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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"> <p class="show_diff"> Showing changes from revision #9 to #10: <ins class="diffins">Added</ins> | <del class="diffdel">Removed</del> | <del class="diffmod">Chan</del><ins class="diffmod">ged</ins> </p> <div class='rightHandSide'> <div class='toc clickDown' tabindex='0'> <h3 id='context'>Context</h3> <h4 id='monoidal_categories'>Monoidal categories</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal categories</a></strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/enriched+monoidal+category'>enriched monoidal category</a>, <a class='existingWikiWord' href='/nlab/show/diff/tensor+category'>tensor category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+diagram'>string diagram</a>, <a class='existingWikiWord' href='/nlab/show/diff/tensor+network'>tensor network</a></p> </li> </ul> <p><strong>With braiding</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+category'>braided monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/balanced+monoidal+category'>balanced monoidal category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/twist'>twist</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/category+with+duals'>category with duals</a> (list of them)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/dualizable+object'>dualizable object</a> (what they have)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/rigid+monoidal+category'>rigid monoidal category</a>, a.k.a. <a class='existingWikiWord' href='/nlab/show/diff/rigid+monoidal+category'>autonomous category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/pivotal+category'>pivotal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spherical+category'>spherical category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/ribbon+category'>ribbon category</a>, a.k.a. <a class='existingWikiWord' href='/nlab/show/diff/ribbon+category'>tortile category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/symmetric+monoidal+dagger-category'>symmetric monoidal dagger-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/trace'>trace</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/closed+monoidal+category'>closed monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/cartesian+closed+category'>cartesian closed category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+category'>closed category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/cartesian+monoidal+category'>cartesian monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/semicartesian+monoidal+category'>semicartesian monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+category+with+diagonals'>monoidal category with diagonals</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/multicategory'>multicategory</a></p> </li> </ul> <p><strong>Semisimplicity</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/semisimple+category'>semisimple category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/fusion+category'>fusion category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/modular+tensor+category'>modular tensor category</a></p> </li> </ul> <p><strong>Morphisms</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>monoidal functor</a></p> <p>(<a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>lax</a>, <a class='existingWikiWord' href='/nlab/show/diff/oplax+monoidal+functor'>oplax</a>, <a class='existingWikiWord' href='/nlab/show/diff/monoidal+functor'>strong</a> <a class='existingWikiWord' href='/nlab/show/diff/bilax+monoidal+functor'>bilax</a>, <a class='existingWikiWord' href='/nlab/show/diff/Frobenius+monoidal+functor'>Frobenius</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+functor'>braided monoidal functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/monoid+in+a+monoidal+category'>monoid in a monoidal category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/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/diff/tensor+product'>tensor product</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/closed+monoidal+structure+on+presheaves'>closed monoidal structure on presheaves</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Day+convolution'>Day convolution</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/coherence+and+strictification+for+monoidal+categories'>coherence theorem for monoidal categories</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/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/diff/monoidal+bicategory'>monoidal 2-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/braided+monoidal+2-category'>braided monoidal 2-category</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+bicategory'>monoidal bicategory</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/cartesian+bicategory'>cartesian bicategory</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+monoidal+n-category'>k-tuply monoidal n-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/little+cubes+operad'>little cubes operad</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/monoidal+%28infinity%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/symmetric+monoidal+%28infinity%2C1%29-category'>symmetric monoidal (∞,1)-category</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+double+category'>compact double category</a></p> </li> </ul> </div> <h4 id='group_theory'>Group Theory</h4> <div class='hide'> <p><strong><a class='existingWikiWord' href='/nlab/show/diff/group+theory'>group theory</a></strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/group'>group</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/group+object'>group object</a>, <a class='existingWikiWord' href='/nlab/show/diff/groupoid+object+in+an+%28infinity%2C1%29-category'>group object in an (∞,1)-category</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/abelian+group'>abelian group</a>, <a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/super+abelian+group'>super abelian group</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/action'>group action</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-action'>∞-action</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/diff/infinity-representation'>∞-representation</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/progroup'>progroup</a></li> <li><a class='existingWikiWord' href='/nlab/show/diff/homogeneous+space'>homogeneous space</a></li> </ul> <p><strong>Classical groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/general+linear+group'>general linear group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/unitary+group'>unitary group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+unitary+group'>special unitary group</a>. <a class='existingWikiWord' href='/nlab/show/diff/projective+unitary+group'>projective unitary group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/orthogonal+group'>orthogonal group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/special+orthogonal+group'>special orthogonal group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symplectic+group'>symplectic group</a></p> </li> </ul> <p><strong>Finite groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/finite+group'>finite group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/symmetric+group'>symmetric group</a>, <a class='existingWikiWord' href='/nlab/show/diff/cyclic+group'>cyclic group</a>, <a class='existingWikiWord' href='/nlab/show/diff/braid+group'>braid group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/classification+of+finite+simple+groups'>classification of finite simple groups</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/sporadic+finite+simple+group'>sporadic finite simple groups</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Monster+group'>Monster group</a>, <a class='existingWikiWord' href='/nlab/show/diff/Mathieu+group'>Mathieu group</a></li> </ul> </li> </ul> <p><strong>Group schemes</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/algebraic+group'>algebraic group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/abelian+variety'>abelian variety</a></p> </li> </ul> <p><strong>Topological groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/topological+group'>topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+topological+group'>compact topological group</a>, <a class='existingWikiWord' href='/nlab/show/diff/locally+compact+topological+group'>locally compact topological group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/maximal+compact+subgroup'>maximal compact subgroup</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/string+group'>string group</a></p> </li> </ul> <p><strong>Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Lie+group'>Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/compact+Lie+group'>compact Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Kac-Moody+group'>Kac-Moody group</a></p> </li> </ul> <p><strong>Super-Lie groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/supergroup'>super Lie group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/super+Euclidean+group'>super Euclidean group</a></p> </li> </ul> <p><strong>Higher groups</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/crossed+module'>crossed module</a>, <a class='existingWikiWord' href='/nlab/show/diff/strict+2-group'>strict 2-group</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/n-group'>n-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group'>∞-group</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/simplicial+group'>simplicial group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/crossed+complex'>crossed complex</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/k-tuply+groupal+n-groupoid'>k-tuply groupal n-groupoid</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/spectrum'>spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/circle+n-group'>circle n-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/string+2-group'>string 2-group</a>, <a class='existingWikiWord' href='/nlab/show/diff/fivebrane+6-group'>fivebrane Lie 6-group</a></p> </li> </ul> <p><strong>Cohomology and Extensions</strong></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+cohomology'>group cohomology</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/group+extension'>group extension</a>,</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/infinity-group+extension'>∞-group extension</a>, <a class='existingWikiWord' href='/nlab/show/diff/Ext'>Ext-group</a></p> </li> </ul> <p><strong>Related concepts</strong></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/quantum+group'>quantum group</a></li> </ul> </div> </div> </div> <h1 id='contents'>Contents</h1> <div class='maruku_toc'><ul><li><a href='#definition'>Definition</a></li><li><a href='#properties'>Properties</a><ul><li><a href='#relation_to_picard_group'>Relation to Picard group</a></li></ul></li><li><a href='#related_concepts'>Related concepts</a></li><li><a href='#references'>References</a></li></ul></div> <h2 id='definition'>Definition</h2> <p>The <em><a class='existingWikiWord' href='/nlab/show/diff/Picard+groupoid'>Picard groupoid</a></em> <math class='maruku-mathml' display='inline' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_1' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>PIC</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>PIC(\mathcal{C}, \otimes)</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/diff/monoidal+category'>monoidal category</a> <math class='maruku-mathml' display='inline' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_2' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/diff/core+groupoid'>core</a> of its <a class='existingWikiWord' href='/nlab/show/diff/full+subcategory'>full subcategory</a> on those <a class='existingWikiWord' href='/nlab/show/diff/object'>objects</a> that are <a class='existingWikiWord' href='/nlab/show/diff/invertible+object'>invertible objects</a> under the <a class='existingWikiWord' href='/nlab/show/diff/tensor+product'>tensor product</a>. This inherits the monoidal structure from <math class='maruku-mathml' display='inline' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_3' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math> and hence becomes a <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-group</a>. This is the <strong>Picard 2-group</strong> of <math class='maruku-mathml' display='inline' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_4' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(\mathcal{C}, \otimes)</annotation></semantics></math>.</p> <p>In geometric contexts this is also called the <em><a class='existingWikiWord' href='/nlab/show/diff/Picard+stack'>Picard stack</a></em>.</p> <h2 id='properties'>Properties</h2> <h3 id='relation_to_picard_group'>Relation to Picard group</h3> <p>The <a class='existingWikiWord' href='/nlab/show/diff/decategorification'>decategorification</a> of the Picard 2-group, hence the <a class='existingWikiWord' href='/nlab/show/diff/group'>group</a> of connected components, is the ordinary <a class='existingWikiWord' href='/nlab/show/diff/Picard+group'>Picard group</a> <math class='maruku-mathml' display='inline' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_5' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pic</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>Pic(\mathcal{C}, \otimes)</annotation></semantics></math>.</p> <div class='maruku-equation'><math class='maruku-mathml' display='block' id='mathml_98aadc6fb45a7869a7f8166ed9cbd2d9263dc3ee_6' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pic</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>π</mi> <mn>0</mn></msub><mi>PIC</mi><mo stretchy='false'>(</mo><mi>𝒞</mi><mo>,</mo><mo>⊗</mo><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Pic(\mathcal{C}, \otimes) \simeq \pi_0 PIC(\mathcal{C}, \otimes) \,. </annotation></semantics></math></div> <h2 id='related_concepts'>Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Picard+3-group'>Picard 3-group</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/diff/Picard+infinity-group'>Picard ∞-group</a></p> </li> </ul> <h2 id='references'>References</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/diff/Alexei+Davydov'>Alexei Davydov</a>, <a class='existingWikiWord' href='/nlab/show/diff/Dmitri+Nikshych'>Dmitri Nikshych</a>, <em>Braided Picard groups and graded extensions of braided tensor categories</em>, (<a href='https://arxiv.org/abs/2006.08022'>arXiv:2006.08022</a>)</li> </ul><ins class='diffins'> </ins><ins class='diffins'><p>Early discussion of <a class='existingWikiWord' href='/nlab/show/diff/2-group'>2-groups</a> as Picard 2-groups:</p></ins><ins class='diffins'> </ins><ins class='diffins'><ul> <li id='Solian72'> <p><a class='existingWikiWord' href='/nlab/show/diff/Alexandru+Solian'>Alexandru Solian</a>, <em>Groupe dans une catégorie</em>, C. R. Acad. Sc. Paris <strong>275</strong> (1972) [[ark:/12148/bpt6k57310477/f7](https://gallica.bnf.fr/ark:/12148/bpt6k57310477/f7.item)]</p> </li> <li id='Solian80'> <p><a class='existingWikiWord' href='/nlab/show/diff/Alexandru+Solian'>Alexandru Solian</a>, <em>Coherence in categorical groups</em>, Communications in Algebra <strong>9</strong> 10 (1980) 1039-1057 [[doi:10.1080/00927878108822631](https://doi.org/10.1080/00927878108822631)]</p> </li> </ul></ins> <p> </p> </div> <div class="revisedby"> <p> Last revised on August 4, 2023 at 09:35:07. 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