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tensor category in nLab
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<div id="Content"> <h1 id="pageName"> <span style="float: left; margin: 0.5em 0.25em -0.25em 0"> <svg xmlns="http://www.w3.org/2000/svg" width="1.872em" height="1.8em" viewBox="0 0 190 181"> <path fill="#226622" d="M72.8 145c-1.6 17.3-15.7 10-23.6 20.2-5.6 7.3 4.8 15 11.4 15 11.5-.2 19-13.4 26.4-20.3 3.3-3 8.2-4 11.2-7.2a14 14 0 0 0 2.9-11.1c-1.4-9.6-12.4-18.6-16.9-27.2-5-9.6-10.7-27.4-24.1-27.7-17.4-.3-.4 26 4.7 30.7 2.4 2.3 5.4 4.1 7.3 6.9 1.6 2.3 2.1 5.8-1 7.2-5.9 2.6-12.4-6.3-15.5-10-8.8-10.6-15.5-23-26.2-31.8-5.2-4.3-11.8-8-18-3.7-7.3 4.9-4.2 12.9.2 18.5a81 81 0 0 0 30.7 23c3.3 1.5 12.8 5.6 10 10.7-2.5 5.2-11.7 3-15.6 1.1-8.4-3.8-24.3-21.3-34.4-13.7-3.5 2.6-2.3 7.6-1.2 11.1 2.8 9 12.2 17.2 20.9 20.5 17.3 6.7 34.3-8 50.8-12.1z"/> <path fill="#a41e32" d="M145.9 121.3c-.2-7.5 0-19.6-4.5-26-5.4-7.5-12.9-1-14.1 5.8-1.4 7.8 2.7 14.1 4.8 21.3 3.4 12 5.8 29-.8 40.1-3.6-6.7-5.2-13-7-20.4-2.1-8.2-12.8-13.2-15.1-1.9-2 9.7 9 21.2 12 30.1 1.2 4 2 8.8 6.4 10.3 6.9 2.3 13.3-4.7 17.7-8.8 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href="/nlab/show/HomePage" accesskey="H" title="Home page">Home Page</a> | <a href="/nlab/all_pages" accesskey="A" title="List of all pages">All Pages</a> | <a href="/nlab/latest_revisions" accesskey="U" title="Latest edits and page creations">Latest Revisions</a> | <a href="https://nforum.ncatlab.org/discussion/10656/#Item_8" title="Discuss this page in its dedicated thread on the nForum" style="color: black">Discuss this page</a> | <form accept-charset="utf-8" action="/nlab/search" id="navigationSearchForm" method="get"> <fieldset class="search"><input type="text" id="searchField" name="query" value="Search" style="display:inline-block; float: left;" onfocus="this.value == 'Search' ? this.value = '' : true" onblur="this.value == '' ? this.value = 'Search' : true" /></fieldset> </form> <span id='navEnd'></span> </div> <div id="revision"> <html xmlns="http://www.w3.org/1999/xhtml" xmlns:svg="http://www.w3.org/2000/svg" xml:lang="en" lang="en"> <head><meta http-equiv="Content-type" content="application/xhtml+xml;charset=utf-8" /><title>Contents</title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <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="category_theory">Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></strong></p> <h2 id="sidebar_concepts">Concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category">category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/functor">functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cat">Cat</a></p> </li> </ul> <h2 id="sidebar_universal_constructions">Universal constructions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/universal+construction">universal construction</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/representable+functor">representable functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor">adjoint functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/limit">limit</a>/<a class="existingWikiWord" href="/nlab/show/colimit">colimit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/weighted+limit">weighted limit</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/end">end</a>/<a class="existingWikiWord" href="/nlab/show/coend">coend</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+extension">Kan extension</a></p> </li> </ul> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+construction">Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+functor+theorem">adjoint functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">monadicity theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+lifting+theorem">adjoint lifting theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Gabriel-Ulmer+duality">Gabriel-Ulmer duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/small+object+argument">small object argument</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Freyd-Mitchell+embedding+theorem">Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+type+theory+and+category+theory">relation between type theory and category theory</a></p> </li> </ul> <h2 id="sidebar_extensions">Extensions</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/sheaf+and+topos+theory">sheaf and topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/enriched+category+theory">enriched category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> </ul> <h2 id="sidebar_applications">Applications</h2> <ul> <li><a class="existingWikiWord" href="/nlab/show/applications+of+%28higher%29+category+theory">applications of (higher) category theory</a></li> </ul> <div> <p> <a href="/nlab/edit/category+theory+-+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='#properties'>Properties</a></li> <ul> <li><a href='#tannaka_theory_delignes_theorem_superrepresentation_theory'>Tannaka theory, Deligne’s theorem, super-representation theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="Idea">Idea</h2> <p>Some authors use “tensor category” essentially as a synonym for (<a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric</a>) <a class="existingWikiWord" href="/nlab/show/monoidal+categories">monoidal categories</a> (e.g. <a href="#Davydov98">Davydov 1998</a>, <a href="#KashiwaraSchapira06">Kashiwara & Schapira 2006, Def. 4.2.1</a>).</p> <p>These days, a <em>tensor category</em> is usually understood to be a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a> equipped with further “<a class="existingWikiWord" href="/nlab/show/linear+algebra">linear algebraic</a>” <a class="existingWikiWord" href="/nlab/show/properties">properties</a> and <a class="existingWikiWord" href="/nlab/show/structure">structure</a>, hence with <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal</a>-<a class="existingWikiWord" href="/nlab/show/structure">structure</a> given by a kind of <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> in the original sense (i.e. actually being a universal <a class="existingWikiWord" href="/nlab/show/bilinear+map">bilinear map</a> of sorts) whence the name.</p> <p>Conventions differ, but at the very least one means</p> <ul> <li>a <a class="existingWikiWord" href="/nlab/show/monoidal+category">monoidal category</a>,</li> </ul> <p>which is at times required to be</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+category">symmetric monoidal</a></p> <p>(e.g <a href="#Deligne90">Deligne 1990</a> in (2.1.1); <a href="#Davydov98">Davydov 1998</a> says “tensor category” for <a class="existingWikiWord" href="/nlab/show/symmetric+monoidal+categories">symmetric monoidal categories</a> and “quasitensor category” for <a class="existingWikiWord" href="/nlab/show/braided+monoidal+categories">braided monoidal categories</a>),</p> </li> <li> <p>(<a class="existingWikiWord" href="/nlab/show/Ab">Ab</a>, <a class="existingWikiWord" href="/nlab/show/tensor+product+of+abelian+groups"><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></a>)-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a> or (<a class="existingWikiWord" href="/nlab/show/Vect">Vect</a>,<a class="existingWikiWord" href="/nlab/show/tensor+product+of+vector+spaces"><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></a>)-<a class="existingWikiWord" href="/nlab/show/enriched+category">enriched</a>,</p> <p>to make an <a class="existingWikiWord" href="/nlab/show/enriched+monoidal+category">enriched monoidal category</a></p> </li> </ul> <p>and, in addition, often</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/additive+category">additive</a> (symmetric) monoidal, so that the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <a class="existingWikiWord" href="/nlab/show/preserved+limit">preserves</a> <a class="existingWikiWord" href="/nlab/show/finite+product">finite</a> <a class="existingWikiWord" href="/nlab/show/direct+sums">direct sums</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/abelian+category">abelian</a> (symmetric) monoidal, in which the <a class="existingWikiWord" href="/nlab/show/tensor+product">tensor product</a> <a class="existingWikiWord" href="/nlab/show/preserved+colimit">preserves</a> <a class="existingWikiWord" href="/nlab/show/finite+colimits">finite colimits</a> in separate arguments,</p> </li> <li> <p>with <a class="existingWikiWord" href="/nlab/show/dual+objects">dual objects</a>, making a <a class="existingWikiWord" href="/nlab/show/rigid+monoidal+category">rigid monoidal category</a>.</p> </li> </ul> <h2 id="properties">Properties</h2> <h3 id="tannaka_theory_delignes_theorem_superrepresentation_theory">Tannaka theory, Deligne’s theorem, super-representation theory</h3> <p><a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a> (<a href="#Deligne02">Deligne 02</a>) establishes <a class="existingWikiWord" href="/nlab/show/Tannaka+duality">Tannaka duality</a> between sufficiently well-behaved linear tensor categories in <a class="existingWikiWord" href="/nlab/show/characteristic+zero">characteristic zero</a> and <a class="existingWikiWord" href="/nlab/show/supergroups">supergroups</a>, realizing these tensor categories as <a class="existingWikiWord" href="/nlab/show/categories+of+representations">categories of representations</a> of these supergroups.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/fusion+category">fusion category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+triangulated+category">tensor triangulated category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+%28%E2%88%9E%2C1%29-category">tensor (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-algebraic+geometry">2-algebraic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/tensor+network">tensor network</a></p> </li> </ul> <h2 id="references">References</h2> <ul> <li id="Deligne90"> <p><a class="existingWikiWord" href="/nlab/show/Pierre+Deligne">Pierre Deligne</a>, section 2 of: <em><a class="existingWikiWord" href="/nlab/show/Cat%C3%A9gories+Tannakiennes">Catégories Tannakiennes</a></em>, Grothendieck Festschrift, vol. II, Birkhäuser Progress in Math. <strong>87</strong> (1990) 111-195 (<a href="https://publications.ias.edu/sites/default/files/60_categoriestanna.pdf">pdf</a>)</p> </li> <li id="Davydov98"> <p><a class="existingWikiWord" href="/nlab/show/Alexei+Davydov">Alexei Davydov</a>: <em>Monoidal categories and functors</em>, Chapter 1 in: <em>Monoidal Categories</em>, J. Math. Sci.(New York) <strong>88</strong> (1998) 457-519 [<a href="https://doi.org/10.1007/BF02365309">doi:10.1007/BF02365309</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Bojko+Bakalov">Bojko Bakalov</a>, <a class="existingWikiWord" href="/nlab/show/Alexander+Kirillov">Alexander Kirillov</a>, <em>Lectures on tensor categories and modular functors</em>, University Lecture Series <strong>21</strong>, Amer. Math. Soc. (2001) [<a href="http://www.math.stonybrook.edu/~kirillov/tensor/tensor.html">webpage</a>, <a href="https://bookstore.ams.org/view?ProductCode=ULECT/21">ams:ulect/21</a>, <a href="http://math.bu.edu/people/jackwalt/research/tqft-seminar/refs/tensor_cat_mod_func.pdf">pdf</a>]</p> <blockquote> <p>(focus on <a class="existingWikiWord" href="/nlab/show/Reshetikhin-Turaev+construction">Reshetikhin-Turaev construction</a> of <a class="existingWikiWord" href="/nlab/show/modular+functors">modular functors</a> from <a class="existingWikiWord" href="/nlab/show/modular+tensor+categories">modular tensor categories</a>)</p> </blockquote> </li> <li id="KashiwaraSchapira06"> <p><a class="existingWikiWord" href="/nlab/show/Masaki+Kashiwara">Masaki Kashiwara</a>, <a class="existingWikiWord" href="/nlab/show/Pierre+Schapira">Pierre Schapira</a>, Section 4 of: <em><a class="existingWikiWord" href="/nlab/show/Categories+and+Sheaves">Categories and Sheaves</a></em>, Grundlehren der Mathematischen Wissenschaften <strong>332</strong>, Springer (2006) [<a href="https://link.springer.com/book/10.1007/3-540-27950-4">doi:10.1007/3-540-27950-4</a>, <a href="https://www.maths.ed.ac.uk/~v1ranick/papers/kashiwara2.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Damien+Calaque">Damien Calaque</a>, <a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <em>Lectures on tensor categories</em>, IRMA Lectures in Mathematics and Theoretical Physics 12, 1-38 (2008) (<a href="https://arxiv.org/abs/math/0401246">arXiv:math/0401246</a>)</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>Topics in Lie theory and Tensor categories – 9 Tensor categories</em>, Lecture notes (spring 2009) (<a href="http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/MIT18_769S09_lec09.pdf">pdf</a> <a href="http://ocw.mit.edu/courses/mathematics/18-769-topics-in-lie-theory-tensor-categories-spring-2009/lecture-notes/">web</a>)</p> </li> <li id="EGNO15"> <p><a class="existingWikiWord" href="/nlab/show/Pavel+Etingof">Pavel Etingof</a>, <a class="existingWikiWord" href="/nlab/show/Shlomo+Gelaki">Shlomo Gelaki</a>, <a class="existingWikiWord" href="/nlab/show/Dmitri+Nikshych">Dmitri Nikshych</a>, <a class="existingWikiWord" href="/nlab/show/Victor+Ostrik">Victor Ostrik</a>, <em>Tensor Categories</em>, AMS Mathematical Surveys and Monographs <strong>205</strong> (2015) [<a href="https://bookstore.ams.org/surv-205">ISBN:978-1-4704-3441-0</a>, <a href="http://www-math.mit.edu/~etingof/egnobookfinal.pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alexei+Davydov">Alexei Davydov</a>: <em>Tensor categories</em>, in <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2311.05789">arXiv:2311.05789</a>]</p> </li> </ul> <p><a class="existingWikiWord" href="/nlab/show/Deligne%27s+theorem+on+tensor+categories">Deligne's theorem on tensor categories</a> 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>)</li> </ul> <p>Review in:</p> <ul> <li id="Ostrik04"><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>)</li> </ul> <p>On <a class="existingWikiWord" href="/nlab/show/quotient+object">quotients</a> of tensor categories:</p> <ul> <li>Zhenbang Zuo, Gongxiang Liu. <em>Quotient Category of a Multiring Category</em> (2024). (<a href="https://arxiv.org/abs/2403.06244">arXiv:2403.06244</a>).</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 24, 2024 at 17:08:20. 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