufficient conditions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> such that the converse holds. Together with the <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> (prop <a class="maruku-ref" href="#EmbeddingOfSmoothManifoldsIntoRAlgebras"></a>), this means that <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is “more algebraic” than it might superficially seem, hence that its “algebraic deformation” to <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> is more natura than it might superficially seem:</p> <div class="num_prop"> <h6 id="proposition_11">Proposition</h6> <p><strong>(<a class="existingWikiWord" href="/nlab/show/smooth+Serre-Swan+theorem">smooth Serre-Swan theorem</a>, <a href="smooth+Serre-Swan+theorem#Nestruev03">Nestruev 03</a>)</strong></p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, then the construction which sends a smooth <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">V \to X</annotation></semantics></math> to its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Γ</mi> <mi>X</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma_X(V)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/sections">sections</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msubsup><mi>VectBund</mi> <mi>X</mi> <mi>fin</mi></msubsup><mover><mo>⟶</mo><mo>≃</mo></mover><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msubsup><mi>Mod</mi> <mi>proj</mi> <mrow><mi>fin</mi><mspace width="thinmathspace"></mspace><mi>gen</mi></mrow></msubsup></mrow><annotation encoding="application/x-tex"> VectBund_X^{fin} \stackrel{\simeq}{\longrightarrow} C^\infty(X) Mod_{proj}^{fin\,gen} </annotation></semantics></math></div> <p>between that of smooth <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> of finite <a class="existingWikiWord" href="/nlab/show/rank">rank</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and that of <a class="existingWikiWord" href="/nlab/show/finitely+generated+object">finitely generated</a> <a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> over the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℝ</mi></mrow><annotation encoding="application/x-tex">\mathbb{R}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> </div> <p>One may turn the <a class="existingWikiWord" href="/nlab/show/Serre-Swan+theorem">Serre-Swan theorem</a> around to regard for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> in some <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), the <a class="existingWikiWord" href="/nlab/show/modules">modules</a> over <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> as “generalized vector bundles” over the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Affines"></a>). These “generalized vector bundles” are called “<a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a>” over affines. Specified to the case that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{C} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a>, this hence yields a concept of <strong>super vector bundles</strong>.</p> <p>We now state the relevant definitions and constructions formally.</p> <div class="num_defn" id="ModulesInMonoidalCategory"> <h6 id="definition_23">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>), and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), then a <strong>left <a class="existingWikiWord" href="/nlab/show/module+object">module object</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> is</p> <ol> <li> <p>an <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">N \in \mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>N</mi><mo>⟶</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">\rho \;\colon\; A \otimes N \longrightarrow N</annotation></semantics></math> (called the <em><a class="existingWikiWord" href="/nlab/show/action">action</a></em>);</p> </li> </ol> <p>such that</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mn>1</mn><mo>⊗</mo><mi>N</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ℓ</mi></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ 1 \otimes N &\overset{e \otimes id}{\longrightarrow}& A \otimes N \\ & {}_{\mathllap{\ell}}\searrow & \downarrow^{\mathrlap{\rho}} \\ && N } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℓ</mi></mrow><annotation encoding="application/x-tex">\ell</annotation></semantics></math> is the left unitor isomorphism of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>.</p> </li> <li> <p>(action property) the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>N</mi></mtd> <mtd><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>a</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>N</mi></mrow></msub></mrow></munderover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>N</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>ρ</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>N</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd></mtd> <mtd><mover><mo>⟶</mo><mi>ρ</mi></mover></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ (A\otimes A) \otimes N &\underoverset{\simeq}{a_{A,A,N}}{\longrightarrow}& A \otimes (A \otimes N) &\overset{A \otimes \rho}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes N}}\downarrow && && \downarrow^{\mathrlap{\rho}} \\ A \otimes N &\longrightarrow& &\overset{\rho}{\longrightarrow}& N } \,, </annotation></semantics></math></div></li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module objects</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (N_1, \rho_1) \longrightarrow (N_2, \rho_2) </annotation></semantics></math></div> <p>is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>f</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>N</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> f\;\colon\; N_1 \longrightarrow N_2 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, such that the following <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commutes</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>ρ</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><msub><mi>N</mi> <mn>1</mn></msub></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><msub><mi>N</mi> <mn>2</mn></msub></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A\otimes N_1 &\overset{A \otimes f}{\longrightarrow}& A\otimes N_2 \\ {}^{\mathllap{\rho_1}}\downarrow && \downarrow^{\mathrlap{\rho_2}} \\ N_1 &\underset{f}{\longrightarrow}& N_2 } \,. </annotation></semantics></math></div> <p>For the resulting <strong><a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a></strong> of left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module homomorphisms between them, we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A Mod(\mathcal{C}) \,. </annotation></semantics></math></div></div> <p>The following degenerate example turns out to be important for the general development of the theory below.</p> <div class="num_example" id="EveryObjectIsModuleOverTensorUnit"> <h6 id="example_20">Example</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>) with the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in a monoidal category</a> via example <a class="maruku-ref" href="#MonoidGivenByTensorUnit"></a>, then the left <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a></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>C</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mn>1</mn><mo>⊗</mo><mi>C</mi><mo>⟶</mo><mi>C</mi></mrow><annotation encoding="application/x-tex"> \ell_C \;\colon\; 1\otimes C \longrightarrow C </annotation></semantics></math></div> <p>makes every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">C \in \mathcal{C}</annotation></semantics></math> into a left module, according to def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>, over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math>. The action property holds due to lemma <a class="maruku-ref" href="#kel1"></a>. This gives an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi><mo>≃</mo><mn>1</mn><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{C} \simeq 1 Mod(\mathcal{C}) </annotation></semantics></math></div> <p>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 the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> over its tensor unit.</p> </div> <div class="num_example" id="RingsAreMonoidsInAb"> <h6 id="example_21">Example</h6> <p>The classical subject of <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, not necessarily over <a class="existingWikiWord" href="/nlab/show/ground+fields">ground fields</a>, is the above general concepts of monoids and their modules specialized to the ambient <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> being the category <a class="existingWikiWord" href="/nlab/show/Ab">Ab</a> of <a class="existingWikiWord" href="/nlab/show/abelian+groups">abelian groups</a> regarded as a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> via the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups">tensor product of abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>ℤ</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_{\mathbb{Z}}</annotation></semantics></math> (whose <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> is the additive group of <a class="existingWikiWord" href="/nlab/show/integers">integers</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}</annotation></semantics></math>):</p> <ol> <li> <p>A <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ab</mi><mo>,</mo><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) is equivalently a <a class="existingWikiWord" href="/nlab/show/ring">ring</a>.</p> </li> <li> <p>A <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ab</mi><mo>,</mo><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) is equivalently a <a class="existingWikiWord" href="/nlab/show/commutative+ring">commutative ring</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>.</p> </li> <li> <p>An <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module+object">module object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ab</mi><mo>,</mo><msub><mo>⊗</mo> <mi>ℤ</mi></msub><mo>,</mo><mi>ℤ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(Ab, \otimes_{\mathbb{Z}}, \mathbb{Z})</annotation></semantics></math> (def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>) is equivalently an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a>;</p> </li> <li> <p>The tensor product of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module objects (def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a>) is the standard <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of module objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>Ab</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R Mod(Ab)</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a>) is the standard <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Mod</annotation></semantics></math>.</p> </li> </ol> </div> <div class="num_example" id="ModulesOverGGroupRing"> <h6 id="example_22">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/discrete+group">discrete group</a> and write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math> for its <a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a> over 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>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo stretchy="false">[</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">k[G]</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> in <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> are equivalently <a class="existingWikiWord" href="/nlab/show/linear+representations">linear representations</a> 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>.</p> </div> <div class="num_prop" id="MonoidModuleOverItself"> <h6 id="proposition_12">Proposition</h6> <p>In the situation of def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>, the monoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu, e)</annotation></semantics></math> canonically becomes a left module over itself by setting <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>≔</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">\rho \coloneqq \mu</annotation></semantics></math>. More generally, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>∈</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">C \in \mathcal{C}</annotation></semantics></math> any object, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">A \otimes C</annotation></semantics></math> naturally becomes a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module by setting:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>C</mi><mo stretchy="false">)</mo><munderover><mo>⟶</mo><mo>≃</mo><mrow><msubsup><mi>a</mi> <mrow><mi>A</mi><mo>,</mo><mi>A</mi><mo>,</mo><mi>C</mi></mrow> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msubsup></mrow></munderover><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo stretchy="false">)</mo><mo>⊗</mo><mi>C</mi><mover><mo>⟶</mo><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mover><mi>A</mi><mo>⊗</mo><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; A \otimes (A \otimes C) \underoverset{\simeq}{a^{-1}_{A,A,C}}{\longrightarrow} (A \otimes A) \otimes C \overset{\mu \otimes id}{\longrightarrow} A \otimes C \,. </annotation></semantics></math></div> <p>The <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules of this form are called <strong><a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a></strong>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/free+functor">free functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math> constructing free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules is <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to 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>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math> which sends a module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N,\rho)</annotation></semantics></math> to the underlying object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>N</mi></mrow><annotation encoding="application/x-tex">U(N,\rho) \coloneqq N</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>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><munderover><mo>⊥</mo><munder><mo>⟶</mo><mi>U</mi></munder><mover><mo>⟵</mo><mi>F</mi></mover></munderover><mi>𝒞</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A Mod(\mathcal{C}) \underoverset {\underset{U}{\longrightarrow}} {\overset{F}{\longleftarrow}} {\bot} \mathcal{C} \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_6">Proof</h6> <p>A homomorphism out of a free <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module is a morphism 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> of the form</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><mi>A</mi><mo>⊗</mo><mi>C</mi><mo>⟶</mo><mi>N</mi></mrow><annotation encoding="application/x-tex"> f \;\colon\; A\otimes C \longrightarrow N </annotation></semantics></math></div> <p>fitting into the diagram (where we are notationally suppressing the <a class="existingWikiWord" href="/nlab/show/associator">associator</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,. </annotation></semantics></math></div> <p>Consider the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>C</mi><munderover><mo>⟶</mo><mo>≃</mo><mrow><msub><mi>ℓ</mi> <mi>C</mi></msub></mrow></munderover><mn>1</mn><mo>⊗</mo><mi>C</mi><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover><mi>A</mi><mo>⊗</mo><mi>C</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>N</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \tilde f \;\colon\; C \underoverset{\simeq}{\ell_C}{\longrightarrow} 1 \otimes C \overset{e\otimes id}{\longrightarrow} A \otimes C \overset{f}{\longrightarrow} N \,, </annotation></semantics></math></div> <p>i.e. the restriction of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to the unit “in” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. By definition, this fits into a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a> of the form (where we are now notationally suppressing the <a class="existingWikiWord" href="/nlab/show/associator">associator</a> and the <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>f</mi></mrow></munder></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\underset{id \otimes f}{\longrightarrow}& A \otimes N } \,. </annotation></semantics></math></div> <p>Pasting this square onto the top of the previous one yields</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mi>f</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>N</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>C</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>N</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes C &\overset{id \otimes \tilde f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{id \otimes e \otimes id}}\downarrow && \downarrow^{\mathrlap{=}} \\ A \otimes A \otimes C &\overset{A \otimes f}{\longrightarrow}& A \otimes N \\ {}^{\mathllap{\mu \otimes id}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A \otimes C &\underset{f}{\longrightarrow}& N } \,, </annotation></semantics></math></div> <p>where now the left vertical composite is the identity, by the unit law in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> is uniquely determined by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> via the relation</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><mi>ρ</mi><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>A</mi></msub><mo>⊗</mo><mover><mi>f</mi><mo stretchy="false">˜</mo></mover><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> f = \rho \circ (id_A \otimes \tilde f) \,. </annotation></semantics></math></div> <p>This natural bijection between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>f</mi><mo stretchy="false">˜</mo></mover></mrow><annotation encoding="application/x-tex">\tilde f</annotation></semantics></math> establishes the adjunction.</p> </div> <div class="num_defn" id="TensorProductOfModulesOverCommutativeMonoidObject"> <h6 id="definition_24">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> denoted <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>τ</mi></mrow><annotation encoding="application/x-tex">\tau</annotation></semantics></math> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>, def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>), given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N_1, \rho_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N_2, \rho_2)</annotation></semantics></math> two left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module+objects">module objects</a> (def.<a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), then</p> <ol> <li> <p>the <strong><a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_1 \otimes_A N_2</annotation></semantics></math> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><munderover><mphantom><mi>AAAA</mi></mphantom><munder><mo>⟶</mo><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>∘</mo><mo stretchy="false">(</mo><msub><mi>τ</mi> <mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><mi>A</mi></mrow></msub><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mover></munderover><msub><mi>N</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mi>coeq</mi></mover><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> N_1 \otimes A \otimes N_2 \underoverset {\underset{\rho_{1}\circ (\tau_{N_1,A} \otimes N_2)}{\longrightarrow}} {\overset{N_1 \otimes \rho_2}{\longrightarrow}} {\phantom{AAAA}} N_1 \otimes N_1 \overset{coeq}{\longrightarrow} N_1 \otimes_A N_2 </annotation></semantics></math></div> <p>and if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \otimes (-)</annotation></semantics></math> preserves these coequalizers, then this is equipped with the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-action induced from the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">N_1</annotation></semantics></math></p> </li> <li> <p>the <strong>function module</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">hom_A(N_1,N_2)</annotation></semantics></math> is, if it exists, the <a class="existingWikiWord" href="/nlab/show/equalizer">equalizer</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>equ</mi></mover><mi>hom</mi><mo stretchy="false">(</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><munderover><mphantom><mi>AAAAAA</mi></mphantom><munder><mo>⟶</mo><mrow><mi>hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ρ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∘</mo><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></munder><mover><mo>⟶</mo><mrow><mi>hom</mi><mo stretchy="false">(</mo><msub><mi>ρ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow></mover></munderover><mi>hom</mi><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> hom_A(N_1, N_2) \overset{equ}{\longrightarrow} hom(N_1, N_2) \underoverset {\underset{hom(A \otimes N_1, \rho_2)\circ (A \otimes(-))}{\longrightarrow}} {\overset{hom(\rho_1,N_2)}{\longrightarrow}} {\phantom{AAAAAA}} hom(A \otimes N_1, N_2) \,. </annotation></semantics></math></div> <p>equipped with the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-action that is induced by the left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_2</annotation></semantics></math> via</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mrow><mi>A</mi><mo>⊗</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><mrow><mi>A</mi><mo>⊗</mo><mi>hom</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⊗</mo><mi>X</mi><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>ev</mi></mrow></mover><mi>A</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mover><msub><mi>N</mi> <mn>2</mn></msub></mrow></mfrac><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{ A \otimes hom(X,N_2) \longrightarrow hom(X,N_2) }{ A \otimes hom(X,N_2) \otimes X \overset{id \otimes ev}{\longrightarrow} A \otimes N_2 \overset{\rho_2}{\longrightarrow} N_2 } \,. </annotation></semantics></math></div></li> </ol> </div> <p>(e.g. <a href="#HoveyShipleySmith00">Hovey-Shipley-Smith 00, lemma 2.2.2 and lemma 2.2.8</a>)</p> <div class="num_prop" id="MonoidalCategoryOfModules"> <h6 id="proposition_13">Proposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed</a> <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>, def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>), and given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid in</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>). If all <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a> exist in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math>, then the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_A</annotation></semantics></math> from def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a> makes the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A Mod(\mathcal{C})</annotation></semantics></math> into a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod, \otimes_A, A)</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> the object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> itself, regarded as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module via prop. <a class="maruku-ref" href="#MonoidModuleOverItself"></a>.</p> <p>If moreover all <a class="existingWikiWord" href="/nlab/show/equalizers">equalizers</a> exist, then this is a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+category">closed monoidal category</a> (def. <a class="maruku-ref" href="#ClosedMonoidalCategory"></a>) with <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> given by the function modules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>hom</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">hom_A</annotation></semantics></math> of def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a>.</p> </div> <p>(e.g. <a href="#HoveyShipleySmith00">Hovey-Shipley-Smith 00, lemma 2.2.2, lemma 2.2.8</a>)</p> <div class="proof"> <h6 id="proof_sketch">Proof sketch</h6> <p>The associators and braiding for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_{A}</annotation></semantics></math> are induced directly from those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a>. That <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> is the tensor unit for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>⊗</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\otimes_{A}</annotation></semantics></math> follows with the same kind of argument that we give in the proof of example <a class="maruku-ref" href="#FreeModulesTensorProduct"></a> below.</p> </div> <div class="num_example" id="FreeModulesTensorProduct"> <h6 id="example_23">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu,e)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) in a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes, 1)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidalCategory"></a>), the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> (def. <a class="maruku-ref" href="#TensorProductOfModulesOverCommutativeMonoidObject"></a>) of two <a class="existingWikiWord" href="/nlab/show/free+modules">free modules</a> (def. <a class="maruku-ref" href="#MonoidModuleOverItself"></a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A\otimes C_1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><msub><mi>C</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A \otimes C_2</annotation></semantics></math> always exists and is the free module over the tensor product 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> of the two generators:</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>A</mi><mo>⊗</mo><msub><mi>C</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>A</mi><mo>⊗</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><msub><mi>C</mi> <mn>1</mn></msub><mo>⊗</mo><msub><mi>C</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (A \otimes C_1) \otimes_A (A \otimes C_2) \simeq A \otimes (C_1 \otimes C_2) \,. </annotation></semantics></math></div> <p>Hence if <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 all <a class="existingWikiWord" href="/nlab/show/coequalizers">coequalizers</a>, so that the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> 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><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod, \otimes_A, A)</annotation></semantics></math> (prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>) then the free module functor (def. <a class="maruku-ref" href="#MonoidModuleOverItself"></a>) is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>)</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><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> F \;\colon\; (\mathcal{C}, \otimes, 1) \longrightarrow (A Mod, \otimes_A, A) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_7">Proof</h6> <p>It is sufficient to show that the diagram</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><munderover><mphantom><mi>AAAA</mi></mphantom><munder><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>μ</mi></mrow></munder><mover><mo>⟶</mo><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mover></munderover><mi>A</mi><mo>⊗</mo><mi>A</mi><mover><mo>⟶</mo><mi>μ</mi></mover><mi>A</mi></mrow><annotation encoding="application/x-tex"> A \otimes A \otimes A \underoverset {\underset{id \otimes \mu}{\longrightarrow}} {\overset{\mu \otimes id}{\longrightarrow}} {\phantom{AAAA}} A \otimes A \overset{\mu}{\longrightarrow} A </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> diagram (we are notationally suppressing the <a class="existingWikiWord" href="/nlab/show/associators">associators</a>), hence that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi><mo>≃</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">A \otimes_A A \simeq A</annotation></semantics></math>, hence that the claim holds for <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><mn>1</mn></mrow><annotation encoding="application/x-tex">C_1 = 1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>C</mi> <mn>2</mn></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">C_2 = 1</annotation></semantics></math>.</p> <p>To that end, we check the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a>:</p> <p>First observe that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi></mrow><annotation encoding="application/x-tex">\mu</annotation></semantics></math> indeed coequalizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>⊗</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">id \otimes \mu</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">\mu \otimes id</annotation></semantics></math>, since this is just the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> clause in def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>. So for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⟶</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">f \colon A \otimes A \longrightarrow Q</annotation></semantics></math> any other morphism with this property, we need to show that there is a unique morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">\phi \colon A \longrightarrow Q</annotation></semantics></math> which makes this <a class="existingWikiWord" href="/nlab/show/commuting+diagram">diagram commute</a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>ϕ</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Q</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{\phi}} \\ Q } \,. </annotation></semantics></math></div> <p>We claim 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><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><munderover><mo>⟶</mo><mo>≃</mo><mrow><msup><mi>r</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></munderover><mi>A</mi><mo>⊗</mo><mn>1</mn><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mover><mi>A</mi><mo>⊗</mo><mi>A</mi><mover><mo>⟶</mo><mi>f</mi></mover><mi>Q</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; A \underoverset{\simeq}{r^{-1}}{\longrightarrow} A \otimes 1 \overset{id \otimes e}{\longrightarrow} A \otimes A \overset{f}{\longrightarrow} Q \,, </annotation></semantics></math></div> <p>where the first morphism is the inverse of the right <a class="existingWikiWord" href="/nlab/show/unitor">unitor</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>.</p> <p>First to see that this does make the required triangle commute, consider the following pasting composite of <a class="existingWikiWord" href="/nlab/show/commuting+diagrams">commuting diagrams</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msubsup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><msup><mi>r</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msubsup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msubsup><mo stretchy="false">↓</mo> <mo>≃</mo> <mpadded width="0"><mrow><msup><mi>r</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow></mpadded></msubsup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><mi>μ</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>f</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mi>f</mi></munder></mtd> <mtd><mi>Q</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{id \otimes r^{-1}}}_{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{r^{-1}}}_{\simeq} \\ A \otimes A \otimes 1 &\overset{\mu \otimes id}{\longrightarrow}& A \otimes 1 \\ {}^{\mathllap{id \otimes e}}\downarrow && \downarrow^{\mathrlap{id \otimes e} } \\ A \otimes A \otimes A &\overset{\mu \otimes id}{\longrightarrow}& A \otimes A \\ {}^{\mathllap{id \otimes \mu}}\downarrow && \downarrow^{\mathrlap{f}} \\ A \otimes A &\underset{f}{\longrightarrow}& Q } \,. </annotation></semantics></math></div> <p>Here the the top square is the <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturality</a> of the right <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a>, the middle square commutes by the functoriality of the tensor product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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> and the definition of the <a class="existingWikiWord" href="/nlab/show/product+category">product category</a> (def. <a class="maruku-ref" href="#OppositeAndProductOfPointedTopologicallyEnrichedCategory"></a>), while the commutativity of the bottom square is the assumption that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> coequalizes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>id</mi><mo>⊗</mo><mi>μ</mi></mrow><annotation encoding="application/x-tex">id \otimes \mu</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>μ</mi><mo>⊗</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">\mu \otimes id</annotation></semantics></math>.</p> <p>Here the right vertical composite is <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math>, while, by <a href="#UnitalityMonoid">unitality</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu ,e)</annotation></semantics></math>, the left vertical composite is the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, Hence the diagram says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo>∘</mo><mi>μ</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">\phi \circ \mu = f</annotation></semantics></math>, which we needed to show.</p> <p>It remains to see that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</annotation></semantics></math> is the unique morphism with this property for given <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>. For that let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>Q</mi></mrow><annotation encoding="application/x-tex">q \colon A \to Q</annotation></semantics></math> be any other morphism with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∘</mo><mi>μ</mi><mo>=</mo><mi>f</mi></mrow><annotation encoding="application/x-tex"> q\circ \mu = f</annotation></semantics></math>. Then consider the <a class="existingWikiWord" href="/nlab/show/commuting+diagram">commuting diagram</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mn>1</mn></mtd> <mtd><mover><mo>⟵</mo><mo>≃</mo></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>id</mi><mo>⊗</mo><mi>e</mi></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msup><mo>↘</mo> <mo>≃</mo></msup></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>=</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mi>μ</mi></mover></mtd> <mtd><mi>A</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>f</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>q</mi></mpadded></msub></mtd></mtr> <mtr><mtd><mi>Q</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes 1 &\overset{\simeq}{\longleftarrow}& A \\ {}^{\mathllap{id\otimes e}}\downarrow & \searrow^{\simeq} & \downarrow^{\mathrlap{=}} \\ A \otimes A &\overset{\mu}{\longrightarrow}& A \\ {}^{\mathllap{f}}\downarrow & \swarrow_{\mathrlap{q}} \\ Q } \,, </annotation></semantics></math></div> <p>where the top left triangle is the <a href="#UnitalityMonoid">unitality</a> condition and the two isomorphisms are the right <a class="existingWikiWord" href="/nlab/show/unitor">unitor</a> and its inverse. The commutativity of this diagram says that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>=</mo><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">q = \phi</annotation></semantics></math>.</p> </div> <div class="num_defn" id="AAlgebra"> <h6 id="definition_25">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> as in prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>, then a <a class="existingWikiWord" href="/nlab/show/monoid+in+a+monoidal+category">monoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu, e)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) is called an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/associative+algebra">algebra</a></strong>.</p> </div> <div class="num_prop" id="AlgebrasOverAAreMonoidsUnderA"> <h6 id="propposition">Propposition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a> <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mi>A</mi></msub><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A Mod , \otimes_A , A)</annotation></semantics></math> in a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C},\otimes, 1)</annotation></semantics></math> as in prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>, and an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><mi>μ</mi><mo>,</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E,\mu,e)</annotation></semantics></math> (def. <a class="maruku-ref" href="#AAlgebra"></a>), then there is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><msub><mi>Alg</mi> <mi>comm</mi></msub><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>A</mi><mi>Mod</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><msup><mo stretchy="false">)</mo> <mrow><mi>A</mi><mo stretchy="false">/</mo></mrow></msup></mrow><annotation encoding="application/x-tex"> A Alg_{comm}(\mathcal{C}) \coloneqq CMon(A Mod) \simeq CMon(\mathcal{C})^{A/} </annotation></semantics></math></div> <p>between the <a class="existingWikiWord" href="/nlab/show/category+of+commutative+monoids">category of commutative monoids</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/coslice+category">coslice category</a> of commutative monoids in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, hence between commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebras in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> and commutative monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> that are equipped with a homomorphism of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">A \longrightarrow E</annotation></semantics></math>.</p> </div> <p>(e.g. <a href="#EKMM97">EKMM 97, VII lemma 1.3</a>)</p> <div class="proof"> <h6 id="proof_8">Proof</h6> <p>In one direction, consider a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> with unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \;\colon\; A \longrightarrow E</annotation></semantics></math> and product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\mu_{E/A} \colon E \otimes_A E \longrightarrow E</annotation></semantics></math>. There is the underlying product <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>E</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><munderover><mphantom><mi>AAA</mi></mphantom><munder><mo>⟶</mo><mrow></mrow></munder><mover><mo>⟶</mo><mrow></mrow></mover></munderover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mi>coeq</mi></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ E \otimes A \otimes E & \underoverset {\underset{}{\longrightarrow}} {\overset{}{\longrightarrow}} {\phantom{AAA}} & E \otimes E &\overset{coeq}{\longrightarrow}& E \otimes_A E \\ && & {}_{\mathllap{\mu_E}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && && E } \,. </annotation></semantics></math></div> <p>By considering a diagram of such coequalizer diagrams with middle vertical morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_E\circ e_A</annotation></semantics></math>, one find that this is a unit for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</annotation></semantics></math> and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_E, e_E \circ e_A)</annotation></semantics></math> is a commutative monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math>.</p> <p>Then consider the two conditions on the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \colon A \longrightarrow E</annotation></semantics></math>. First of all this is an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module homomorphism, which means that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ρ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> (\star) \;\;\;\;\; \;\;\;\;\; \array{ A \otimes A &\overset{id \otimes e_E}{\longrightarrow}& A \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\rho}} \\ A &\underset{e_E}{\longrightarrow}& E } </annotation></semantics></math></div> <p><a class="existingWikiWord" href="/nlab/show/commuting+diagram">commutes</a>. Moreover it satisfies the unit property</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes_A E &\overset{e_A \otimes id}{\longrightarrow}& E \otimes_A E \\ & {}_{\mathllap{\simeq}}\searrow & \downarrow^{\mathrlap{\mu_{E/A}}} \\ && E } \,. </annotation></semantics></math></div> <p>By forgetting the tensor product over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, the latter gives</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>e</mi><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>≃</mo></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><mo>=</mo></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>≃</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ρ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>E</mi></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes E &\overset{e \otimes id}{\longrightarrow}& E \otimes E \\ \downarrow && \downarrow^{\mathrlap{}} \\ A \otimes_A E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes_A E \\ {}^{\mathllap{\simeq}}\downarrow && \downarrow^{\mathrlap{\mu_{E/A}}} \\ E &=& E } \;\;\;\;\;\;\;\; \simeq \;\;\;\;\;\;\;\; \array{ A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ E &\underset{id}{\longrightarrow}& E } \,, </annotation></semantics></math></div> <p>where the top vertical morphisms on the left the canonical coequalizers, which identifies the vertical composites on the right as shown. Hence this may be <a class="existingWikiWord" href="/nlab/show/pasting">pasted</a> to the square <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>⋆</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\star)</annotation></semantics></math> above, to yield a <a class="existingWikiWord" href="/nlab/show/commuting+square">commuting square</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><mi>id</mi><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>A</mi><mo>⊗</mo><mi>E</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>ρ</mi></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd> <mtd><munder><mo>⟶</mo><mi>id</mi></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mo>=</mo><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mspace width="thickmathspace"></mspace><mrow><mtable><mtr><mtd><mi>A</mi><mo>⊗</mo><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub><mo>⊗</mo><msub><mi>e</mi> <mi>E</mi></msub></mrow></mover></mtd> <mtd><mi>E</mi><mo>⊗</mo><mi>E</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>μ</mi> <mi>A</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>E</mi></msub></mrow></munder></mtd> <mtd><mi>E</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A \otimes A &\overset{id\otimes e_E}{\longrightarrow}& A \otimes E &\overset{e_E \otimes id}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && {}^{\mathllap{\rho}}\downarrow && \downarrow^{\mathrlap{\mu_{E}}} \\ A &\underset{e_E}{\longrightarrow}& E &\underset{id}{\longrightarrow}& E } \;\;\;\;\;\;\;\;\;\; = \;\;\;\;\;\;\;\;\;\; \array{ A \otimes A &\overset{e_E \otimes e_E}{\longrightarrow}& E \otimes E \\ {}^{\mathllap{\mu_A}}\downarrow && \downarrow^{\mathrlap{\mu_E}} \\ A &\underset{e_E}{\longrightarrow}& E } \,. </annotation></semantics></math></div> <p>This shows that the unit <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">e_A</annotation></semantics></math> is a homomorphism of monoids <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>⟶</mo><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo>∘</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu_A, e_A) \longrightarrow (E, \mu_E, e_E\circ e_A)</annotation></semantics></math>.</p> <p>Now for the converse direction, assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>μ</mi> <mi>A</mi></msub><mo>,</mo><msub><mi>e</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A,\mu_A, e_A)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mi>E</mi></msub><mo>,</mo><mi>e</mi><msub><mo>′</mo> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_E, e'_E)</annotation></semantics></math> are two commutative monoids in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{C}, \otimes, 1)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>e</mi> <mi>E</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>→</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">e_E \;\colon\; A \to E</annotation></semantics></math> a monoid homomorphism. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> inherits a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> structure by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mo>⊗</mo><mi>E</mi><mover><mo>⟶</mo><mrow><msub><mi>e</mi> <mi>A</mi></msub><mo>⊗</mo><mi>id</mi></mrow></mover><mi>E</mi><mo>⊗</mo><mi>E</mi><mover><mo>⟶</mo><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow></mover><mi>E</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; A \otimes E \overset{e_A \otimes id}{\longrightarrow} E \otimes E \overset{\mu_E}{\longrightarrow} E \,. </annotation></semantics></math></div> <p>By commutativity and associativity it follows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mi>E</mi></msub></mrow><annotation encoding="application/x-tex">\mu_E</annotation></semantics></math> coequalizes the two induced morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>E</mi><munderover><mphantom><mi>AA</mi></mphantom><mo>⟶</mo><mo>⟶</mo></munderover><mi>E</mi><mo>⊗</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">E \otimes A \otimes E \underoverset{\longrightarrow}{\longrightarrow}{\phantom{AA}} E \otimes E</annotation></semantics></math>. Hence the <a class="existingWikiWord" href="/nlab/show/universal+property">universal property</a> of the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> gives a factorization through some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo lspace="verythinmathspace">:</mo><mi>E</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>E</mi><mo>⟶</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">\mu_{E/A}\colon E \otimes_A E \longrightarrow E</annotation></semantics></math>. This shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>E</mi><mo>,</mo><msub><mi>μ</mi> <mrow><mi>E</mi><mo stretchy="false">/</mo><mi>A</mi></mrow></msub><mo>,</mo><msub><mi>e</mi> <mi>E</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(E, \mu_{E/A}, e_E)</annotation></semantics></math> is a commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-algebra.</p> <p>Finally one checks that these two constructions are inverses to each other, up to isomorphism.</p> </div> <p>When thinking of commutative monoids in some tensor category as <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> to certain <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a>, as in def. <a class="maruku-ref" href="#Affines"></a>, then we are interested in forming <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> and more generally <a class="existingWikiWord" href="/nlab/show/fiber+products">fiber products</a> of these spaces. Dually this is given by [fcoproducts] of commutative monoids and commutative <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>-algebras. The following says that these may be computed just as the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a>:</p> <div class="num_prop" id="CoproductsInCMon"> <h6 id="proposition_14">Proposition</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/symmetric+monoidal+category">symmetric monoidal category</a> such that</p> <ol> <li> <p>it has <a class="existingWikiWord" href="/nlab/show/reflexive+coequalizers">reflexive coequalizers</a></p> </li> <li> <p>which are preserved by the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>𝒞</mi><mo>→</mo><mi>𝒞</mi></mrow><annotation encoding="application/x-tex">A \otimes (-) \colon \mathcal{C} \to \mathcal{C}</annotation></semantics></math> for all objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> 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>.</p> </li> </ol> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding="application/x-tex">f \colon A \to B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">g \colon A \to C</annotation></semantics></math> two morphisms in the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathcal{C})</annotation></semantics></math> of <em><a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a></em> 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> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), the underlying object 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> of the <a class="existingWikiWord" href="/nlab/show/pushout">pushout</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathcal{C})</annotation></semantics></math> coincides with the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in the monoidal category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Mod">Mod</a> (according to prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a>):</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>U</mi><mrow><mo>(</mo><mi>B</mi><msub><mo>⊔</mo> <mi>A</mi></msub><mi>C</mi><mo>)</mo></mrow><mo>≃</mo><mi>B</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>C</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> U\left(B \sqcup_A C\right) \simeq B \otimes_A C \,. </annotation></semantics></math></div> <p>Here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> are regarded as equipped with the canonical <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module structure induced by the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> and <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>, respectively.</p> </div> <p>This appears for instance as (<a class="existingWikiWord" href="/nlab/show/Sketches+of+an+Elephant">Johnstone, page 478, cor. 1.1.9</a>).</p> <div class="num_remark" id="AssumptionsOnCoproductInCMonGenericallySatisfied"> <h6 id="remark_6">Remark</h6> <p>In every <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>) the conditions in prop. <a class="maruku-ref" href="#CoproductsInCMon"></a> are satisfied.</p> </div> <div class="proof"> <h6 id="proof_9">Proof</h6> <p>By definition, every <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> is an <a class="existingWikiWord" href="/nlab/show/abelian+category">abelian category</a> (def. <a class="maruku-ref" href="#AdditiveAndAbelianCategories"></a>). The <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> of two <a class="existingWikiWord" href="/nlab/show/parallel+morphisms">parallel morphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f,g</annotation></semantics></math> in an abelian category is isomorphic to the <a class="existingWikiWord" href="/nlab/show/cokernel">cokernel</a> of the difference <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>−</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f-g</annotation></semantics></math> (formed in the <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a> struture on the <a class="existingWikiWord" href="/nlab/show/hom-space">hom-space</a>). Hence all coequalizers exist, in particlar the <a class="existingWikiWord" href="/nlab/show/split+coequalizers">split coequalizers</a> required in prop. <a class="maruku-ref" href="#CoproductsInCMon"></a>.</p> <p>Moreover, by definition every <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> is a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>. This implies that it is also a <a class="existingWikiWord" href="/nlab/show/closed+monoidal+categories">closed monoidal categories</a>, by prop. <a class="maruku-ref" href="#CompactClosedMonoidalCategory"></a>, and this means that the functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \otimes (-)</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> functors, and such preserve all colimits.</p> </div> <div class="num_prop" id="ProductsInAff"> <h6 id="proposition_15">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>, and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \in CMon(\mathcal{A})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>.</p> <p>Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1, A_2</annotation></semantics></math> two <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>-algebas according to def. <a class="maruku-ref" href="#AAlgebra"></a>, regarded as affine schemes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>,</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∈</mo><mi>Aff</mi><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_1), Spec(A_2) \in Aff(R Mod(\mathcal{C}))</annotation></semantics></math> according to prop. <a class="maruku-ref" href="#MonoidalCategoryOfModules"></a> and def. <a class="maruku-ref" href="#Affines"></a> the <a class="existingWikiWord" href="/nlab/show/Cartesian+product">Cartesian product</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_1)</annotation></semantics></math> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_2)</annotation></semantics></math> exists in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo stretchy="false">(</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aff(R Mod(\mathcal{C}))</annotation></semantics></math> and is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the tensor product algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>R</mi></msub><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \otimes_R A_2</annotation></semantics></math> according to example <a class="maruku-ref" href="#TensorProductOfTwoCommutativeMonoids"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>×</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>R</mi></msub><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spec(A_1) \times Spec(A_2) \simeq Spec(A_1 \otimes_R A_2) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_10">Proof</h6> <p>By prop. <a class="maruku-ref" href="#AlgebrasOverAAreMonoidsUnderA"></a> the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the statement is given by prop. <a class="maruku-ref" href="#CoproductsInCMon"></a>, which does apply, according to remark <a class="maruku-ref" href="#AssumptionsOnCoproductInCMonGenericallySatisfied"></a>.</p> </div> <div class="num_defn" id="ExtensionOfScalars"> <h6 id="proposition_16">Proposition</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/symmetric+monoidal+category">symmetric monoidal category</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A_1, A_2 \in CMon(\mathcal{C})</annotation></semantics></math> be two <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>𝒞</mi></mrow><annotation encoding="application/x-tex">\mathcal{C}</annotation></semantics></math> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) and</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><msub><mi>A</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; A_1 \longrightarrow A_2 </annotation></semantics></math></div> <p>a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>).</p> <p>Then there is a pair of <a class="existingWikiWord" href="/nlab/show/adjoint+functors">adjoint functors</a> between the <a class="existingWikiWord" href="/nlab/show/categories+of+modules">categories of modules</a> (def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟵</mo><mrow><msup><mi>ϕ</mi> <mo>*</mo></msup></mrow></munder><mover><mo>⟶</mo><mrow><msub><mi>ϕ</mi> <mo>*</mo></msub></mrow></mover></munderover><msub><mi>A</mi> <mn>2</mn></msub><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A_1 Mod(\mathcal{C}) \underoverset {\underset{\phi^\ast}{\longleftarrow}} {\overset{\phi_\ast}{\longrightarrow}} {} A_2 Mod(\mathcal{C}) </annotation></semantics></math></div> <p>where</p> <ol> <li> <p>the <a class="existingWikiWord" href="/nlab/show/right+adjoint">right adjoint</a>, called <strong><a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a></strong>, sends an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N, \rho)</annotation></semantics></math> to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>ρ</mi><mo>′</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N,\rho')</annotation></semantics></math> whose <a class="existingWikiWord" href="/nlab/show/action">action</a> is given by precomposition with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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><mi>A</mi> <mn>1</mn></msub><mo>⊗</mo><mi>N</mi><mover><mo>⟶</mo><mrow><mi>ϕ</mi><mo>⊗</mo><mi>id</mi></mrow></mover><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><mi>N</mi><mover><mo>⟶</mo><mi>ρ</mi></mover><mi>N</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_1 \otimes N \stackrel{\phi \otimes id}{\longrightarrow} A_2 \otimes N \stackrel{\rho}{\longrightarrow} N \,. </annotation></semantics></math></div></li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a>, called <strong><a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a></strong> sends an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>N</mi><mo>,</mo><mi>ρ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(N,\rho)</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</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> <mo>*</mo></msub><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo>⊗</mo> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mi>N</mi></mrow><annotation encoding="application/x-tex"> \phi_\ast(N) \coloneqq A_2 \otimes_{A_1} N </annotation></semantics></math></div> <p>(where we are regarding <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math> as a commutative monoid in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">A_1</annotation></semantics></math>-modules via prop. <a class="maruku-ref" href="#AlgebrasOverAAreMonoidsUnderA"></a>) and equipped with the evident <a class="existingWikiWord" href="/nlab/show/action">action</a> induced by the multiplication in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_2</annotation></semantics></math>:</p> </li> </ol> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>A</mi> <mn>2</mn></msub><mo>⊗</mo><msub><mi>A</mi> <mn>2</mn></msub><msub><mo>⊗</mo> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mi>N</mi><mover><mo>⟶</mo><mrow><msub><mi>μ</mi> <mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow></msub><msub><mo>⊗</mo> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mi>N</mi></mrow></mover><msub><mi>A</mi> <mn>2</mn></msub><msub><mo>⊗</mo> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mi>N</mi><mo>=</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A_2 \otimes \phi^\ast(N) = A_2 \otimes A_2 \otimes_{A_1} N \stackrel{\mu_{A_2} \otimes_{A_1} N }{\longrightarrow} A_2 \otimes_{A_1} N = \phi^\ast(N) \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_11">Proof</h6> <p>By prop. <a class="maruku-ref" href="#AlgebrasOverAAreMonoidsUnderA"></a> the <a class="existingWikiWord" href="/nlab/show/adjunction">adjunction</a> in question has 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>A</mi> <mn>1</mn></msub><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟵</mo><mi>U</mi></munder><mover><mo>⟶</mo><mi>F</mi></mover></munderover><msub><mi>A</mi> <mn>2</mn></msub><mi>Mod</mi><mo stretchy="false">(</mo><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mi>Mod</mi><mo>,</mo><msub><mo>⊗</mo> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mo>,</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> A_1 Mod(\mathcal{C}) \underoverset {\underset{U}{\longleftarrow}} {\overset{F}{\longrightarrow}} {} A_2 Mod( (A_1 Mod, \otimes_{A_1}, A_1) ) </annotation></semantics></math></div> <p>and hence the statement follows with prop. <a class="maruku-ref" href="#MonoidModuleOverItself"></a>.</p> </div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>In the dual interpretation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/modules">modules</a> as generalized <a class="existingWikiWord" href="/nlab/show/vector+bundles">vector bundles</a> (namely: <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a>) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Affines"></a>) then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi><mo lspace="verythinmathspace">:</mo><msub><mi>A</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\phi \colon A_1 \to A_2</annotation></semantics></math> becomes a map of spaces</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>ϕ</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(\phi) \colon Spec(A_2) \longrightarrow Spec(A_1) </annotation></semantics></math></div> <p>and then <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> according to prop. <a class="maruku-ref" href="#ExtensionOfScalars"></a> corresponds to the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of vector bundles from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_1)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_2)</annotation></semantics></math>.</p> </div> <h3 id="GroupsAsHopfAlgebras">Super-Groups as super-commutative Hopf algebras</h3> <p>Above we have considered affine spaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Affines"></a>) in <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</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>. Now we discuss what it means to equip these with the stucture of <a class="existingWikiWord" href="/nlab/show/group+objects">group objects</a>, hence to form affine groups 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>.</p> <p>A (possibly) familiar example arises in <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, where one considers groups whose underlying set is promoted to a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and all whose operations (product, inverses) are <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a>. These are of course the <em><a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a></em>.</p> <p>A <strong><a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a></strong> of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <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> on a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> <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/smooth+function">smooth function</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo>×</mo><mi>V</mi><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> \rho \;\colon\; G \times V \longrightarrow V </annotation></semantics></math></div> <p>such that</p> <ol> <li> <p>(<a class="existingWikiWord" href="/nlab/show/linear+map">linearity</a>) for all elements <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> the function</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo lspace="verythinmathspace">:</mo><mi>V</mi><mo>⟶</mo><mi>V</mi></mrow><annotation encoding="application/x-tex"> \rho(g) \colon V \longrightarrow V </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/linear+function">linear function</a></p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>) for <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> the <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>e</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(e)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> function;</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/action">action</a> property) for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><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><annotation encoding="application/x-tex">g_1, g_2 \in G</annotation></semantics></math> any two elements, then acting with them consecutively is the same as acting with their product:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>g</mi> <mn>2</mn></msub><msub><mi>g</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho(g_2) \circ \rho(g_1) = \rho(g_2 g_1) \,. </annotation></semantics></math></div></li> </ol> <p>But here we need to consider groups with more general geometric structure. The key to the generalization is to regard <a class="existingWikiWord" href="/nlab/show/spaces">spaces</a> dually via their <a class="existingWikiWord" href="/nlab/show/algebras+of+functions">algebras of functions</a>.</p> <p>In the above example, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(X)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/smooth+algebra">smooth algebra</a> of <a class="existingWikiWord" href="/nlab/show/smooth+functions">smooth functions</a> on a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. The assignment</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>SmthMfd</mi><mo>↪</mo><msubsup><mi>SmthAlg</mi> <mi>ℝ</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex"> C^\infty(-) \;\colon\; SmthMfd \hookrightarrow SmthAlg_{\mathbb{R}}^{op} </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/embedding+of+smooth+manifolds+into+formal+duals+of+R-algebras">embedding of smooth manifolds into formal duals of R-algebras</a> from prop. <a class="maruku-ref" href="#EmbeddingOfSmoothManifoldsIntoRAlgebras"></a>.</p> <p>Moreover, the functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(-)</annotation></semantics></math> sends <a class="existingWikiWord" href="/nlab/show/Cartesian+products">Cartesian products</a> of smooth manifolds to “completed tensor products” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>⊗</mo> <mi>c</mi></msup></mrow><annotation encoding="application/x-tex">\otimes^c</annotation></semantics></math> of function algebras (namely to the <a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a> of <a class="existingWikiWord" href="/nlab/show/smooth+algebras">smooth algebras</a>, see there)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo>×</mo><mi>Y</mi><mo stretchy="false">)</mo><mo>≃</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>c</mi></msup><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> C^\infty(X \times Y) \simeq C^\infty(X) \otimes^c C^\infty(Y) \,. </annotation></semantics></math></div> <p>Together this means that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X = G</annotation></semantics></math> is equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a>, then the product operation in the group induces a “<a class="existingWikiWord" href="/nlab/show/coproduct">coproduct</a>” operation on its smooth algebra of smooth functions:</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>product</mi></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><mi>G</mi><mo>×</mo><mi>G</mi></mtd> <mtd><mo>⟶</mo></mtd> <mtd><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><msup><mo>⊗</mo> <mi>c</mi></msup><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>⟵</mo></mtd> <mtd><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo lspace="verythinmathspace">:</mo></mtd> <mtd><msup><mi>product</mi> <mo>*</mo></msup><mo>=</mo><mi>coproduct</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ product &\colon& G \times G &\longrightarrow& G \\ && C^\infty(G) \otimes^c C^\infty(G) &\longleftarrow& C^\infty(G) &\colon& product^\ast = coproduct } \,. </annotation></semantics></math></div> <p>Now the <a class="existingWikiWord" href="/nlab/show/associativity">associativity</a> of the group product translates into a corresponding dual property of its dual, called “<a class="existingWikiWord" href="/nlab/show/co-associativity">co-associativity</a>”, and so forth. The resulting algebraic structure is called a <strong><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></strong>.</p> <p>While the explicit definition of a <em><a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></em> may look involved at first sight, Hopf algebras are simply <a class="existingWikiWord" href="/nlab/show/formal+duals">formal duals</a> of <a class="existingWikiWord" href="/nlab/show/groups">groups</a>. Since this perspective is straightforward, we may just as well consider it in the generality of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a>.</p> <p>A simple illustrative archetype of the following construction of commutative Hopf algebroids from homotopy commutative ring spectra is the following situation:</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/finite+set">finite set</a> consider</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><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∘</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>pr</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>pr</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi><mo>×</mo><mi>X</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>s</mi><mo>=</mo><msub><mi>pr</mi> <mn>1</mn></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo><mo stretchy="false">↑</mo><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>t</mi><mo>=</mo><msub><mi>pr</mi> <mn>2</mn></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ X \times X \times X \\ \downarrow^{\mathrlap{\circ = (pr_1, pr_3)}} \\ X \times X \\ {}^{\mathllap{s = pr_1}}\downarrow \uparrow \downarrow^{\mathrlap{t = pr_2}} \\ X } </annotation></semantics></math></div> <p>as the (“<a class="existingWikiWord" href="/nlab/show/codiscrete+groupoid">codiscrete</a>”) <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> as <a class="existingWikiWord" href="/nlab/show/objects">objects</a> and precisely one morphism from every object to every other. Hence the <a class="existingWikiWord" href="/nlab/show/composition">composition</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math>, and the <a class="existingWikiWord" href="/nlab/show/source">source</a> and <a class="existingWikiWord" href="/nlab/show/target">target</a> maps are simply projections as shown. The identity morphism (going upwards in the above diagram) is the <a class="existingWikiWord" href="/nlab/show/diagonal">diagonal</a>.</p> <p>Then consider the image of this structure under forming the <a class="existingWikiWord" href="/nlab/show/free+abelian+groups">free abelian groups</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[X]</annotation></semantics></math>, regarded as <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a> under pointwise multiplication.</p> <p>Since</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo>×</mo><mi>X</mi><mo stretchy="false">]</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[X \times X] \simeq \mathbb{Z}[X] \otimes \mathbb{Z}[X] </annotation></semantics></math></div> <p>this yields a diagram of homomorphisms of commutative rings of the form</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>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↑</mo><mo stretchy="false">↓</mo><mo stretchy="false">↑</mo></mtd></mtr> <mtr><mtd><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X]) \\ \uparrow^{\mathrlap{} } \\ \mathbb{Z}[X] \otimes \mathbb{Z}[X] \\ \uparrow \downarrow \uparrow \\ \mathbb{Z}[X] } </annotation></semantics></math></div> <p>satisfying some obvious conditions. Observe that here</p> <ol> <li> <p>the two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⇉</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[X] \rightrightarrows \mathbb{Z}[X] \otimes \mathbb{Z}[X]</annotation></semantics></math> are <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><mi>f</mi><mo>⊗</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">f \mapsto f \otimes e</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>↦</mo><mi>e</mi><mo>⊗</mo><mi>f</mi></mrow><annotation encoding="application/x-tex">f \mapsto e \otimes f</annotation></semantics></math>, respectively, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>e</mi></mrow><annotation encoding="application/x-tex">e</annotation></semantics></math> denotes the unit element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[X]</annotation></semantics></math>;</p> </li> <li> <p>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>→</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[X] \otimes \mathbb{Z}[X] \to \mathbb{Z}[X]</annotation></semantics></math> is the multiplication in the ring <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}[X]</annotation></semantics></math>;</p> </li> <li> <p>the morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⟶</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>C</mi><mo stretchy="false">]</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mrow><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo></mrow></msub><mo stretchy="false">(</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo>⊗</mo><mi>ℤ</mi><mo stretchy="false">[</mo><mi>X</mi><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbb{Z}[X] \otimes \mathbb{Z}[X] \longrightarrow \mathbb{Z}[X] \otimes \mathbb{Z}[C] \otimes \mathbb{Z}[C] \overset{\simeq}{\longrightarrow} (\mathbb{Z}[X] \otimes \mathbb{Z}[X] ) \otimes_{\mathbb{Z}[X]} (\mathbb{Z}[X] \otimes \mathbb{Z}[X]) </annotation></semantics></math></div> <p>is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>⊗</mo><mi>g</mi><mo>↦</mo><mi>f</mi><mo>⊗</mo><mi>e</mi><mo>⊗</mo><mi>g</mi></mrow><annotation encoding="application/x-tex">f \otimes g \mapsto f \otimes e \otimes g</annotation></semantics></math>.</p> </li> </ol> <p>We now say this again, in generality:</p> <div class="num_defn" id="CommutativeHopfAlgebroid"> <h6 id="definition_26">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>). A <strong><a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a></strong> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})^{op}</annotation></semantics></math> of <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>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, regarded with its <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> structure according to prop. <a class="maruku-ref" href="#ProductsInAff"></a>.</p> </div> <p>(e.g. <a href="commutative+Hopf+algebroid#Ravenel86">Ravenel 86, def. A1.1.1</a>)</p> <p>We unwind def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \in CMon(\mathcal{A})</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>R</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(R)</annotation></semantics></math> for same same object, but regarded as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})^{op}</annotation></semantics></math>.</p> <div class="num_prop" id="CommutativeHopfAlgebroidSpelledOut"> <h6 id="proposition_17">Proposition</h6> <p>An <a class="existingWikiWord" href="/nlab/show/internal+category">internal category</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})^{op}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/diagram">diagram</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})^{op}</annotation></semantics></math> of the form</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>Spec</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow></munder><mi>Spec</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>∘</mo></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Spec</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mi>s</mi></mpadded></msup><mo stretchy="false">↓</mo><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>i</mi></mpadded></msup><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>t</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ Spec(\Gamma) \underset{Spec(A)}{\times} Spec(\Gamma) \\ \downarrow^{\mathrlap{\circ}} \\ Spec(\Gamma) \\ {}^{\mathllap{s}}\downarrow \; \uparrow^{\mathrlap{i}} \downarrow^{\mathrlap{t}} \\ Spec(A) } \,, </annotation></semantics></math></div> <p>(where the <a class="existingWikiWord" href="/nlab/show/fiber+product">fiber product</a> at the top is over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi></mrow><annotation encoding="application/x-tex">s</annotation></semantics></math> on the left and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi></mrow><annotation encoding="application/x-tex">t</annotation></semantics></math> on the right) such that the pairing <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math> defines an <a class="existingWikiWord" href="/nlab/show/associativity+law">associative</a> <a class="existingWikiWord" href="/nlab/show/composition">composition</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/unitality">unital</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>i</mi></mrow><annotation encoding="application/x-tex">i</annotation></semantics></math>. This is an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> if it is furthemore equipped with a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>inv</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Spec</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>Γ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> inv \;\colon\; Spec(\Gamma) \longrightarrow Spec(\Gamma) </annotation></semantics></math></div> <p>acting as assigning <a class="existingWikiWord" href="/nlab/show/inverses">inverses</a> with respect to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∘</mo></mrow><annotation encoding="application/x-tex">\circ</annotation></semantics></math>.</p> <p>The key fact to use now is prop. <a class="maruku-ref" href="#ProductsInAff"></a>: the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of commutative monoids exhibits the <a class="existingWikiWord" href="/nlab/show/cartesian+monoidal+category">cartesian monoidal category</a> structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})^{op}</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>Spec</mi><mo stretchy="false">(</mo><msub><mi>R</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>R</mi> <mn>3</mn></msub><mo stretchy="false">)</mo></mrow></munder><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>R</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>≃</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>R</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mrow><msub><mi>R</mi> <mn>3</mn></msub></mrow></msub><msub><mi>R</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Spec(R_1) \underset{Spec(R_3)}{\times} Spec(R_2) \simeq Spec(R_1 \otimes_{R_3} R_2) \,. </annotation></semantics></math></div> <p>This means that def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a> is equivalently a diagram in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">CMon(\mathcal{A})</annotation></semantics></math> of the form</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>Γ</mi><munder><mo>⊗</mo><mi>A</mi></munder><mi>Γ</mi></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mi>Ψ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Γ</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>η</mi> <mi>L</mi></msub></mrow></mpadded></msup><mo stretchy="false">↑</mo><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>ϵ</mi></mpadded></msup><mspace width="thickmathspace"></mspace><msup><mo stretchy="false">↑</mo> <mpadded width="0"><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mi>A</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma \underset{A}{\otimes} \Gamma \\ \uparrow^{\mathrlap{\Psi}} \\ \Gamma \\ {}^{\mathllap{\eta_L}}\uparrow \downarrow^{\mathrlap{\epsilon}} \; \uparrow^{\mathrlap{\eta_R}} \\ A } </annotation></semantics></math></div> <p>as well as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>c</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Γ</mi><mo>⟶</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex"> c \; \colon \; \Gamma \longrightarrow \Gamma </annotation></semantics></math></div> <p>and satisfying <a class="existingWikiWord" href="/nlab/show/formal+duality">formally dual</a> conditions, spelled out as def. <a class="maruku-ref" href="#CommutativeHopfAlgebroidDefinitionInExplicitComponents"></a> below. Here</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>L</mi></msub><mo>,</mo><mo lspace="0em" rspace="thinmathspace">etaR</mo></mrow><annotation encoding="application/x-tex">\eta_L, \etaR</annotation></semantics></math> are called the left and right <em><a class="existingWikiWord" href="/nlab/show/unit">unit</a> maps</em>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is called the <em>co-unit</em>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">\Psi</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/comultiplication">comultiplication</a></em>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi></mrow><annotation encoding="application/x-tex">c</annotation></semantics></math> is called the <em><a class="existingWikiWord" href="/nlab/show/antipode">antipode</a></em> or <em>conjugation</em></p> </li> </ul> </div> <div class="num_remark" id="HopfAlgebrasAsHopfAlgebroids"> <h6 id="remark_8">Remark</h6> <p>Generally, in a commutative Hopf algebroid, def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a>, the two morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>L</mi></msub><mo>,</mo><msub><mi>η</mi> <mi>R</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>→</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\eta_L, \eta_R\colon A \to \Gamma</annotation></semantics></math> from remark <a class="maruku-ref" href="#CommutativeHopfAlgebroidSpelledOut"></a> need not coincide, they make <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> genuinely into a <a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>, and it is the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of <a class="existingWikiWord" href="/nlab/show/bimodules">bimodules</a> that appears in remark <a class="maruku-ref" href="#CommutativeHopfAlgebroidSpelledOut"></a>. But it may happen that they coincide:</p> <p>An <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub><mover><munder><mo>⟶</mo><mi>t</mi></munder><mover><mo>⟶</mo><mi>s</mi></mover></mover><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_1 \stackrel{\overset{s}{\longrightarrow}}{\underset{t}{\longrightarrow}} \mathcal{G}_0</annotation></semantics></math> for which the <a class="existingWikiWord" href="/nlab/show/domain">domain</a> and <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a> morphisms coincide, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>=</mo><mi>t</mi></mrow><annotation encoding="application/x-tex">s = t</annotation></semantics></math>, is euqivalently a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the <a class="existingWikiWord" href="/nlab/show/slice+category">slice category</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{G}_0</annotation></semantics></math>.</p> <p>Dually, a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mover><munder><mo>⟵</mo><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></munder><mover><mo>⟵</mo><mrow><msub><mi>η</mi> <mi>L</mi></msub></mrow></mover></mover><mi>A</mi></mrow><annotation encoding="application/x-tex">\Gamma \stackrel{\overset{\eta_L}{\longleftarrow}}{\underset{\eta_R}{\longleftarrow}} A</annotation></semantics></math> for which <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\eta_L</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">\eta_R</annotation></semantics></math> happen to coincide is equivalently a <strong>commutative <a class="existingWikiWord" href="/nlab/show/Hopf+algebra">Hopf algebra</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> </div> <p>Writing out the formally dual axioms of an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> as in remark <a class="maruku-ref" href="#CommutativeHopfAlgebroidSpelledOut"></a> yields the following equivalent but maybe more explicit definition of commutative Hopf algebroids, def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a></p> <div class="num_defn" id="CommutativeHopfAlgebroidDefinitionInExplicitComponents"> <h6 id="definition_27">Definition</h6> <p>A <strong><a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a></strong> is</p> <ol> <li> <p>two <a class="existingWikiWord" href="/nlab/show/commutative+rings">commutative rings</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>;</p> </li> <li> <p>ring <a class="existingWikiWord" href="/nlab/show/homomorphisms">homomorphisms</a></p> <ol> <li> <p>(left/right unit)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>L</mi></msub><mo>,</mo><msub><mi>η</mi> <mi>R</mi></msub><mo lspace="verythinmathspace">:</mo><mi>A</mi><mo>⟶</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\eta_L,\eta_R \colon A \longrightarrow \Gamma</annotation></semantics></math>;</p> </li> <li> <p>(comultiplication)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo lspace="verythinmathspace">:</mo><mi>Γ</mi><mo>⟶</mo><mi>Γ</mi><munder><mo>⊗</mo><mi>A</mi></munder><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Psi \colon \Gamma \longrightarrow \Gamma \underset{A}{\otimes} \Gamma</annotation></semantics></math>;</p> </li> <li> <p>(counit)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo lspace="verythinmathspace">:</mo><mi>Γ</mi><mo>⟶</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">\epsilon \colon \Gamma \longrightarrow A</annotation></semantics></math>;</p> </li> <li> <p>(conjugation)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo lspace="verythinmathspace">:</mo><mi>Γ</mi><mo>⟶</mo><mi>Γ</mi></mrow><annotation encoding="application/x-tex">c \colon \Gamma \longrightarrow \Gamma</annotation></semantics></math></p> </li> </ol> </li> </ol> <p>such that</p> <ol> <li> <p>(co-<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>)</p> <ol> <li> <p>(identity morphisms respect source and target)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∘</mo><msub><mi>η</mi> <mi>L</mi></msub><mo>=</mo><mi>ϵ</mi><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon \circ \eta_L = \epsilon \circ \eta_R = id_A</annotation></semantics></math>;</p> </li> <li> <p>(identity morphisms are units for composition)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>Γ</mi></msub><msub><mo>⊗</mo> <mi>A</mi></msub><mi>ϵ</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Ψ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>ϵ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>id</mi> <mi>Γ</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>Ψ</mi><mo>=</mo><msub><mi>id</mi> <mi>Γ</mi></msub></mrow><annotation encoding="application/x-tex">(id_\Gamma \otimes_A \epsilon) \circ \Psi = (\epsilon \otimes_A id_\Gamma) \circ \Psi = id_\Gamma</annotation></semantics></math>;</p> </li> <li> <p>(composition respects source and target)</p> <ol> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>id</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>η</mi> <mi>R</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">\Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>∘</mo><msub><mi>η</mi> <mi>L</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>η</mi> <mi>L</mi></msub><msub><mo>⊗</mo> <mi>A</mi></msub><mi>id</mi><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">\Psi \circ \eta_L = (\eta_L \otimes_A id) \circ \eta_L</annotation></semantics></math></p> </li> </ol> </li> </ol> </li> <li> <p>(co-<a class="existingWikiWord" href="/nlab/show/associativity">associativity</a>) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>Γ</mi></msub><msub><mo>⊗</mo> <mi>A</mi></msub><mi>Ψ</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Ψ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>Ψ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>id</mi> <mi>Γ</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><mi>Ψ</mi></mrow><annotation encoding="application/x-tex">(id_\Gamma \otimes_A \Psi) \circ \Psi = (\Psi \otimes_A id_\Gamma) \circ \Psi</annotation></semantics></math>;</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/inverses">inverses</a>)</p> <ol> <li> <p>(inverting twice is the identity)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∘</mo><mi>c</mi><mo>=</mo><msub><mi>id</mi> <mi>Γ</mi></msub></mrow><annotation encoding="application/x-tex">c \circ c = id_\Gamma</annotation></semantics></math>;</p> </li> <li> <p>(inversion swaps source and target)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∘</mo><msub><mi>η</mi> <mi>L</mi></msub><mo>=</mo><msub><mi>η</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex">c \circ \eta_L = \eta_R</annotation></semantics></math>; <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>=</mo><msub><mi>η</mi> <mi>L</mi></msub></mrow><annotation encoding="application/x-tex">c \circ \eta_R = \eta_L</annotation></semantics></math>;</p> </li> <li> <p>(inverse morphisms are indeed left and right inverses for composition)</p> <p>the morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi></mrow><annotation encoding="application/x-tex">\alpha</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> induced via the <a class="existingWikiWord" href="/nlab/show/coequalizer">coequalizer</a> property of the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> from <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><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-) \cdot c(-)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><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></mrow><annotation encoding="application/x-tex">c(-)\cdot (-)</annotation></semantics></math>, respectively</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>Γ</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>Γ</mi></mtd> <mtd><munderover><mrow></mrow><mo>⟶</mo><mo>⟶</mo></munderover></mtd> <mtd><mi>Γ</mi><mo>⊗</mo><mi>Γ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>coeq</mi></mover></mtd> <mtd><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>Γ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>⋅</mo><mi>c</mi><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msub><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>α</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Γ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{(-)\cdot c(-)}}\downarrow & \swarrow_{\mathrlap{\alpha}} \\ && \Gamma } </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>Γ</mi><mo>⊗</mo><mi>A</mi><mo>⊗</mo><mi>Γ</mi></mtd> <mtd><munderover><mrow></mrow><mo>⟶</mo><mo>⟶</mo></munderover></mtd> <mtd><mi>Γ</mi><mo>⊗</mo><mi>Γ</mi></mtd> <mtd><mover><mo>⟶</mo><mi>coeq</mi></mover></mtd> <mtd><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>Γ</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><mi>c</mi><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></mrow></mpadded></msub><mo stretchy="false">↓</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mi>β</mi></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>Γ</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ \Gamma \otimes A \otimes \Gamma & \underoverset {\longrightarrow} {\longrightarrow} {} & \Gamma \otimes \Gamma & \overset{coeq}{\longrightarrow} & \Gamma \otimes_A \Gamma \\ && {}_{\mathllap{c(-)\cdot (-)}}\downarrow & \swarrow_{\mathrlap{\beta}} \\ && \Gamma } </annotation></semantics></math></div> <p>satisfy</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>α</mi><mo>∘</mo><mi>Ψ</mi><mo>=</mo><msub><mi>η</mi> <mi>L</mi></msub><mo>∘</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\alpha \circ \Psi = \eta_L \circ \epsilon </annotation></semantics></math></p> <p>and</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi><mo>∘</mo><mi>Ψ</mi><mo>=</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>∘</mo><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\beta \circ \Psi = \eta_R \circ \epsilon </annotation></semantics></math>.</p> </li> </ol> </li> </ol> </div> <p>e.g. (<a href="commutative+Hopf+algebroid#Ravenel86">Ravenel 86, def. A1.1.1</a>)</p> <p>By internalizing all of the above from <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> 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>, we obtain the concept of <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a>:</p> <div class="num_defn" id="Supergroup"> <h6 id="definition_28">Definition</h6> <p>An affine algebraic <em><a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a></em> <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> is equivalently</p> <ul> <li> <p>a pointed, one-object <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> in the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>=</mo><mi>CMon</mi><mo stretchy="false">(</mo><msub><mi>sVect</mi> <mi>k</mi></msub><msup><mo stretchy="false">)</mo> <mi>op</mi></msup></mrow><annotation encoding="application/x-tex">Aff(sVect) = CMon(sVect_k)^{op}</annotation></semantics></math> (def. <a class="maruku-ref" href="#Affines"></a>) of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> from def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a></p> </li> <li> <p>the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of a <strong><a class="existingWikiWord" href="/nlab/show/super-commutative+Hopf+algebra">super-commutative Hopf algebra</a></strong>, namely a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebra">commutative Hopf algebra</a> (prop. <a class="maruku-ref" href="#CommutativeHopfAlgebroidSpelledOut"></a>, remark <a class="maruku-ref" href="#HopfAlgebrasAsHopfAlgebroids"></a>).</p> </li> </ul> </div> <p>We will often just say “supergroup” for short in the following. If <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> is the corresponding <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebra">supercommutative Hopf algebra</a> then we also write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(H)</annotation></semantics></math> for this supergroup.</p> <p>The following asks that the parity involution (def. <a class="maruku-ref" href="#ParityAutomorphism"></a>) on a supergroup is an <a class="existingWikiWord" href="/nlab/show/inner+automorphism">inner automorphism</a>:</p> <div class="num_defn" id="InnerParity"> <h6 id="definition_29">Definition</h6> <p>An <strong>inner parity</strong> of a <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a> <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>, def. <a class="maruku-ref" href="#Supergroup"></a> is an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><msub><mi>G</mi> <mi>even</mi></msub></mrow><annotation encoding="application/x-tex">\epsilon \in G_{even}</annotation></semantics></math> such that</p> <ol> <li> <p>it is <a class="existingWikiWord" href="/nlab/show/involution">involutive</a> i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mn>2</mn></msup><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\epsilon^2 = 1</annotation></semantics></math></p> </li> <li> <p>its <a class="existingWikiWord" href="/nlab/show/adjoint+action">adjoint action</a> on <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> is the parity involution of def. <a class="maruku-ref" href="#ParityAutomorphism"></a>.</p> </li> </ol> <p>Dually this mean that an inner pariy is an algebra homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ϵ</mi> <mo>*</mo></msup><mo lspace="verythinmathspace">:</mo><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">\epsilon^\ast \colon\mathcal{O}(G) \to k</annotation></semantics></math> such that</p> <ol> <li> <p>the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mi>Ψ</mi></mover><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>ϵ</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>k</mi></msub><msup><mi>ϵ</mi> <mo>*</mo></msup></mrow></mover><mi>k</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>k</mi><mo>≃</mo><mi>k</mi></mrow><annotation encoding="application/x-tex"> \mathcal{O}(G) \stackrel{\Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k \epsilon^\ast}{\longrightarrow} k \otimes_k k \simeq k </annotation></semantics></math></div> <p>is the counit of the Hopf algebra (hence the formal dual of the neutral element)</p> </li> <li> <p>the parity involution <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mo>≃</mo></mover><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(G) \stackrel{\simeq}{\longrightarrow} \mathcal{O}(G)</annotation></semantics></math> conincides with the composite</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mi>id</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>Ψ</mi><mo stretchy="false">)</mo><mo>∘</mo><mi>Ψ</mi></mrow></mover><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mover><mo>⟶</mo><mrow><msup><mi>ϵ</mi> <mo>*</mo></msup><msub><mo>⊗</mo> <mi>k</mi></msub><mi>id</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>c</mi><mo>∘</mo><msup><mi>ϵ</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow></mover><mi>k</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><msub><mo>⊗</mo> <mi>k</mi></msub><mi>k</mi></mrow><annotation encoding="application/x-tex"> \mathcal{O}(G) \stackrel{(id \otimes_k \Psi) \circ \Psi}{\longrightarrow} \mathcal{O}(G) \otimes_k \mathcal{O}(G) \otimes_k \mathcal{O}(G) \stackrel{\epsilon^\ast \otimes_k id \otimes_k (c \circ \epsilon^\ast)}{\longrightarrow} k \otimes_k \mathcal{O}(G) \otimes_k k </annotation></semantics></math></div></li> </ol> </div> <p>(<a href="#Deligne02">Deligne 02, 0.3</a>)</p> <div class="num_example" id="InnerParityInBosonicSupergroup"> <h6 id="example_24">Example</h6> <p>For <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> an ordinary affine <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, regarded as a supergroup with trivial odd-graded part, then every element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∈</mo><mi>Z</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\epsilon \in Z(G)</annotation></semantics></math> in the <a class="existingWikiWord" href="/nlab/show/center">center</a> defines an inner parity, def. <a class="maruku-ref" href="#InnerParity"></a>.</p> </div> <p>(<a href="#Deligne02">Deligne 02, 0.4 i)</a>)</p> <div class="num_remark"> <h6 id="remark_9">Remark</h6> <p>In view of remak <a class="maruku-ref" href="#InnerParityInBosonicSupergroup"></a>, specifying an involutive central element in an ordinary group is a faint shadow of genuine supergroup structure. In fact such pairs are being referred to as “supergroups” in (<a href="#Mueger06">Müger 06</a>).</p> </div> <p>Demanding the existence of inner parity is not actually a restriction of the theory:</p> <div class="num_example"> <h6 id="example_25">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> any supergroup, def. <a class="maruku-ref" href="#Supergroup"></a>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>=</mo><mo stretchy="false">{</mo><mi>id</mi><mo>,</mo><mi>par</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 = \{id,par\}</annotation></semantics></math> acting on it by parity involution, def. <a class="maruku-ref" href="#ParityAutomorphism"></a> then the <a class="existingWikiWord" href="/nlab/show/semidirect+product+group">semidirect product group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⋉</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbb{Z}_2 \ltimes G</annotation></semantics></math> has inner parity, def. <a class="maruku-ref" href="#InnerParity"></a>, given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>≔</mo><mi>par</mi><mo>∈</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>↪</mo><msub><mi>ℤ</mi> <mn>2</mn></msub><mo>⋉</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\epsilon \coloneqq par \in \mathbb{Z}_2 \hookrightarrow \mathbb{Z}_2 \ltimes G</annotation></semantics></math>.</p> </div> <p>(<a href="#Deligne02">Deligne 02, 0.4 ii)</a>)</p> <h3 id="LinearRepresentationsAsComodules">Linear super-representations as Comodules</h3> <div class="num_defn" id="CommutativeHopfAlgebroidComodule"> <h6 id="definition_30">Definition</h6> <p>Given a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (def. <a class="maruku-ref" href="#CommutativeHopfAlgebroidDefinitionInExplicitComponents"></a>) in some <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>), then a <strong>left <a class="existingWikiWord" href="/nlab/show/comodule">comodule</a></strong> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> is</p> <ol> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module+object">module object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>) i;</p> </li> <li> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/module">module</a> <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> (co-action)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ψ</mi> <mi>N</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>N</mi><mo>⟶</mo><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>N</mi></mrow><annotation encoding="application/x-tex">\Psi_N \;\colon\; N \longrightarrow \Gamma \otimes_A N</annotation></semantics></math>;</p> </li> </ol> <p>such that</p> <ol> <li> <p>(co-<a class="existingWikiWord" href="/nlab/show/unitality">unitality</a>)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>ϵ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>id</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Ψ</mi> <mi>N</mi></msub><mo>=</mo><msub><mi>id</mi> <mi>N</mi></msub></mrow><annotation encoding="application/x-tex">(\epsilon \otimes_A id_N) \circ \Psi_N = id_N</annotation></semantics></math>;</p> </li> <li> <p>(co-action property)</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ψ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>id</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Ψ</mi> <mi>N</mi></msub><mo>=</mo><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>Γ</mi></msub><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>Ψ</mi> <mi>N</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>Ψ</mi> <mi>N</mi></msub></mrow><annotation encoding="application/x-tex">(\Psi \otimes_A id_N) \circ \Psi_N = (id_\Gamma \otimes_A \Psi_N)\circ \Psi_N</annotation></semantics></math>.</p> </li> </ol> <p>A <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> between comodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_1 \to N_2</annotation></semantics></math> is a homomorphism of underlying <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-modules making <a class="existingWikiWord" href="/nlab/show/commuting+diagrams">commuting diagrams</a> with the co-action morphism. Write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mi>CoMod</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \Gamma CoMod(\mathcal{A}) </annotation></semantics></math></div> <p>for the resulting <a class="existingWikiWord" href="/nlab/show/category">category</a> of (left) comodules over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>. Analogously there are right comodules.</p> </div> <div class="num_example" id="ComoduleStructureOnGroundRing"> <h6 id="example_26">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Γ</mi><mo>,</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Gamma,A)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroid">commutative Hopf algebroid</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> becomes a left <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math>-comodule (def. <a class="maruku-ref" href="#CommutativeHopfAlgebroidComodule"></a>) with coaction given by the right unit</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></mover><mi>Γ</mi><mo>≃</mo><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \,. </annotation></semantics></math></div></div> <div class="proof"> <h6 id="proof_12">Proof</h6> <p>The required co-unitality property is the dual condition in def. <a class="maruku-ref" href="#CommutativeHopfAlgebroidDefinitionInExplicitComponents"></a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>=</mo><msub><mi>id</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex"> \epsilon \circ \eta_R = id_A </annotation></semantics></math></div> <p>of the fact in def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a> that identity morphisms respect sources:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>id</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>A</mi><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></mover><mi>Γ</mi><mo>≃</mo><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi><mover><mo>⟶</mo><mrow><mi>ϵ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>id</mi></mrow></mover><mi>A</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi><mo>≃</mo><mi>A</mi></mrow><annotation encoding="application/x-tex"> id \;\colon\; A \overset{\eta_R}{\longrightarrow} \Gamma \simeq \Gamma \otimes_A A \overset{\epsilon \otimes_A id}{\longrightarrow} A \otimes_A A \simeq A </annotation></semantics></math></div> <p>The required co-action property is the dual condition</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Ψ</mi><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub><mo>=</mo><mo stretchy="false">(</mo><mi>id</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>η</mi> <mi>R</mi></msub><mo stretchy="false">)</mo><mo>∘</mo><msub><mi>η</mi> <mi>R</mi></msub></mrow><annotation encoding="application/x-tex"> \Psi \circ \eta_R = (id \otimes_A \eta_R) \circ \eta_R </annotation></semantics></math></div> <p>of the fact in def. <a class="maruku-ref" href="#CommutativeHopfAlgebroid"></a> that composition of morphisms in a groupoid respects sources</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>A</mi></mtd> <mtd><mover><mo>⟶</mo><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></mover></mtd> <mtd><mi>Γ</mi></mtd></mtr> <mtr><mtd><msup><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mi>η</mi> <mi>R</mi></msub></mrow></mpadded></msup><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>Ψ</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Γ</mi><mo>≃</mo><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>A</mi></mtd> <mtd><munder><mo>⟶</mo><mrow><mi>id</mi><msub><mo>⊗</mo> <mi>A</mi></msub><msub><mi>η</mi> <mi>R</mi></msub></mrow></munder></mtd> <mtd><mi>Γ</mi><msub><mo>⊗</mo> <mi>A</mi></msub><mi>Γ</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ A &\overset{\eta_R}{\longrightarrow}& \Gamma \\ {}^{\mathllap{\eta_R}}\downarrow && \downarrow^{\mathrlap{\Psi}} \\ \Gamma \simeq \Gamma \otimes_A A &\underset{id \otimes_A \eta_R}{\longrightarrow}& \Gamma \otimes_A \Gamma } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="TensorProductOfComodulesOverAHopfAlgebra"> <h6 id="definition_31">Definition</h6> <p>Given two comodules <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_1, N_2</annotation></semantics></math> over a <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebra">commutative Hopf algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi></mrow><annotation encoding="application/x-tex">\Gamma</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, then their <strong><a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a></strong> is the the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+modules">tensor product of modules</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">N_1 \otimes_k N_2</annotation></semantics></math> equipped with the following co-action</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>2</mn></msub><mover><mo lspace="0em" rspace="thinmathspace">longightarrow</mo><mrow><mi>Ψ</mi><mn>1</mn><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>Ψ</mi> <mn>2</mn></msub></mrow></mover><mi>Γ</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><mi>Γ</mi><mo>⊗</mo><msub><mi>N</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow></mrow></mover><mi>Γ</mi><msub><mo>⊗</mo> <mi>k</mi></msub><mi>Γ</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><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><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>id</mi> <mrow><msub><mi>N</mi> <mn>1</mn></msub></mrow></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>id</mi> <mrow><msub><mi>N</mi> <mn>2</mn></msub></mrow></msub></mrow></mover><mi>Γ</mi><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>1</mn></msub><msub><mo>⊗</mo> <mi>k</mi></msub><msub><mi>N</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> N_1 \otimes_k N_2 \overset{\Psi1 \otimes_k \Psi_2}{\longightarrow} \Gamma \otimes_k N_1 \otimes_k \Gamma \otimes N_2 \overset{}{\longrightarrow} \Gamma \otimes_k \Gamma \otimes_k N_1 \otimes_k N_2 \overset{((-)\cdot (-)) \otimes_k id_{N_1} \otimes_k id_{N_2} }{\longrightarrow} \Gamma \otimes_k N_1 \otimes_k N_2 \,. </annotation></semantics></math></div></div> <p>This is the <a class="existingWikiWord" href="/nlab/show/formal+dual">formal dual</a> of the <a class="existingWikiWord" href="/nlab/show/tensor+product+of+representations">tensor product of representations</a>, the action on which is induced by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>×</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>Δ</mi> <mi>G</mi></msub><mo>×</mo><mi>id</mi></mrow></mover><mi>G</mi><mo>×</mo><mi>G</mi><mo>×</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub><mo>≃</mo><mi>G</mi><mo>×</mo><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><mi>G</mi><mo>×</mo><msub><mi>V</mi> <mn>1</mn></msub><mover><mo>⟶</mo><mrow><msub><mi>ρ</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>ρ</mi> <mn>2</mn></msub></mrow></mover><msub><mi>V</mi> <mn>1</mn></msub><mo>×</mo><msub><mi>V</mi> <mn>2</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> G \times V_1 \times V_2 \overset{\Delta_G \times id}{\longrightarrow} G \times G \times V_1 \times V_2 \simeq G \times V_1 \times G \times V_1 \overset{\rho_1 \times \rho_2}{\longrightarrow} V_1 \times V_2 \,. </annotation></semantics></math></div> <p>Under the tensor product of co-modules (def. <a class="maruku-ref" href="#TensorProductOfComodulesOverAHopfAlgebra"></a>), these form a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal category</a> (def. <a class="maruku-ref" href="#SymmetricMonoidalCategory"></a>).</p> <div class="num_defn" id="Superrepresentation"> <h6 id="definition_32">Definition</h6> <p>A <em><a class="existingWikiWord" href="/nlab/show/linear+representation">linear representation</a></em> of a <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a> <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>, def. <a class="maruku-ref" href="#Supergroup"></a>, with inner parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math>, def. <a class="maruku-ref" href="#InnerParity"></a>, is</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/comodule">comodule</a> <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> (def. <a class="maruku-ref" href="#CommutativeHopfAlgebroidComodule"></a>) in <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>) for the <a class="existingWikiWord" href="/nlab/show/formal+dual">formally dual</a> <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebra">commutative Hopf algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(G)</annotation></semantics></math>.</li> </ul> <p>such that</p> <ul> <li>the innr parity element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> acts as the identity on <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> and by multiplicatin with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math> on <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>.</li> </ul> </div> <div class="num_example"> <h6 id="example_27">Example</h6> <p>For <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> an ordinary (affine algebraic) group regarded as a supergroup with trivial odd-graded part, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi><mo>=</mo><mi>e</mi></mrow><annotation encoding="application/x-tex">\epsilon = e </annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/neutral+element">neutral element</a> taken as the inner parity, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(G,\epsilon)</annotation></semantics></math> in the sense of def. <a class="maruku-ref" href="#Superrepresentation"></a> is just the ordinary <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> 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>.</p> </div> <p>(<a href="#Deligne02">Deligne 02, 0.4 i)</a>)</p> <div class="num_prop" id="RegularTensorCategoriesOfSuperrepresentations"> <h6 id="proposition_18">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Rep(G,\epsilon)</annotation></semantics></math> of def. <a class="maruku-ref" href="#Superrepresentation"></a> of an affine algebraic <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a> <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>, def. <a class="maruku-ref" href="#Supergroup"></a>, with inner parity <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> (def. <a class="maruku-ref" href="#InnerParity"></a>) on finite-dimensional super vector spaces (example <a class="maruku-ref" href="#FiniteDimensionalSuperVectorSpaces"></a>) and equippd with the tensor product of comodules from def. <a class="maruku-ref" href="#TensorProductOfComodulesOverAHopfAlgebra"></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/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>) of subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>).</p> </div> <p>(<a href="#Deligne02">Deligne 02, 1.21</a>)</p> <p>Moreover, any finite dimensional <a class="existingWikiWord" href="/nlab/show/faithful+representation">faithful representation</a> (which always exists, <a href="faithful+representation#AlgebraicGroupHasFinDimFaithfulRepresentations">prop.</a>) serves as an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>⊗</mo></mrow><annotation encoding="application/x-tex">\otimes</annotation></semantics></math>-generator (def. <a class="maruku-ref" href="#FiniteTensorGeneration"></a>).</p> <p>See (<a href="faithful+representation#AnyFinDimRepOfAffineAlgebraicGroupOverFieldIsSubquotientsOfFaithfulRep">this prop.</a>).</p> <h3 id="FiberFunctors">Super Fiber functors and their automorphism supergroups</h3> <p>The first step in exhibiting a given <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> as being a <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> is to exhibit its objects as having an <a class="existingWikiWord" href="/nlab/show/forgetful+functor">underlying</a> representation space of sorts, and then an <a class="existingWikiWord" href="/nlab/show/action">action</a> represented on that space. Hence a necessary condition on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is that there exists a <a class="existingWikiWord" href="/nlab/show/forgetful+functor">forgetful functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; \mathcal{A} \longrightarrow \mathcal{V} </annotation></semantics></math></div> <p>to some other <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>, such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> satisfies a list of properties, in particular it should be a <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+functor">symmetric</a> <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a>.</p> <p>Such functors are called <em><a class="existingWikiWord" href="/nlab/show/fiber+functors">fiber functors</a></em>. The idea is that we think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/bundle">bundle</a> over <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">\mathcal{V}</annotation></semantics></math>, and over each <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mi>𝒱</mi></mrow><annotation encoding="application/x-tex">V \in \mathcal{V}</annotation></semantics></math> we find the <a class="existingWikiWord" href="/nlab/show/fiber">fiber</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ω</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\omega^{-1}(V)</annotation></semantics></math> of that bundle, consisting of all those objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> whose underlying object in the given <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>.</p> <p>The main point of <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> of tensor categories is the observation that if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> of some <a class="existingWikiWord" href="/nlab/show/group">group</a> <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>, then <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> also <a class="existingWikiWord" href="/nlab/show/action">acts</a> by <a class="existingWikiWord" href="/nlab/show/automorphisms">automorphisms</a> on that <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> (i.e. via <a class="existingWikiWord" href="/nlab/show/natural+isomorphisms">natural isomorphisms</a> of functors). In good cases then this may be turned around, and the full <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a> of a fiber functor is identified with the group <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> for which the objects in its fibers are <a class="existingWikiWord" href="/nlab/show/representations">representations</a>, this is the process of <a class="existingWikiWord" href="/nlab/show/Tannaka+reconstruction">Tannaka reconstruction</a>.</p> <p>There are slight variants on what one requires of a fiber functor. For the present purpose we fix the following definition</p> <div class="num_defn" id="FiberFunctor"> <h6 id="definition_33">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> be 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>-<a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>) such that</p> <ol> <li> <p>all <a class="existingWikiWord" href="/nlab/show/object+of+finite+length">objects have finite length</a>;</p> </li> <li> <p>all <a class="existingWikiWord" href="/nlab/show/hom+spaces">hom spaces</a> are of <a class="existingWikiWord" href="/nlab/show/finite+number">finite</a> <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>.</p> </li> </ol> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒯</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">R \in CMon(Ind(\mathcal{T}))</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/commutative+monoid+in+a+symmetric+monoidal+category">commutative monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>) in the category of <a class="existingWikiWord" href="/nlab/show/ind-objects">ind-objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi></mrow><annotation encoding="application/x-tex">\mathcal{T}</annotation></semantics></math> (prop. <a class="maruku-ref" href="#IndObjectsInATensorCategory"></a>).</p> <p>Then a <strong><a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> over <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></strong> is 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><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒯</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; \mathcal{A} \longrightarrow R Mod(Ind(\mathcal{T})) </annotation></semantics></math></div> <p>from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> to the <a class="existingWikiWord" href="/nlab/show/category+of+modules">category of</a> <a class="existingWikiWord" href="/nlab/show/module+objects">module objects</a> over <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> (def. <a class="maruku-ref" href="#ModulesInMonoidalCategory"></a>) in the <a class="existingWikiWord" href="/nlab/show/category+of+ind-objects">category of ind-objects</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Ind</mi><mo stretchy="false">(</mo><mi>𝒯</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ind(\mathcal{T})</annotation></semantics></math> (def. <a class="maruku-ref" href="#IndObjectsInATensorCategory"></a>), which is</p> <ol> <li> <p>a <a class="existingWikiWord" href="/nlab/show/braided+monoidal+functor">braided</a> <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a>;</p> </li> <li> <p>an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a> in both variables.</p> </li> </ol> <p>If here <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒯</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{T} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sFinDimVect">sFinDimVect</a> (def. <a class="maruku-ref" href="#FiniteDimensionalSuperVectorSpaces"></a>), then this is called a <strong>super fiber functor</strong>.</p> </div> <p>(<a href="#Deligne02">Deligne 02, 3.1</a>)</p> <div class="num_defn" id="TannakianCategory"> <h6 id="definition_34">Definition</h6> <p>A <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) is called</p> <ol> <li> <p>a <strong>neutral <a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a></strong> if it admits a fiber functor (def. <a class="maruku-ref" href="#FiberFunctor"></a>) to <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> (example <a class="maruku-ref" href="#VectAsAMonoidalCategory"></a>) (<a href="#DeligneMilne12">Deligne-Milne 12, def. 2.19</a>)</p> </li> <li> <p>a <strong>neutral super <a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a></strong> if it admits a fiber functor (def. <a class="maruku-ref" href="#FiberFunctor"></a>) to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>sVect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">sVect_k</annotation></semantics></math> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>)</p> </li> <li> <p>(not needed here) a general <strong><a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a></strong> if the <a class="existingWikiWord" href="/nlab/show/stack">stack</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Aff</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">Aff_k</annotation></semantics></math> which sends <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>∈</mo><mi>CRing</mi></mrow><annotation encoding="application/x-tex">R \in CRing</annotation></semantics></math> to the groupoid of fiber functors to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi><mi>Proj</mi><mo>↪</mo><mi>R</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">R Proj \hookrightarrow R Mod</annotation></semantics></math> (<a class="existingWikiWord" href="/nlab/show/projective+modules">projective modules</a> over <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>) is an affine <a class="existingWikiWord" href="/nlab/show/gerbe">gerbe</a> such that its category of representations is equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (<a href="#DeligneMilne12">Deligne-Milne 12, def. 3.7</a>).</p> </li> </ol> </div> <p>Given a super fiber functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><msub><mi>sVect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex">\omega \colon \mathcal{A} \to sVect_k</annotation></semantics></math> (def. <a class="maruku-ref" href="#FiberFunctor"></a>) there is an evident notion of its <a class="existingWikiWord" href="/nlab/show/automorphism+group">automorphism group</a>: a <a class="existingWikiWord" href="/nlab/show/homomorphism">homomorphism</a> between <a class="existingWikiWord" href="/nlab/show/functors">functors</a> is a <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>, and that between <a class="existingWikiWord" href="/nlab/show/monoidal+functors">monoidal functors</a> is a <a class="existingWikiWord" href="/nlab/show/monoidal+natural+transformation">monoidal natural transformation</a>, according to def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>, and this is an <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> of functors if it is a <a class="existingWikiWord" href="/nlab/show/natural+automorphism">natural automorphism</a>. We write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> Aut(\omega) \in Grp </annotation></semantics></math></div> <p>for this automorphism group.</p> <p>So far this is a group without geometric structure (a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>). But it is naturally equipped with <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a> (super-<a class="existingWikiWord" href="/nlab/show/algebraic+geometry">algebraic geometry</a>) exhibited by a rule for what the geometrically parameterized families of its elements are. (For exposition of this perspective see at <em><a class="existingWikiWord" href="/nlab/show/motivation+for+sheaves%2C+cohomology+and+higher+stacks">motivation for sheaves, cohomology and higher stacks</a></em>).</p> <p>Concretely, this means that for each <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> with corresponding affine <a class="existingWikiWord" href="/nlab/show/super+scheme">super scheme</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Affines"></a>, def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a>) we are to say what the set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(A)) \in Set </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math>-parameterized elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(\omega)</annotation></semantics></math> is. In fact, under parameter-wise multiplication in the group, any such set must inherit group structure, so that we should have not one discrete group, but a system of them, labeled by supercommutative superalgebras:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>∈</mo><mi>Grp</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(A)) \in Grp \,. </annotation></semantics></math></div> <p>Moreover, if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>A</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">A_1 \longrightarrow A_2</annotation></semantics></math> is an algebra homomorphism, hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> Spec(A_2) \longrightarrow Spec(A_1) </annotation></semantics></math></div> <p>a map of affine super schemes according to def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a>, then there should be a group homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>⟵</mo><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(A_2)) \longleftarrow \underline{Aut}(\omega)(Spec(A_1)) </annotation></semantics></math></div> <p>that expresses how a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_1)</annotation></semantics></math>-parameterized family of elements of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(\omega)</annotation></semantics></math> becomes a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>A</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A_1)</annotation></semantics></math>-parameterized family, under this map.</p> <p>For a minimum of consistency, this assignment must be such that the identity map on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(A)</annotation></semantics></math> induces the identity on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{Aut}(\omega)(Spec(A))</annotation></semantics></math>, and that the composite of two maps of affine superschemes goes to the correspondng composite group homomorphisms.</p> <p>In conclusion, this says that an algebraic supergeometric structure on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(\omega)</annotation></semantics></math> is the datum of a <a class="existingWikiWord" href="/nlab/show/presheaf">presheaf</a> of groups, hence of 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><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>≃</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \;\colon\; Aff(sVect)^{op} \simeq CMon(sVect) \longrightarrow Grp </annotation></semantics></math></div> <p>such that the underlying points are those of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Aut(\omega)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>Aut</mi><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(k)) \simeq Aut(\omega) \,. </annotation></semantics></math></div> <div class="num_defn" id="RepresentableAutomorphismGroup"> <h6 id="definition_35">Definition</h6> <p>We say that a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><msup><mo stretchy="false">)</mo> <mi>op</mi></msup><mo>=</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>Grp</mi></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \;\colon\; Aff(sVect)^{op} = CMon(sVect) \longrightarrow Grp </annotation></semantics></math></div> <p>is <strong><a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a></strong> if there exists a <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebra">supercommutative Hopf algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math>, hence an affine <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(H)</annotation></semantics></math> (def. <a class="maruku-ref" href="#Supergroup"></a>) and a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a> with the <a class="existingWikiWord" href="/nlab/show/hom+functor">hom functor</a> into <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Spec</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Spec(H)</annotation></semantics></math>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>H</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>H</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H,-) = Hom_{Aff(sVect)}(-,Spec(H)) \,. </annotation></semantics></math></div></div> <div class="num_defn" id="AutomorphismGroupOfFiberFunctor"> <h6 id="definition_36">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>ℬ</mi></mrow><annotation encoding="application/x-tex">\omega \colon \mathcal{A} \to \mathcal{B}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> (def <a class="maruku-ref" href="#FiberFunctor"></a>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in CMon(\mathcal{B})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mover><mo>⟶</mo><mi>ω</mi></mover><mi>ℬ</mi><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} \mathcal{B} \stackrel{A \otimes(-)}{\longrightarrow} A Mod(\mathcal{B}) </annotation></semantics></math></div> <p>for its image under <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">1 \to A</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (prop. <a class="maruku-ref" href="#ExtensionOfScalars"></a>).</p> <p>With this, the <strong>automorphism group</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Aff</mi><mo stretchy="false">(</mo><mi>ℬ</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \in PSh(Aff(\mathcal{B})) </annotation></semantics></math></div> <p>is defined to be the functor which on objects assigns the <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> of <a class="existingWikiWord" href="/nlab/show/natural+automorphisms">natural automorphisms</a> of the image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\omega_A</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> under <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> as above</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≔</mo><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(A)) \coloneqq Aut(\omega_{A}) </annotation></semantics></math></div> <p>and which to a homomorphism of algebras</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><msub><mi>A</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>A</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \phi \;\colon\; A_1 \longrightarrow A_2 </annotation></semantics></math></div> <p>assigns the action of the <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a>-functor along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϕ</mi></mrow><annotation encoding="application/x-tex">\phi</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><mi>ϕ</mi> <mo>*</mo></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mrow><msub><mi>A</mi> <mn>1</mn></msub></mrow></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mrow><msub><mi>A</mi> <mn>2</mn></msub></mrow></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \phi_\ast \;\colon\; Aut(\omega_{A_1}) \longrightarrow Aut(\omega_{A_2}) \,. </annotation></semantics></math></div> <p>This is clearly a presheaf, by <a class="existingWikiWord" href="/nlab/show/functor">functoriality</a> of <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a>.</p> </div> <p>Specializing def. <a class="maruku-ref" href="#AutomorphismGroupOfFiberFunctor"></a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ℬ</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathcal{B} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>), where a commutative monoid is a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> (def. <a class="maruku-ref" href="#SupercommutativeSuperalgebra"></a>) it reads as follows:</p> <div class="num_example" id="AutomorphismSuperGroupOfSuperFiberFunctor"> <h6 id="example_28">Example</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>sVect</mi></mrow><annotation encoding="application/x-tex">\omega \colon \mathcal{A} \to sVect</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> (def <a class="maruku-ref" href="#FiberFunctor"></a>).</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in CMon(sVect)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a> (def. <a class="maruku-ref" href="#MonoidsInMonoidalCategory"></a>), write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mover><mo>⟶</mo><mi>ω</mi></mover><mi>sVect</mi><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect) </annotation></semantics></math></div> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in CMon(sVect)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/supercommutative+algebra">supercommutative algebra</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>A</mi></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mover><mo>⟶</mo><mi>ω</mi></mover><mi>sVect</mi><mover><mo>⟶</mo><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega_A \;\colon\; \mathcal{A} \stackrel{\omega}{\longrightarrow} sVect \stackrel{A \otimes(-)}{\longrightarrow} A Mod(sVect) </annotation></semantics></math></div> <p>for its image under <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> (prop. <a class="maruku-ref" href="#MonoidModuleOverItself"></a>).</p> <p>With this, the <strong>automorphism super-group</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>PSh</mi><mo stretchy="false">(</mo><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \in PSh(Aff(sVect)) </annotation></semantics></math></div> <p>is defined by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>Spec</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≔</mo><mi>Aut</mi><mo stretchy="false">(</mo><msub><mi>ω</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega)(Spec(A)) \coloneqq Aut(\omega_{A}) \,. </annotation></semantics></math></div></div> <div class="num_prop" id="AutomorphismSupergroupOfFiberFunctorIsRepresentable"> <h6 id="proposition_19">Proposition</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> an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a>, and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> 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/tensor+category">tensor category</a> equipped with a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>, then its automorphism supergroup (def. <a class="maruku-ref" href="#AutomorphismSuperGroupOfSuperFiberFunctor"></a>) is <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> (def. <a class="maruku-ref" href="#RepresentableAutomorphismGroup"></a>): there exists a <a class="existingWikiWord" href="/nlab/show/supercommutative+Hopf+algebra">supercommutative Hopf algebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>H</mi> <mi>ω</mi></msub></mrow><annotation encoding="application/x-tex">H_\omega</annotation></semantics></math> and 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><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Hom</mi> <mrow><mi>CMon</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>ω</mi></msub><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo>=</mo><msub><mi>Hom</mi> <mrow><mi>Aff</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>ω</mi></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \simeq Hom_{CMon(sVect)}(H_\omega,-) = Hom_{Aff(sVect)}(-, Spec(H_\omega)) \,, </annotation></semantics></math></div> <p>which, with the <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> understood, we write simply as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>Spec</mi><mo stretchy="false">(</mo><msub><mi>H</mi> <mi>ω</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \simeq Spec(H_\omega) \,. </annotation></semantics></math></div></div> <p>(<a href="#Deligne90">Deligne 90, prop. 8.11</a>)</p> <p>The following says that in fact all homomorphisms between fiber functors are necessarily <a class="existingWikiWord" href="/nlab/show/isomorphisms">isomorphisms</a>:</p> <div class="num_lemma" id="MonoidalTransformationBetweenFiberFunctorIsIso"> <h6 id="lemma_2">Lemma</h6> <p>Every <a class="existingWikiWord" href="/nlab/show/monoidal+natural+transformation">monoidal natural transformation</a> (def. <a class="maruku-ref" href="#LaxMonoidalFunctor"></a>) between two <a class="existingWikiWord" href="/nlab/show/fiber+functors">fiber functors</a> (def. <a class="maruku-ref" href="#FiberFunctor"></a>) is an <a class="existingWikiWord" href="/nlab/show/isomorphism">isomorphism</a> (i.e. a <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">natural isomorphism</a>).</p> </div> <p>(<a href="#Deligne90">Deligne 90, 8.11 (ii)</a>, <a href="#Deligne02">Deligne 02, lemma 3.2</a>)</p> <div class="num_defn" id="FundamentalSupergroup"> <h6 id="definition_37">Definition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a> and regard the <a class="existingWikiWord" href="/nlab/show/identity">identity</a> functor on it as a fiber functor (def. <a class="maruku-ref" href="#FiberFunctor"></a>). Then the automorphism group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>id</mi> <mi>𝒜</mi></msub></mrow><annotation encoding="application/x-tex">id_{\mathcal{A}}</annotation></semantics></math> according to def. <a class="maruku-ref" href="#AutomorphismSuperGroupOfSuperFiberFunctor"></a> is called the <strong>fundamental group</strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, denoted:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><msub><mi>id</mi> <mi>𝒜</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi(\mathcal{A}) \coloneqq \underline{Aut}(id_{\mathcal{A}}) </annotation></semantics></math></div></div> <p>(<a href="#Deligne90">Deligne 90, 8.12, 8.13</a>)</p> <div class="num_example" id="FundamentalGroupOfCategoryOfSuperVectorSpaces"> <h6 id="example_29">Example</h6> <p>The fundamental group (def. <a class="maruku-ref" href="#FundamentalSupergroup"></a>) of the <a class="existingWikiWord" href="/nlab/show/category+of+super+vector+spaces">category of super vector spaces</a> <a class="existingWikiWord" href="/nlab/show/sVect">sVect</a> (def. <a class="maruku-ref" href="#CategoryOfSuperVectorSpaces"></a>) is <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>sVect</mi><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(sVect) \simeq \mathbb{Z}/2 \,. </annotation></semantics></math></div> <p>The non-trivial element in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi(sVect)</annotation></semantics></math> acts on any super-vector space as the <a class="existingWikiWord" href="/nlab/show/endomorphism">endomorphism</a> which is the identity on even graded elements, and multiplication by <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 stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(-1)</annotation></semantics></math> on odd graded elements.</p> </div> <p>(<a href="#Deligne90">Deligne 90, 8.14 iv)</a>)</p> <div class="num_prop" id="AutOfFibFuncIsImagUnderFibFuncOfFundamentalGroup"> <h6 id="proposition_20">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> 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/tensor+category">tensor category</a> equipped with a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>sVect</mi></mrow><annotation encoding="application/x-tex">\omega \colon \mathcal{A} \to sVect</annotation></semantics></math> (def. <a class="maruku-ref" href="#FiberFunctor"></a>), then the automorphism supergroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is the image under the super fiber functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> of the fundamental group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, according to def. <a class="maruku-ref" href="#FundamentalSupergroup"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \underline{Aut}(\omega) \simeq \omega(\pi(\mathcal{A})) \,. </annotation></semantics></math></div> <p>Here on the right we are using that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">strong monoidal functor</a> so that it preserves <a class="existingWikiWord" href="/nlab/show/commutative+monoids">commutative monoids</a> as well as <a class="existingWikiWord" href="/nlab/show/comonoids">comonoids</a> by prop. <a class="maruku-ref" href="#MonoidsPreservedByLaxMonoidalFunctor"></a>, hence preserves <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebras">commutative Hopf algebras</a>.</p> </div> <p>(<a href="#Deligne90">Deligne 90 (8.13.1)</a>)</p> <div class="num_prop" id="HomomorphismFromPiT2ToEtaPiT1"> <h6 id="proposition_21">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒜</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>𝒜</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_1, \mathcal{A}_2</annotation></semantics></math> be 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>-<a class="existingWikiWord" href="/nlab/show/tensor+categories">tensor categories</a> and let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>η</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><msub><mi>𝒜</mi> <mn>1</mn></msub><mo>⟶</mo><msub><mi>𝒜</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex"> \eta \;\colon\; \mathcal{A}_1 \longrightarrow \mathcal{A}_2 </annotation></semantics></math></div> <p>be a <a class="existingWikiWord" href="/nlab/show/functor">functor</a> which is <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, <a class="existingWikiWord" href="/nlab/show/strong+monoidal+functor">monoidal</a> and <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>. Then there is induced a canonical group homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><msub><mi>𝒜</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>⟶</mo><mi>η</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><msub><mi>𝒜</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi(\mathcal{A}_2) \longrightarrow \eta(\pi(\mathcal{A}_1)) </annotation></semantics></math></div> <p>from the fundamental group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒜</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_1</annotation></semantics></math> (def. <a class="maruku-ref" href="#FundamentalSupergroup"></a>) to the image under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> of the fundamental group of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒜</mi> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\mathcal{A}_2</annotation></semantics></math>.</p> </div> <p>(<a href="#Deligne90">Deligne 90, 8. 15. 2</a>)</p> <h3 id="superexterior_powers_and_schur_functors">Super-exterior powers and Schur functors</h3> <p>A <a class="existingWikiWord" href="/nlab/show/finite+dimensional+vector+space">finite dimensional vector space</a> <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> has the property that a high enough <a class="existingWikiWord" href="/nlab/show/alternating+power">alternating power</a> of it vanishes <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mo>∧</mo> <mi>n</mi></msup><mi>V</mi><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">\wedge^n V = 0</annotation></semantics></math>, namely this is the case for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>></mo><mi>dim</mi><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">n \gt dim(V)</annotation></semantics></math>, and hence this vanishing is just another reflection of the finiteness of the <a class="existingWikiWord" href="/nlab/show/dimension">dimension</a> of <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>. For a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> <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> of degreewise finite dimension an analog statement is still true, but one needs to form not just alternating powers but also <a class="existingWikiWord" href="/nlab/show/symmetric+powers">symmetric powers</a> (prop. <a class="maruku-ref" href="#SchurFunctorAnnihilatingFiniteDimensionalSuperVectorSpace"></a> below), in fact one needs to apply a generalization of both of these constructions, a <em><a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a></em>.</p> <p>The operation of forming <a class="existingWikiWord" href="/nlab/show/symmetric+powers">symmetric powers</a> and <a class="existingWikiWord" href="/nlab/show/alternating+powers">alternating powers</a> makes sense in every <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a>. Moreover, these operations are the two extreme cases of the more general concept of <a class="existingWikiWord" href="/nlab/show/Schur+functors">Schur functors</a>: Given any <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> and given any choice of <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">V_\lambda</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Σ</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">\Sigma_n</annotation></semantics></math>, then one consider the <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><msup><mo>⊗</mo> <mi>n</mi></msup></mrow></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">S_\lambda(X^{\otimes^n})</annotation></semantics></math> of the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-fold <a class="existingWikiWord" href="/nlab/show/tensor+power">tensor power</a> that is <a class="existingWikiWord" href="/nlab/show/invariant">invariant</a> under this action.</p> <p>The first step in the proof of the main theorem (theorem <a class="maruku-ref" href="#TheTheorem"></a> below) is the proposition (prop. <a class="maruku-ref" href="#LengthOfObjectIsBounded"></a> below) that all objects that have subexponential growth of length (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>) are actually annihilated by some <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> for the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a>.</p> <div class="num_defn" id="SchurFunctor"> <h6 id="definition_38">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝒜</mi><mo>,</mo><mo>⊗</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathcal{A},\otimes)</annotation></semantics></math> 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/tensor+category">tensor category</a> as in def.<a class="maruku-ref" href="#TensorCategory"></a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/object">object</a>, for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/partition">partition</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>, regarded as a <a class="existingWikiWord" href="/nlab/show/Young+diagram">Young diagram</a> and hence as a <a class="existingWikiWord" href="/nlab/show/representation">representation</a> of the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">V_\lambda</annotation></semantics></math>, say that the value of the <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">S_\lambda</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</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><mrow><mtable displaystyle="true" columnalign="right left right left right left right left right left" columnspacing="0em"><mtr><mtd><msub><mi>S</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≔</mo><mo stretchy="false">(</mo><msub><mi>V</mi> <mi>λ</mi></msub><mo>⊗</mo><msup><mi>X</mi> <mrow><msub><mo>⊗</mo> <mi>n</mi></msub></mrow></msup><msup><mo stretchy="false">)</mo> <mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow></msup></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mrow><mo>(</mo><mfrac><mn>1</mn><mrow><mi>n</mi><mo>!</mo></mrow></mfrac><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo><mrow><mi>g</mi><mo>∈</mo><msub><mi>S</mi> <mi>n</mi></msub></mrow></munder><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>)</mo></mrow><mrow><mo>(</mo><msub><mi>V</mi> <mi>λ</mi></msub><mo>⊗</mo><msup><mi>X</mi> <mrow><msub><mo>⊗</mo> <mi>n</mi></msub></mrow></msup><mo>)</mo></mrow></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} S_{\lambda}(X) & \coloneqq (V_\lambda \otimes X^{\otimes_n})^{S_n} \\ & = \left( \frac{1}{n!} \underset{g\in S_n}{\sum} \rho(g) \right) \left( V_\lambda \otimes X^{\otimes_n} \right) \end{aligned} </annotation></semantics></math></div> <p>where</p> <ul> <li> <p><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><msup><mo stretchy="false">)</mo> <mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">(-)^{S_n}</annotation></semantics></math> is the subobject of <a class="existingWikiWord" href="/nlab/show/invariants">invariants</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> elements;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">V_\lambda</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/irreducible+representation">irreducible representation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> corresponding to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/diagonal+action">diagonal action</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow><annotation encoding="application/x-tex">S_n</annotation></semantics></math> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>λ</mi></msub><mo>⊗</mo><msup><mi>X</mi> <mrow><msub><mo>⊗</mo> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">V_\lambda \otimes X^{\otimes_n}</annotation></semantics></math>, coming from the canonical <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> action on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>X</mi> <mrow><msub><mo>⊗</mo> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">X^{\otimes_n}</annotation></semantics></math>;</p> </li> <li> <p><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><msup><mo stretchy="false">)</mo> <mrow><msub><mi>S</mi> <mi>n</mi></msub></mrow></msup></mrow><annotation encoding="application/x-tex">(-)^{S_n}</annotation></semantics></math> denotes the subspace of <a class="existingWikiWord" href="/nlab/show/invariants">invariants</a> under the action <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math></p> </li> <li> <p>the second expression just rewrites the invariants as the image of all elements under <a class="existingWikiWord" href="/nlab/show/group+averaging">group averaging</a>.</p> </li> </ul> </div> <p>(<a href="#Deligne02">Deligne 02, 1.4</a>)</p> <div class="num_example"> <h6 id="example_30">Example</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>=</mo><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda = (n)</annotation></semantics></math>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">V_{(n)} = k</annotation></semantics></math> equipped with the trivial action of the symmetric group. In this case the corresponding <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> (def. <a class="maruku-ref" href="#SchurFunctor"></a>) forms the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>th <a class="existingWikiWord" href="/nlab/show/symmetric+power">symmetric power</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mo stretchy="false">(</mo><mi>n</mi><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>Sym</mi> <mi>n</mi></msup><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{(n)}(X) = Sym^n(X) \,. </annotation></semantics></math></div> <p>For the dual case where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi><mo>=</mo><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\lambda = (1,1, \cdots, 1)</annotation></semantics></math> then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo>=</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">V_{(1,1,\cdots, 1)} = k</annotation></semantics></math> equipped with the action by multiplication with the <a class="existingWikiWord" href="/nlab/show/signature+of+a+permutation">signature of a permutation</a> and the corresponding <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> forms the <a class="existingWikiWord" href="/nlab/show/alternating+power">alternating power</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mrow><mo stretchy="false">(</mo><mn>1</mn><mo>,</mo><mn>1</mn><mo>,</mo><mi>⋯</mi><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mo>∧</mo> <mi>n</mi></msup><mi>X</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_{(1,1, \cdots, 1)}(X) = \wedge^n X \,. </annotation></semantics></math></div></div> <div class="num_prop" id="SchurFunctorAnnihilatingFiniteDimensionalSuperVectorSpace"> <h6 id="proposition_22">Proposition</h6> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" 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></mrow><annotation encoding="application/x-tex">V = V_{even} \oplus V_{odd}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> of degreewise finite dimension <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>even</mi></msub><mo>,</mo><msub><mi>d</mi> <mi>odd</mi></msub><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">d_{even}, d_{odd} \in \mathbb{N}</annotation></semantics></math>. Then there exists a <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>λ</mi></msub></mrow><annotation encoding="application/x-tex">S_\lambda</annotation></semantics></math> (def. <a class="maruku-ref" href="#SchurFunctor"></a>) that annihilates <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>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>V</mi><mo stretchy="false">)</mo><mo>≃</mo><mn>0</mn><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> S_\lambda(V) \simeq 0 \,. </annotation></semantics></math></div> <p>Specifically, this is the case precisely if the corresponding <a class="existingWikiWord" href="/nlab/show/Young+tableau">Young tableau</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>λ</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[\lambda]</annotation></semantics></math> satifies</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>λ</mi><mo stretchy="false">]</mo><mo>⊂</mo><mrow><mo>{</mo><mo stretchy="false">(</mo><mi>i</mi><mo>,</mo><mi>j</mi><mo stretchy="false">)</mo><mspace width="thickmathspace"></mspace><mo stretchy="false">|</mo><mspace width="thickmathspace"></mspace><mi>i</mi><mo>≤</mo><msub><mi>d</mi> <mi>even</mi></msub><mo>,</mo><mi>j</mi><mo>≤</mo><msub><mi>d</mi> <mi>odd</mi></msub><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> [\lambda] \subset \left\{ (i,j)\;\vert\; i \leq d_{even}, j \leq d_{odd} \right\} \,. </annotation></semantics></math></div></div> <p>(<a href="#Deligne02">Deligne 02, corollary 1.9</a>)</p> <h2 id="Statement">Statement of the theorem</h2> <div class="num_theorem" id="TheTheorem"> <h6 id="theorem">Theorem</h6> <p>Every <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/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) such that</p> <ol> <li> <p><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 an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> (e.g. the field of <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>)</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is of subexponential growth according to def. <a class="maruku-ref" href="#SubexponentialGrowth"></a></p> </li> </ol> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is a neutral super <a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a> (def. <a class="maruku-ref" href="#TannakianCategory"></a>) and there exists</p> <ol> <li> <p>an affine algebraic <a class="existingWikiWord" href="/nlab/show/supergroup">supergroup</a> <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> (def. <a class="maruku-ref" href="#Supergroup"></a>) whose <a class="existingWikiWord" href="/nlab/show/algebra+of+functions">algebra of functions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒪</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{O}(G)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/finitely+generated+object">finitely generated</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>-algebra.</p> </li> <li> <p>a tensor-<a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≃</mo><mi>Rep</mi><mo stretchy="false">(</mo><mi>G</mi><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \simeq Rep(G,\epsilon) \,. </annotation></semantics></math></div> <p>between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> and the <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of representations</a> 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> of finite dimension, according to def. <a class="maruku-ref" href="#Superrepresentation"></a> and prop. <a class="maruku-ref" href="#RegularTensorCategoriesOfSuperrepresentations"></a>.</p> </li> </ol> </div> <p>(<a href="#Deligne02">Deligne 02, theorem 0.6</a>)</p> <h2 id="proof_13">Proof</h2> <p>We outline key steps of the proof of theorem <a class="maruku-ref" href="#TheTheorem"></a>, given in <a href="#Deligne02">Deligne 02</a>.</p> <p>Throughout, let <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> be an <a class="existingWikiWord" href="/nlab/show/algebraically+closed+field">algebraically closed field</a> of <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> (for instance the <a class="existingWikiWord" href="/nlab/show/complex+numbers">complex numbers</a>).</p> <p>The proof proceeds in three main steps:</p> <ol> <li> <p><strong>Proposition <a class="maruku-ref" href="#LengthOfObjectIsBounded"></a></strong> states that in 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/tensor+category">tensor category</a> an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is of subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>) precisely if there exists a <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> that annihilates it, hence if some power of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, skew-symmetrized in sme variables and symmetrized in others, vanishes.</p> <p>This proposition is where the <a class="existingWikiWord" href="/nlab/show/symmetric+group">symmetric group</a> and its <a class="existingWikiWord" href="/nlab/show/permutation">permutation</a> <a class="existingWikiWord" href="/nlab/show/action">action</a> on <a class="existingWikiWord" href="/nlab/show/tensor+powers">tensor powers</a> appears, from just a kind of finite-dimensionality assumption.</p> </li> <li> <p><strong>Proposition <a class="maruku-ref" href="#SchurFinitenessImpliesExistenceOfSuperFiberFunctor"></a></strong> in turn says that if every object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is annihilated by some <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a>, then there exists a super <a class="existingWikiWord" href="/nlab/show/fiber+functor">fiber functor</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> over some <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>, hence then every object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> has underlying it a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a> with some extra structure.</p> <p>This proposition is where <a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a> proper appears.</p> </li> <li> <p><strong>Proposition <a class="maruku-ref" href="#DeligneTannakaReconstruction"></a></strong> states that every <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/tensor+category">tensor category</a> equipped with a super fiber functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo lspace="verythinmathspace">:</mo><mi>𝒜</mi><mo>→</mo><mi>sVect</mi></mrow><annotation encoding="application/x-tex">\omega \colon \mathcal{A} \to sVect</annotation></semantics></math>, is equivalent to the category of super-representations of the automorphism supergroup of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math>.</p> <p>This proposition is the instance of general <a class="existingWikiWord" href="/nlab/show/Tannaka+reconstruction">Tannaka reconstruction</a> applied to the case of fiber functors with values in super vector spaces. This is where the “supersymmetry” supergroup is extracted.</p> </li> </ol> <div class="num_prop" id="LengthOfObjectIsBounded"> <h6 id="proposition_23">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> 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/tensor+category">tensor category</a> (def. <a class="maruku-ref" href="#TensorCategory"></a>), then the following are equivalent:</p> <ol> <li> <p>the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> has subexponential growth (def. <a class="maruku-ref" href="#SubexponentialGrowth"></a>);</p> </li> <li> <p>for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">X \in \mathcal{A}</annotation></semantics></math> there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">n \in \mathbb{N}</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/partition">partition</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>λ</mi></mrow><annotation encoding="application/x-tex">\lambda</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> such that the corresponding value of the <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a>, def. <a class="maruku-ref" href="#SchurFunctor"></a>, on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> vanishes: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>S</mi> <mi>λ</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">S_\lambda(X) = 0</annotation></semantics></math>.</p> </li> </ol> </div> <p>(<a href="#Deligne02">Deligne 02, prop. 05</a>)</p> <div class="num_prop" id="SchurFinitenessImpliesExistenceOfSuperFiberFunctor"> <h6 id="proposition_24">Proposition</h6> <p>If for every object of 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/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) there exists a <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> (def. <a class="maruku-ref" href="#SchurFunctor"></a>) that annihilates it, then there exists a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> (def. <a class="maruku-ref" href="#FiberFunctor"></a>) over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>, hence then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> is a neutral super <a class="existingWikiWord" href="/nlab/show/Tannakian+category">Tannakian category</a> (def. <a class="maruku-ref" href="#TannakianCategory"></a>).</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><mi>sVect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; \mathcal{A} \longrightarrow sVect \,. </annotation></semantics></math></div></div> <p>(<a href="#Deligne02">Deligne 02, prop. 2.1 “résultat clé de l’article”, together with prop. 4.5</a>)</p> <div class="proof"> <h6 id="proof_idea">Proof idea</h6> <p>First (<a href="#Deligne02">Deligne 02, middle of p. 16</a>) consider the <a class="existingWikiWord" href="/nlab/show/tensor+category">tensor category</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>𝒜</mi><mo>≔</mo><mo stretchy="false">(</mo><msup><mi>𝒜</mi> <mrow><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow></msup><mo>,</mo><msub><mi>τ</mi> <mi>super</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> s \mathcal{A} \coloneqq (\mathcal{A}^{\mathbb{Z}/2}, \tau_{super} ) </annotation></semantics></math></div> <p>which 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>-graded objects of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, and whose <a class="existingWikiWord" href="/nlab/show/braiding">braiding</a> is given on 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> of homogeneous degree by that of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> multiplied with <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><msup><mo stretchy="false">)</mo> <mrow><mi>deg</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mi>deg</mi><mo stretchy="false">(</mo><mi>Y</mi><mo stretchy="false">)</mo></mrow></msup></mrow><annotation encoding="application/x-tex">(-1)^{deg(X) deg(Y)}</annotation></semantics></math>.</p> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mn>1</mn><mo>¯</mo></mover></mrow><annotation encoding="application/x-tex">\overline{1}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/tensor+unit">tensor unit</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, regarded in even degree and in odd degree in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">s \mathcal{A}</annotation></semantics></math>, respectively.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>∈</mo><mi>CMon</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \in CMon(\mathcal{A})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/commutative+monoid">commutative monoid</a>, write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>≃</mo><mn>1</mn><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><munderover><mrow></mrow><munder><mo>⟵</mo><mi>U</mi></munder><mover><mo>⟶</mo><mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">)</mo> <mi>A</mi></msub><mo>≔</mo><mi>F</mi></mrow></mover></munderover><mi>A</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \simeq 1 Mod(\mathcal{A}) \underoverset{\underset{U}{\longleftarrow}}{\overset{(-)_A \coloneqq F}{\longrightarrow}} {} A Mod(\mathcal{A}) </annotation></semantics></math></div> <p>for the <a class="existingWikiWord" href="/nlab/show/extension+of+scalars">extension of scalars</a> operation <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>⊗</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">A \otimes(-)</annotation></semantics></math>, <a class="existingWikiWord" href="/nlab/show/left+adjoint">left adjoint</a> to <a class="existingWikiWord" href="/nlab/show/restriction+of+scalars">restriction of scalars</a> (prop. <a class="maruku-ref" href="#ExtensionOfScalars"></a>).</p> <p>Show then that the condition that an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is annihilated by some <a class="existingWikiWord" href="/nlab/show/Schur+functor">Schur functor</a> is equivalent to the existence of an algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>A</mi></msub><mo>≃</mo><msup><mn>1</mn> <mi>p</mi></msup><mo>⊕</mo><msup><mover><mn>1</mn><mo>¯</mo></mover> <mi>q</mi></msup></mrow><annotation encoding="application/x-tex"> X_A \simeq 1^p \oplus \overline{1}^q </annotation></semantics></math></div> <p>for some <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>,</mo><mi>q</mi><mo>∈</mo><mi>ℕ</mi></mrow><annotation encoding="application/x-tex">p,q \in \mathbb{N}</annotation></semantics></math>, hence that each such object is <em><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-locally</em> a <a class="existingWikiWord" href="/nlab/show/super+vector+space">super vector space</a>.</p> <p>(<a href="#Deligne02">Deligne 02, prop. 2.9</a>).</p> <p>Moreover, for each <a class="existingWikiWord" href="/nlab/show/short+exact+sequence">short exact sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><mi>X</mi><mo>→</mo><mi>Y</mi><mo>→</mo><mi>Z</mi><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to X \to Y \to Z \to 0 </annotation></semantics></math></div> <p>in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">s \mathcal{A}</annotation></semantics></math>, there exists an algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> such that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>0</mn><mo>→</mo><msub><mi>X</mi> <mi>A</mi></msub><mo>→</mo><msub><mi>Y</mi> <mi>A</mi></msub><mo>→</mo><msub><mi>Z</mi> <mi>A</mi></msub><mo>→</mo><mn>0</mn></mrow><annotation encoding="application/x-tex"> 0 \to X_A \to Y_A \to Z_A \to 0 </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/split+exact+sequence">split exact sequence</a>, (hence every short exact sequence is <em>locally</em> split).</p> <p>(<a href="#Deligne90">Deligne 90, 7.14</a>, Deligne 02, rappel 2.102))</p> <p>Now (<a href="Deligne02">Deligne 02, middle of p. 17</a>) let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math> be the commutative monoid which is the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> of commutative monoids (example <a class="maruku-ref" href="#TensorProductOfTwoCommutativeMonoids"></a>) over all <a class="existingWikiWord" href="/nlab/show/isomorphism+classes">isomorphism classes</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> and of <a class="existingWikiWord" href="/nlab/show/short+exact+sequences">short exact sequences</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> of choices of commutative monoids for which these objects/exact sequencs are locally split, as above.</p> <p>Then for an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-module <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math>, write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(N)</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/subobject">subobject</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>N</mi></mrow><annotation encoding="application/x-tex">N</annotation></semantics></math> inside <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>sVect</mi><mo>≃</mo><mi>Ind</mi><mo stretchy="false">⟨</mo><mn>1</mn><mo>,</mo><mover><mn>1</mn><mo>¯</mo></mover><mo stretchy="false">⟩</mo><mo>↪</mo><mi>s</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">sVect \simeq Ind\langle 1, \overline{1}\rangle \hookrightarrow s \mathcal{A}</annotation></semantics></math>.</p> <p>Check (<a href="Deligne02">Deligne 02, bottom of p. 17</a>) that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(A)</annotation></semantics></math> inherits the structure of a commutative monoid, and that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(N)</annotation></semantics></math> inherits the structure of a module over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(N)</annotation></semantics></math>.</p> <p>Set</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>≔</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> R \coloneqq \rho(A) \,. </annotation></semantics></math></div> <p>Hence for every object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>≔</mo><mi>ρ</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega(X) \coloneqq \rho(X_A) </annotation></semantics></math></div> <p>has the structure of an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math>-module. By <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>-local splitness of all short exact sequence, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/exact+functor">exact functor</a>.</p> <p>Hence</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>s</mi><mi>𝒜</mi><mo>⟶</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; s \mathcal{A} \longrightarrow R Mod(sVect) </annotation></semantics></math></div> <p>is a super fiber functor on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">s\mathcal{A}</annotation></semantics></math> over <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>. This restricts to a super fiber functor over <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> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math>, regarded as the sub-category of even-graded objects in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">s \mathcal{A}</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>𝒜</mi><mo>↪</mo><mi>s</mi><mi>𝒜</mi><mover><mo>⟶</mo><mi>ω</mi></mover><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \hookrightarrow s \mathcal{A} \overset{\omega}{\longrightarrow} R Mod(sVect) \,, </annotation></semantics></math></div> <p>Finally check (<a href="#Deligne02">Deligne 02, prop. 4.5</a>) that if 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/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) admits a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> (def. <a class="maruku-ref" href="#FiberFunctor"></a>) over a <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>R</mi></mrow><annotation encoding="application/x-tex">R</annotation></semantics></math> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mo>⟶</mo><mi>R</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \longrightarrow R Mod(sVect) </annotation></semantics></math></div> <p>then it also admits a super fiber functor over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> itself, i.e. a fiber functor to <a class="existingWikiWord" href="/nlab/show/sVect">sVect</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><mo>⟶</mo><mi>k</mi><mi>Mod</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>sVect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \longrightarrow k Mod(sVect) \simeq sVect \,. </annotation></semantics></math></div> <p>This is argued by expressing <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> as an <a class="existingWikiWord" href="/nlab/show/inductive+limit">inductive limit</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>R</mi><mo>=</mo><msub><munder><mi>lim</mi><mo>⟶</mo></munder> <mi>β</mi></msub><msub><mi>R</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex"> R = \underset{\longrightarrow}{\lim}_\beta R_\beta </annotation></semantics></math></div> <p>over <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebras">supercommutative superalgebras</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">R_\beta</annotation></semantics></math> of finite type over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> and observing (…) that there exists <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>β</mi></mrow><annotation encoding="application/x-tex">\beta</annotation></semantics></math> such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>β</mi></msub></mrow><annotation encoding="application/x-tex">\omega_\beta</annotation></semantics></math> is still a fiber functor and such that there exists an algebra homomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>R</mi> <mi>β</mi></msub><mo>→</mo><mi>k</mi></mrow><annotation encoding="application/x-tex">R_\beta \to k</annotation></semantics></math>.</p> <p>Finally then the fiber functor in question is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ω</mi> <mi>β</mi></msub><msub><mo>⊗</mo> <mrow><msub><mi>R</mi> <mi>β</mi></msub></mrow></msub><mi>k</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><msub><mi>sVect</mi> <mi>k</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \omega_\beta \otimes_{R_\beta} k \;\colon\; \mathcal{A} \longrightarrow sVect_k \,. </annotation></semantics></math></div></div> <div class="num_prop" id="DeligneTannakaReconstruction"> <h6 id="proposition_25">Proposition</h6> <p>For every <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/tensor+category">tensor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> (def. <a class="maruku-ref" href="#TensorCategory"></a>) and a <a class="existingWikiWord" href="/nlab/show/fiber+functor">super fiber functor</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math> (def. <a class="maruku-ref" href="#FiberFunctor"></a>)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ω</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>𝒜</mi><mo>⟶</mo><msub><mi>sVect</mi> <mi>k</mi></msub></mrow><annotation encoding="application/x-tex"> \omega \;\colon\; \mathcal{A} \longrightarrow sVect_k </annotation></semantics></math></div> <p>then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ω</mi></mrow><annotation encoding="application/x-tex">\omega</annotation></semantics></math> induces an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi><mover><mo>⟶</mo><mo>≃</mo></mover><mi>Rep</mi><mo stretchy="false">(</mo><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>,</mo><mi>ϵ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathcal{A} \stackrel{\simeq}{\longrightarrow} Rep( \underline{Aut}(\omega),\epsilon) </annotation></semantics></math></div> <p>of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒜</mi></mrow><annotation encoding="application/x-tex">\mathcal{A}</annotation></semantics></math> with the <a class="existingWikiWord" href="/nlab/show/category+of+representations">category of finite dimensional representations</a>, according to def. <a class="maruku-ref" href="#Superrepresentation"></a> and prop. <a class="maruku-ref" href="#RegularTensorCategoriesOfSuperrepresentations"></a>, of the automorphism supergroup <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\underline{Aut}(\omega)</annotation></semantics></math> (example. <a class="maruku-ref" href="#AutomorphismSuperGroupOfSuperFiberFunctor"></a>, prop. <a class="maruku-ref" href="#AutomorphismSupergroupOfFiberFunctorIsRepresentable"></a>) of the super fiber functor, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ϵ</mi></mrow><annotation encoding="application/x-tex">\epsilon</annotation></semantics></math> is te image of the unique nontrivial element in</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>≃</mo><mi>ℤ</mi><mo stretchy="false">/</mo><mn>2</mn></mrow><annotation encoding="application/x-tex"> \underline{Aut}(sVect) \simeq \mathbb{Z}/2 </annotation></semantics></math></div> <p>(according to example <a class="maruku-ref" href="#FundamentalGroupOfCategoryOfSuperVectorSpaces"></a>) under the group homomorphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>π</mi><mo stretchy="false">(</mo><mi>sVect</mi><mo stretchy="false">)</mo><mo>⟶</mo><mi>ω</mi><mo stretchy="false">(</mo><mi>π</mi><mo stretchy="false">(</mo><mi>𝒜</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>Aut</mi><mo>̲</mo></munder><mo stretchy="false">(</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \pi(sVect) \longrightarrow \omega(\pi(\mathcal{A})) \simeq \underline{Aut}(\omega) </annotation></semantics></math></div> <p>from prop. <a class="maruku-ref" href="#HomomorphismFromPiT2ToEtaPiT1"></a> and using the isomorphism from prop. <a class="maruku-ref" href="#AutOfFibFuncIsImagUnderFibFuncOfFundamentalGroup"></a>.</p> </div> <p>This is the main <a class="existingWikiWord" href="/nlab/show/Tannaka+reconstruction">Tannaka reconstruction</a> theorem (<a href="#Deligne90">Deligne 90, 8.17</a>) specialized to super fiber functors (<a href="#Deligne90">Deligne 90, 8.19</a>).</p> <h2 id="references">References</h2> <p>The theorem is due to</p> <ul> <li id="Deligne02"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em>Catégorie Tensorielle</em>, Moscow Math. Journal 2 (2002) no. 2, 227-248. (<a href="https://www.math.ias.edu/files/deligne/Tensorielles.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/DeligneTensorCategories02.pdf" title="pdf">pdf</a>)</li> </ul> <p>building on the general results on <a class="existingWikiWord" href="/nlab/show/Tannakian+categories">Tannakian categories</a> in</p> <ul> <li id="Deligne90"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <em><a class="existingWikiWord" href="/nlab/show/Cat%C3%A9gories+Tannakiennes">Catégories Tannakiennes</a></em>, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. 87 (1990) pp. 111-195 (<a href="https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf">pdf</a>)</li> </ul> <p>which are reviewed and further generalized in</p> <ul> <li id="DeligneMilne12"><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, <a class="existingWikiWord" href="/nlab/show/James+Milne">James Milne</a>, <em>Tannakian categories</em>, 2012 (<a href="http://www.jmilne.org/math/xnotes/tc.pdf">pdf</a>)</li> </ul> <p>Review is in</p> <ul> <li id="Ostrik04"> <p><a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>Tensor categories (after P. Deligne)</em> (<a href="http://arxiv.org/abs/math/0401347">arXiv:math/0401347</a>)</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, Shlomo Gelaki, Dmitri Nikshych, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, section 9.11 in <em>Tensor categories</em>, Mathematical Surveys and Monographs, Volume 205, American Mathematical Society, 2015 (<a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a>)</p> </li> </ul> <p>Further discussion in view of the theory of <a class="existingWikiWord" href="/nlab/show/triangular+Hopf+algebras">triangular Hopf algebras</a> is in</p> <ul> <li id="EtingofGelaki02"><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <em>The classification of finite-dimensional triangular Hopf algebras over an algebraically closed field of characteristic 0</em> (<a href="https://arxiv.org/abs/math/0202258">arXiv:0202258</a>)</li> </ul> <p>Tannaka duality for ordinary compact groups regarded as supergroups (hence equipped with “inner parity”, def. <a class="maruku-ref" href="#InnerParity"></a>, here just being an involutive central element) is discussed in</p> <ul> <li id="Mueger06"><a class="existingWikiWord" href="/nlab/show/Michael+M%C3%BCger">Michael Müger</a>, <em>Abstract Duality Theory for Symmetric Tensor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo></mrow><annotation encoding="application/x-tex">*</annotation></semantics></math>-Categories</em> appendix (<a href="http://www.math.ru.nl/~mueger/PDF/16.pdf">pdf</a>), in <a class="existingWikiWord" href="/nlab/show/Hans+Halvorson">Hans Halvorson</a>, <em>Algebraic quantum field theory</em> (<a href="http://arxiv.org/abs/math-ph/0602036">arXiv:math-ph/0602036</a>), in J. Butterfield & J. Earman (eds.) <em>Handbook of Philosophy and Physics</em></li> </ul> <p>as a proof of <a class="existingWikiWord" href="/nlab/show/Doplicher-Roberts+reconstruction">Doplicher-Roberts reconstruction</a></p> <p>Commutative algebra internal to symmetric monoidal categories is discussed in</p> <ul> <li id="EKMM97"> <p><a class="existingWikiWord" href="/nlab/show/Anthony+Elmendorf">Anthony Elmendorf</a>, <a class="existingWikiWord" href="/nlab/show/Igor+Kriz">Igor Kriz</a>, <a class="existingWikiWord" href="/nlab/show/Michael+Mandell">Michael Mandell</a>, <a class="existingWikiWord" href="/nlab/show/Peter+May">Peter May</a>, <em>Rings, modules and algebras in stable homotopy theory</em>, AMS 1997, 2014</p> </li> <li id="HoveyShipleySmith00"> <p><a class="existingWikiWord" href="/nlab/show/Mark+Hovey">Mark Hovey</a>, <a class="existingWikiWord" href="/nlab/show/Brooke+Shipley">Brooke Shipley</a>, <a class="existingWikiWord" href="/nlab/show/Jeff+Smith">Jeff Smith</a>, <em>Symmetric spectra</em>, J. Amer. Math. Soc. 13 (2000), 149-208 (<a href="http://arxiv.org/abs/math/9801077">arXiv:math/9801077</a>)</p> </li> </ul> <p>and specifically for <a class="existingWikiWord" href="/nlab/show/commutative+Hopf+algebroids">commutative Hopf algebroids</a> in</p> <ul> <li id="Ravenel86"><a class="existingWikiWord" href="/nlab/show/Doug+Ravenel">Doug Ravenel</a>, chapter 2 and appendix A.1 of <em><a class="existingWikiWord" href="/nlab/show/Complex+cobordism+and+stable+homotopy+groups+of+spheres">Complex cobordism and stable homotopy groups of spheres</a></em>, 1986 (<a href="http://www.math.rochester.edu/people/faculty/doug/mybooks/ravenelA1.pdf">pdf</a>)</li> </ul> <p>(These authors are motivated by the application of the general theory of algebra in monoidal categories to “<a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a>” (“<a class="existingWikiWord" href="/nlab/show/brave+new+algebra">brave new algebra</a>”) in <a class="existingWikiWord" href="/nlab/show/stable+homotopy+theory">stable homotopy theory</a>. This happens to also be a version of <a class="existingWikiWord" href="/nlab/show/supercommutative+superalgebra">supercommutative superalgebra</a>, see at <em><a class="existingWikiWord" href="/nlab/show/Introduction+to+Stable+homotopy+theory">Introduction to Stable homotopy theory</a></em> the section <a href="Introduction+to+Stable+homotopy+theory+--+1-2#HomotopyRingSpectra">1-2 Homotopy commutative ring spectra</a>.)</p> <p>For an attempt to generalize Deligne’s theorem to <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a>, see</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>On symmetric fusion categories in positive characteristic</em>, (<a href="https://arxiv.org/abs/1503.01492">arXiv:1503.01492</a>)</li> </ul> <p>This was realised for <a class="existingWikiWord" href="/nlab/show/Frobenius+exact+category">Frobenius exact</a> tensor categories in <a class="existingWikiWord" href="/nlab/show/positive+characteristic">positive characteristic</a> in:</p> <ul> <li id="CoulembierEtingofOstrik21"><a class="existingWikiWord" href="/nlab/show/Kevin+Coulembier">Kevin Coulembier</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>On Frobenius exact symmetric tensor categories</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo></mrow><annotation encoding="application/x-tex">[</annotation></semantics></math><a href="https://arxiv.org/abs/2107.02372">arXiv:2107.02372</a><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">]</annotation></semantics></math></li> </ul> <p>where <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> and <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 replaced by more exotic targets.</p> <p>Discussion relating to <a class="existingWikiWord" href="/nlab/show/2-rings">2-rings</a> and the <a class="existingWikiWord" href="/nlab/show/spin-statistics+theorem">spin-statistics theorem</a> is in</p> <ul> <li id="JohnsonFreyd15"><a class="existingWikiWord" href="/nlab/show/Theo+Johnson-Freyd">Theo Johnson-Freyd</a>, <em>Spin, statistics, orientations, unitarity</em>, Algebraic and Geometric Topology (<a href="https://arxiv.org/abs/1507.06297">arXiv:1507.06297</a>)</li> </ul> <p>On Deligne categories:</p> <ul> <li id="Hu24">Serina Hu. <em>An Introduction to Deligne Categories</em> (2024). (<a href="https://arxiv.org/abs/2404.08689">arXiv:2404.08689</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 24, 2024 at 15:45:58. 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