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It does not yet have a dedicated thread; feel free to create one, giving it the same name as the title of this page" 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></title></head> <body> <div class="rightHandSide"> <div class="toc clickDown" tabindex="0"> <h3 id="context">Context</h3> <h4 id="higher_geometry">Higher geometry</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></strong> / <strong><a class="existingWikiWord" href="/nlab/show/derived+geometry">derived geometry</a></strong></p> <p><strong>Ingredients</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+topos+theory">higher topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></p> </li> </ul> <p><strong>Concepts</strong></p> <ul> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">little</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/structured+%28%E2%88%9E%2C1%29-topos">structured (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometry+%28for+structured+%28%E2%88%9E%2C1%29-toposes%29">geometry (for structured (∞,1)-toposes)</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+scheme">generalized scheme</a></p> </li> </ul> </li> <li> <p><strong>geometric <a class="existingWikiWord" href="/nlab/show/big+and+little+toposes">big</a> <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/function+algebras+on+%E2%88%9E-stacks">function algebras on ∞-stacks</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/geometric+%E2%88%9E-stacks">geometric ∞-stacks</a></li> </ul> </li> </ul> <p><strong>Constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</a>, <a class="existingWikiWord" href="/nlab/show/free+loop+space+object">free loop space object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid in a locally ∞-connected (∞,1)-topos</a> / <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+of+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">of a locally ∞-connected (∞,1)-topos</a></p> </li> </ul> <p><strong>Examples</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+algebraic+geometry">derived algebraic geometry</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%C3%A9tale+%28%E2%88%9E%2C1%29-site">étale (∞,1)-site</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Hochschild cohomology</a> of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/dg-scheme">dg-scheme</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/schematic+homotopy+type">schematic homotopy type</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+noncommutative+geometry">derived noncommutative geometry</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/noncommutative+geometry">noncommutative geometry</a></li> </ul> </li> <li> <p>derived smooth geometry</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a>, <a class="existingWikiWord" href="/nlab/show/differential+topology">differential topology</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derived+smooth+manifold">derived smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/dg-manifold">dg-manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+symplectic+geometry">higher symplectic geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Klein+geometry">higher Klein geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+Cartan+geometry">higher Cartan geometry</a></p> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Isbell+duality">Isbell duality</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Jones' theorem</a>, <a class="existingWikiWord" href="/nlab/show/Hochschild+cohomology">Deligne-Kontsevich conjecture</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+geometric+stacks">Tannaka duality for geometric stacks</a></p> </li> </ul> </div></div> <h4 id="cohesive_toposes">Cohesive <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Toposes</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></strong></p> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesive+homotopy+type+theory">cohesive homotopy type theory</a></strong></p> <p><strong>Backround</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/motivation+for+cohesive+toposes">motivation for cohesive toposes</a></p> </li> </ul> <p><strong>Definition</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+topos">locally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+topos">connected topos</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+topos">strongly connected topos</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">strongly ∞-connected (∞,1)-topos</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+topos">totally connected topos</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">totally ∞-connected (∞,1)-topos</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+topos">local topos</a> / <a class="existingWikiWord" href="/nlab/show/local+%28%E2%88%9E%2C1%29-topos">local (∞,1)-topos</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+topos">cohesive topos</a> / <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a></p> </li> </ul> <p><strong>Presentation over a site</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+connected+site">locally connected site</a> / <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+site">locally ∞-connected site</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/connected+site">connected site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-connected+site">∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strongly+connected+site">strongly connected site</a> / <a class="existingWikiWord" href="/nlab/show/strongly+%E2%88%9E-connected+site">strongly ∞-connected site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/totally+connected+site">totally connected site</a> / <a class="existingWikiWord" href="/nlab/show/totally+%E2%88%9E-connected+site">totally ∞-connected site</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/local+site">local site</a> / <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-local+site">∞-local site</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/cohesive+site">cohesive site</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-cohesive+site">∞-cohesive site</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/discrete+%E2%88%9E-groupoid">discrete ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/discrete+space">discrete space</a>, <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, <a class="existingWikiWord" href="/nlab/show/discrete+groupoid">discrete groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/D-topological+%E2%88%9E-groupoid">D-topological ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/topological+space">topological space</a>, <a class="existingWikiWord" href="/nlab/show/topological+groupoid">topological groupoid</a>, <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>, <a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a>, <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>, <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+2-groupoid">Lie 2-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebra">∞-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superalgebra">superalgebra</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/smooth+super+%E2%88%9E-groupoid">smooth super ∞-groupoid</a></strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/superpoint">superpoint</a>, <a class="existingWikiWord" href="/nlab/show/supergeometry">supergeometry</a>, <a class="existingWikiWord" href="/nlab/show/supermanifold">supermanifold</a>, <a class="existingWikiWord" href="/nlab/show/super+Lie+group">super Lie group</a>, <a class="existingWikiWord" href="/nlab/show/super+%E2%88%9E-groupoid">super ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/synthetic+differential+super+%E2%88%9E-groupoid">synthetic differential super ∞-groupoid</a></li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/cohesion+of+global-+over+G-equivariant+homotopy+theory">cohesion of global- over G-equivariant homotopy theory</a></strong></p> </li> </ul> </div></div> </div> </div> <h1 id='section_table_of_contents'>Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#GeneralIdea'>General idea</a></li> <li><a href='#InHigherPrequantumGeometryMotivationAndSurvey'>In higher prequantum geometry: motivation and survey</a></li> <ul> <li><a href='#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices'>Ordinary prequantum geometry in terms of automorphisms in slices</a></li> <li><a href='#TheNeedForHigherPrequantumBundles'>The need for higher prequantum bundles</a></li> <li><a href='#brief_recollection_higher_geometry'>Brief recollection: Higher geometry</a></li> <li><a href='#higher_atiyah_groupoids'>Higher Atiyah groupoids</a></li> <li><a href='#the_central_theorem_quantomorphism_group_extensions'>The central theorem: Quantomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group extensions</a></li> <li><a href='#examples__and__as_heisenberg_groups'>Examples: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math> as Heisenberg <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</a></li> </ul> </ul> <li><a href='#Definition'>Definition</a></li> <li><a href='#properties'>Properties</a></li> <ul> <li><a href='#equivalence_of_atiyahgroupoid_bisections_to_slice_automorphisms'>Equivalence of Atiyah-groupoid bisections to slice automorphisms</a></li> <ul> <li><a href='#propostition'>Propostition</a></li> </ul> <li><a href='#SequencesOfInclusionsOfGroupsOfBisections'>Sequences of inclusions of Atiyah-bisection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</a></li> </ul> <li><a href='#Examples'>Examples</a></li> <ul> <li><a href='#TheTraditionalAtiyagLieGroupoid'>The traditional Atiyah Lie groupoid</a></li> <li><a href='#TheTraditionalCourantLie2Algebroid'>The traditional Courant Lie 2-algebroid</a></li> <li><a href='#TheTraditionalQuantomorphismGroup'>The traditional quantomorphism group</a></li> <li><a href='#TheQuantomorphismNGroups'>The quantomorphism <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>-groups</a></li> <li><a href='#TheTraditionalHeisenbergGroup'>The traditional Heisenberg group</a></li> <li><a href='#TheHeisenbergnGroup'>The Heisenberg <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>-group</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>We discuss a refinement of the traditional notion of <em><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoids">Atiyah Lie groupoids</a></em> (the <a class="existingWikiWord" href="/nlab/show/Lie+groupoids">Lie groupoids</a> which are the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroids">Atiyah Lie algebroids</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/principal+bundles">principal bundles</a>) from <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a> and generally to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <ul> <li> <p><em><a href="#GeneralIdea">General idea</a></em></p> </li> <li> <p><em><a href="#InHigherPrequantumGeometryMotivationAndSurvey">In higher prequantum geometry: motivation and survey</a></em></p> </li> </ul> <h3 id="GeneralIdea">General idea</h3> <p>Briefly, 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 <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> 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/principal+%E2%88%9E-bundle">principal ∞-bundle</a>, its <em>higher Atiyah groupoid</em> is the <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">At(P)</annotation></semantics></math> such that the</p> <ul> <li> <p>object of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> is <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> </li> <li> <p>object of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> is the collection of all <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>-equivariant maps between all pairs of <a class="existingWikiWord" href="/nlab/show/fibers">fibers</a> of <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>.</p> </li> </ul> <p>In these vague words this is precisely the same description as for the traditional Atiyah groupoid. Definition <a class="maruku-ref" href="#HigherAtiyahGroupoid"></a> below makes precise what this means in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>.</p> <p>Besides generalizing the traditional definition to <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>, the notion of higher Atiyah groupoids also generalizes from <a class="existingWikiWord" href="/nlab/show/concrete+objects">concrete objects</a> such as <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> to general objects in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> (general <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a>, not necessarily “supported on points”), notably to <em><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a></em> for <a class="existingWikiWord" href="/nlab/show/cocycles">cocycles</a> in <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a>. For instance if we assume that the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a> and consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi mathvariant="normal">Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \in \mathrm{Grp}(\mathbf{H})</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/sylleptic+%E2%88%9E-group">sylleptic ∞-group</a></em>, then there is the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_{\mathrm{conn}}</annotation></semantics></math> of <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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a></em> and this is itself again a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a>. A <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> is equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}^2\mathbb{G})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> “without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>”. For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = U(1)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a> in <a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoids">smooth ∞-groupoids</a>, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math> classifies <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> without 2-form part: in parts of the literature this is known as “<a class="existingWikiWord" href="/nlab/show/bundle+gerbes">bundle gerbes</a> with connective structure but without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>”.</p> <p>So the general definition considered here assigns a higher Atiyah groupoid to a “bundle gerbe with connective structure but no <a class="existingWikiWord" href="/nlab/show/curving">curving</a>”. It turns out that this is the <em><a class="existingWikiWord" href="/nlab/show/Courant+2-groupoid">Courant 2-groupoid</a></em> which <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrates</a> the <a class="existingWikiWord" href="/nlab/show/standard+Courant+Lie+2-algebroid">standard Courant Lie 2-algebroid</a> traditionally induced by this data.</p> <p>The notion of higher Atiyah groupoids is more general still: the definition does not really require that the object fed into the construction is a plain <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a>. It may notably also be a genuine <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> (hence <em>with</em> “<a class="existingWikiWord" href="/nlab/show/curving">curving</a>”). We show below that the corresponding higher Atiyah groupoid is that groupoid object whose <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> is the <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a> of the principal <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-connection regarded as a <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a>.</p> <p>In summary, the <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometric</a> generalization of the notion of Atiyah groupoids unifies all three of the traditional notions of <a class="existingWikiWord" href="/nlab/show/Atiyah+groupoid">Atiyah groupoid</a>, of <a class="existingWikiWord" href="/nlab/show/Courant+2-groupoid">Courant 2-groupoid</a> and of <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> and refines each of these to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a>:</th><th>standard <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th>groupoid version of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>:</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_1"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_2"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_3"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbb{G}_{conn}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> without top-degree connection form</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a></td></tr> </tbody></table> </div> <p>At the same time the definition of higher Atiyah groupoids in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>, def. <a class="maruku-ref" href="#HigherAtiyahGroupoid"></a> below, is very simple, almost tautological, identifying it as a very fundamental notion in <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a>.</p> <h3 id="InHigherPrequantumGeometryMotivationAndSurvey">In higher prequantum geometry: motivation and survey</h3> <p>Higher Atiyah groupoids play a central role in <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a>.</p> <blockquote> <p>under construction</p> </blockquote> <h4 id="OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">Ordinary prequantum geometry in terms of automorphisms in slices</h4> <p>A sequence of time-honored traditional concepts in <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>/<a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/gauge+group">gauge group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebras">Lie algebras</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebra">Poisson Lie algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;">twisted <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a></td></tr> </tbody></table> <p>For instance in the <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electrically</a> <a class="existingWikiWord" href="/nlab/show/charged+particle">charted</a> <a class="existingWikiWord" href="/nlab/show/particle">particle</a> <a class="existingWikiWord" href="/nlab/show/sigma-model">sigma-model</a> we have a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+bundle">prequantum circle bundle</a> <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> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</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">\nabla</annotation></semantics></math> on a <a class="existingWikiWord" href="/nlab/show/cotangent+bundle">cotangent bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>=</mo><msup><mi>T</mi> <mo>*</mo></msup><mi>Y</mi></mrow><annotation encoding="application/x-tex">X = T^* Y</annotation></semantics></math> which is essentially the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of the <a class="existingWikiWord" href="/nlab/show/electromagnetic+field">electromagnetic field</a>-bundle on <a class="existingWikiWord" href="/nlab/show/target+space">target</a> <a class="existingWikiWord" href="/nlab/show/spacetime">spacetime</a> <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>. Its <em><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></em> is the group of <a class="existingWikiWord" href="/nlab/show/diffeomorphisms">diffeomorphisms</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>P</mi></mrow><annotation encoding="application/x-tex">P \stackrel{\simeq}{\to} P</annotation></semantics></math> of the total space of the prequantum bundle which preserve the connection (also called the <em><a class="existingWikiWord" href="/nlab/show/contactomorphism">contactomorphism</a></em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P,\nabla)</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/regular+contact+manifold">regular contact manifold</a>). For the following it is convenient to say this using the language of <em><a class="existingWikiWord" href="/nlab/show/moduli+stacks">moduli stacks</a></em>: we may regard <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 <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <a class="existingWikiWord" href="/nlab/show/sheaf">sheaf</a> on the <a class="existingWikiWord" href="/nlab/show/site">site</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> (a “<a class="existingWikiWord" href="/nlab/show/smooth+space">smooth space</a>”) and then moreover as a <a class="existingWikiWord" href="/nlab/show/representable+functor">representable</a> <a class="existingWikiWord" href="/nlab/show/stack">stack</a> on this site (a “<a class="existingWikiWord" href="/nlab/show/smooth+groupoid">smooth groupoid</a>”) and make use of the tautological existence of the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a>, which we write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)_{conn}</annotation></semantics></math> (we don’t need further details right now, but they can be found for instance at <em><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></em> for details). By definition this is such that for any <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 map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}U(1)_{conn}</annotation></semantics></math> is equivalently a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> and such that a <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo>→</mo><msub><mo>∇</mo> <mn>2</mn></msub></mrow><annotation encoding="application/x-tex">\eta \colon \nabla_1 \to \nabla_2</annotation></semantics></math> between two such maps is equivalently a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between two such connections. With this formulation a <a class="existingWikiWord" href="/nlab/show/quantomorphism">quantomorphism</a> of the <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</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">\nabla</annotation></semantics></math> is equivalently a diagram of the form as on the right of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & \searrow &\swArrow_{\eta}& \swarrow \\ && \mathbf{B}U(1)_{conn} } \right\} </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-category">(2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>, namely a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</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>X</mi><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \stackrel{\simeq}{\to} X</annotation></semantics></math> of the base space of the bundle together with a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connections">principal connections</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msup><mi>ϕ</mi> <mo>*</mo></msup><mo>∇</mo><mover><mo>→</mo><mo>≃</mo></mover><mo>∇</mo></mrow><annotation encoding="application/x-tex">\eta \colon \phi^* \nabla \stackrel{\simeq}{\to} \nabla</annotation></semantics></math>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> is naturally an (<a class="existingWikiWord" href="/nlab/show/infinite-dimensional+Lie+group">infinite dimensional</a>) <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a>. Its <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> is the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>. If <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 equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> itself (notably if it is a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>), then the sub-Lie algebra of that on the <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">invariant vectors</a> is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+Lie+algebra">Heisenberg Lie algebra</a> and the Lie group corresponding to that is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a>.</p> <p>One also says that a triangular diagram as above is an autoequivalence of the “modulating” map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}U(1)_{conn}</annotation></semantics></math> in the <em><a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-category">slice (2,1)-category</a></em> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)_{conn}</annotation></semantics></math>.</p> <p>Such autoequivalences in slices are familiar from basic concepts of <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a> theory. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>=</mo><mo stretchy="false">(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mover><mo>→</mo><mo>→</mo></mover><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} = (\mathcal{G}_1 \stackrel{\to}{\to} \mathcal{G}_0)</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>, we may regard the inclusion of its manifold of objects as an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> being a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>p</mi> <mi>𝒢</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo>→</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">p_\mathcal{G} \colon\mathcal{G}_0 \to \mathcal{G}</annotation></semantics></math>. Regarding this atlas as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-category">slice (2,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/stacks">stacks</a>/<a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> over <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">\mathcal{G}</annotation></semantics></math>, its autoequivalences are diagrams as on the right of</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>𝒢</mi></msub><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mi>ϕ</mi></mover></mtd> <mtd></mtd> <mtd><msub><mi>𝒢</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>𝒢</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(p_{\mathcal{G}}) = \left\{ \array{ \mathcal{G}_0 &&\stackrel{\phi}{\to}&& \mathcal{G}_0 \\ & \searrow &\swArrow_\eta & \swarrow \\ && \mathcal{G} } \right\} \,. </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</a> <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>𝒢</mi> <mn>0</mn></msub><mover><mo>→</mo><mo>≃</mo></mover><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\phi \colon \mathcal{G}_0 \stackrel{\simeq}{\to} \mathcal{G}_0</annotation></semantics></math> of the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/objects">objects</a> equipped with 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><mi>η</mi></mrow><annotation encoding="application/x-tex">\eta</annotation></semantics></math> whose component map is a <a class="existingWikiWord" href="/nlab/show/smooth+function">smooth function</a> that assigns to each point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>q</mi><mo>∈</mo><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">q \in \mathcal{G}_0</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> 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">\mathcal{G}</annotation></semantics></math> of the form <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>η</mi> <mi>q</mi></msub><mo lspace="verythinmathspace">:</mo><mi>q</mi><mo>→</mo><mi>ϕ</mi><mo stretchy="false">(</mo><mi>q</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta_q \colon q \to \phi(q)</annotation></semantics></math>. This collection of data is known as a <em><a class="existingWikiWord" href="/nlab/show/bisection">bisection</a></em> of a <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>. Bisections naturally form a group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><msub><mi>p</mi> <mi>𝒢</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{BiSect}(p_{\mathcal{G}})</annotation></semantics></math> , which is all the more manifest if we understand them as autoequivalences of the atlas in the slice, called the <a class="existingWikiWord" href="/nlab/show/group+of+bisections">group of bisections</a>.</p> <p>This perspective of regarding maps of <a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a> as objects in the slice over their codomain (an elementary step in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>/<a class="existingWikiWord" href="/nlab/show/%28infinity%2C1%29-topos+theory">higher topos theory</a>, but not common in traditional differential geometry) turns out to be useful and drives all of the refinements, generalizations and theorems that we discuss in the following: we will see that higher <a class="existingWikiWord" href="/nlab/show/prequantum+geometry">prequantum geometry</a> is essentially the geometry insice <a class="existingWikiWord" href="/nlab/show/slice+%28infinity%2C1%29-topos">higher slice categories</a> of <a class="existingWikiWord" href="/nlab/show/infinity-stack">higher stacks</a> over <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">higher moduli stacks</a> of <a class="existingWikiWord" href="/nlab/show/principal+infinity-connection">higher principal connections</a>.</p> <p>Before we get there, notice the following…</p> <h4 id="TheNeedForHigherPrequantumBundles">The need for higher prequantum bundles</h4> <p>The tools of <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> mainly apply to <a class="existingWikiWord" href="/nlab/show/quantum+mechanics">quantum mechanics</a> and only partially to <a class="existingWikiWord" href="/nlab/show/quantum+field+theory">quantum field theory</a>. In particular in the context of <em><a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a></em> in <a class="existingWikiWord" href="/nlab/show/dimension">dimension</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> a <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a> over the (<a class="existingWikiWord" href="/nlab/show/phase+space">phase</a>-)space of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a> is to be refined (de-<a class="existingWikiWord" href="/nlab/show/transgression">transgressed</a>) to a <em><a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a></em> over the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</a>. Therefore in order to apply <a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a> to <a class="existingWikiWord" href="/nlab/show/extended+prequantum+field+theory">extended prequantum field theory</a> to obtain <a class="existingWikiWord" href="/nlab/show/extended+quantum+field+theory">extended quantum field theory</a> we first need extended/higher prequantum geometry.</p> <p>For instance the <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 3-bundle</a> for standard <a class="existingWikiWord" href="/nlab/show/3d+Chern-Simons+theory">3d</a> <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> <a class="existingWikiWord" href="/nlab/show/Chern-Simons+theory">Chern-Simons theory</a> is modulated by the differential <a class="existingWikiWord" href="/nlab/show/smooth+first+fractional+Pontryagin+class">smooth first fractional Pontryagin class</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>Spin</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>forget</mi><mspace width="thickmathspace"></mspace><mi>connections</mi></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>1</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>realization</mi></mtd></mtr> <mtr><mtd><mi>B</mi><mi>Spin</mi></mtd> <mtd><mover><mo>→</mo><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mi>p</mi> <mn>1</mn></msub></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>4</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}Spin_{conn} &\stackrel{\tfrac{1}{2}\hat \mathbf{p}_1}{\to}& \mathbf{B}^3 U(1)_{conn} \\ \downarrow && \downarrow & forget \; connections \\ \mathbf{B}Spin &\stackrel{\tfrac{1}{2}\mathbf{p}_1}{\to}& \mathbf{B}^3 U(1) \\ \downarrow && \downarrow & geometric\;realization \\ B Spin &\stackrel{\tfrac{1}{2}p_1}{\to}& K(\mathbb{Z},4) } \,, </annotation></semantics></math></div> <p>modulating/clsasifying the universal <em><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+3-bundle+with+connection">Chern-Simons circle 3-bundle with connection</a></em> (also known as a <em><a class="existingWikiWord" href="/nlab/show/bundle+2-gerbe">bundle 2-gerbe</a></em>) over the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</a> of <a class="existingWikiWord" href="/nlab/show/field+%28physics%29">fields</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>-Chern-Simons theory, which is the moduli stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}G_{conn}</annotation></semantics></math> 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/principal+connection">principal connection</a>.</p> <p>Similarly the <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum 7-bundle</a> for <a class="existingWikiWord" href="/nlab/show/7d+Chern-Simons+theory">7d Chern-Simons theory</a> on <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <a class="existingWikiWord" href="/nlab/show/principal+infinity-connections">principal 2-connections</a> is given by the differential <a class="existingWikiWord" href="/nlab/show/smooth+second+fractional+Pontryagin+class">smooth second fractional Pontryagin class</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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>forget</mi><mspace width="thickmathspace"></mspace><mi>connections</mi></mtd></mtr> <mtr><mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mstyle mathvariant="bold"><mi>p</mi></mstyle> <mn>2</mn></msub></mrow></mover></mtd> <mtd><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>7</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mi>geometric</mi><mspace width="thickmathspace"></mspace><mi>realization</mi></mtd></mtr> <mtr><mtd><mi>B</mi><mi>String</mi></mtd> <mtd><mover><mo>→</mo><mrow><mfrac><mn>1</mn><mn>6</mn></mfrac><msub><mi>p</mi> <mn>2</mn></msub></mrow></mover></mtd> <mtd><mi>K</mi><mo stretchy="false">(</mo><mi>ℤ</mi><mo>,</mo><mn>8</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{B}String_{conn} &\stackrel{\frac{1}{6}\hat \mathbf{p}_2}{\to}& \mathbf{B}^7 U(1)_{conn} \\ \downarrow && \downarrow & forget\; connections \\ \mathbf{B}String &\stackrel{\frac{1}{6}\mathbf{p}_2}{\to}& \mathbf{B}^7 U(1) \\ \downarrow && \downarrow & geometric\; realization \\ B String &\stackrel{\frac{1}{6}p_2}{\to}& K(\mathbb{Z},8) } \,, </annotation></semantics></math></div> <p>modulating/classifying the universal <em><a class="existingWikiWord" href="/nlab/show/Chern-Simons+circle+7-bundle+with+connection">Chern-Simons circle 7-bundle with connection</a></em> over the moduli 2-stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>String</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}String_{conn}</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <a class="existingWikiWord" href="/nlab/show/principal+infinity-connection">principal 2-connections</a>.</p> <p>Therefore we want to lift the <a href="#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">above</a> table of traditional notions to <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>…</p> <h4 id="brief_recollection_higher_geometry">Brief recollection: Higher geometry</h4> <p>In order to say this, clearly we need some basics of <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</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></mtd> <mtd></mtd> <mtd><mi>Groupoids</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><msup><mo>↘</mo> <mpadded width="0"><mi>nerve</mi></mpadded></msup></mtd></mtr> <mtr><mtd><mi>Categories</mi></mtd> <mtd></mtd> <mtd></mtd> <mtd></mtd> <mtd><mi>Kan</mi><mi>complexes</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mo>−</mo><mi>Categories</mi></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ && Groupoids \\ & \swarrow && \searrow^{\mathrlap{nerve}} \\ Categories && && Kan complexes \\ & \searrow && \swarrow \\ && (\infty,1)-Categories } \,. </annotation></semantics></math></div> <p>Important construction principle for <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categories">(∞,1)-categories</a>: <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</a>. 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/category">category</a> with some subset of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>W</mi><mo>↪</mo><mi>Mor</mi><mo stretchy="false">(</mo><mi>𝒞</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">W \hookrightarrow Mor(\mathcal{C})</annotation></semantics></math> declared to be “<a class="existingWikiWord" href="/nlab/show/weak+equivalences">weak equivalences</a>”, the <a class="existingWikiWord" href="/nlab/show/simplicial+localization">simplicial localization</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>W</mi></msub><mi>𝒞</mi><mo>∈</mo><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo><mi>Cat</mi></mrow><annotation encoding="application/x-tex"> L_W \mathcal{C} \in (\infty,1)Cat </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/universal+construction">universal</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-category obtained from <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> by universally turning each weak equivalence into an actual <a class="existingWikiWord" href="/nlab/show/homotopy+equivalence">homotopy equivalence</a> in the sense of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>In particular 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">C</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/site">site</a>, assumed for simplicity to have <a class="existingWikiWord" href="/nlab/show/point+of+a+topos">enough points</a>. Declare then that in the <a class="existingWikiWord" href="/nlab/show/functor+category">functor category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Func(C^{op}, KanCplx)</annotation></semantics></math>, hence in <a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a>-valued presheaves, the weak equivalences are the <a class="existingWikiWord" href="/nlab/show/stalk">stalkwise</a> <a class="existingWikiWord" href="/nlab/show/homotopy+equivalences">homotopy equivalences</a> of Kan complexes. Then</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo><msub><mi>Sh</mi> <mn>∞</mn></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>L</mi> <mi>W</mi></msub><mi>Func</mi><mo stretchy="false">(</mo><msup><mi>C</mi> <mi>op</mi></msup><mo>,</mo><mi>KanCplx</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H} \coloneqq Sh_{\infty}(C) \coloneqq L_{W} Func(C^{op}, KanCplx) </annotation></semantics></math></div> <p>is called the <em><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></em> of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-sheaves">(∞,1)-sheaves</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stacks">∞-stacks</a> 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">C</annotation></semantics></math>.</p> <p>An <a class="existingWikiWord" href="/nlab/show/A-%E2%88%9E+algebra">A-∞ algebra</a>-object <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> in such an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos such that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\pi_0(G)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/group">group</a> is called an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> “with geometric structure as encoded by the test spaces <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 canonical source of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups are the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+products">homotopy fiber products</a> of point inclusions <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">* \to X</annotation></semantics></math> of any object X, the <a class="existingWikiWord" href="/nlab/show/loop+space+object">loop space object</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><mi>X</mi><mo>≔</mo><mo>*</mo><munder><mo>×</mo><mi>X</mi></munder><mo>*</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Omega X \coloneqq {*} \underset{X}{\times} {*} \,. </annotation></semantics></math></div> <p>In fact this are <em>all</em> the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> that there are, up to equivalence: forimg <a class="existingWikiWord" href="/nlab/show/loop+space+objects">loop space objects</a> is an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mover><munderover><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo>≃</mo></munderover><mover><mo>←</mo><mi>Ω</mi></mover></mover><msubsup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo>≥</mo><mn>1</mn></mrow> <mrow><mo>*</mo><mo stretchy="false">/</mo></mrow></msubsup></mrow><annotation encoding="application/x-tex"> Grp(\mathbf{H}) \stackrel{\overset{\Omega}{\leftarrow}}{\underoverset{\mathbf{B}}{\simeq}{\to}} \mathbf{H}^{*/}_{\geq 1} </annotation></semantics></math></div> <p>between <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a> and <a class="existingWikiWord" href="/nlab/show/pointed+object">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected+object+in+an+%28%E2%88%9E%2C1%29-topos">connected</a> objects. The inverse equivalence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}</annotation></semantics></math> is the <em><a class="existingWikiWord" href="/nlab/show/delooping">delooping</a></em> operation.</p> <p>We say that such an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-topos <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is <em><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a></em> if it is equipped with an <a class="existingWikiWord" href="/nlab/show/adjoint+triple">adjoint triple</a> of <a class="existingWikiWord" href="/nlab/show/idempotent+monad">idempotent</a> (co)/<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-monads">(∞,1)-monads</a></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/shape+modality">shape modality</a></th><th></th><th><a class="existingWikiWord" href="/nlab/show/flat+modality">flat modality</a></th><th></th><th><a class="existingWikiWord" href="/nlab/show/sharp+modality">sharp modality</a></th></tr></thead><tbody><tr><td style="text-align: left;">idemp. monad</td><td style="text-align: left;"></td><td style="text-align: left;">idemp. comonad</td><td style="text-align: left;"></td><td style="text-align: left;">idemp. monad</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>Π</mi></mrow><annotation encoding="application/x-tex">\Pi</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><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</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><mo>♭</mo></mrow><annotation encoding="application/x-tex">\flat</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><mo>⊣</mo></mrow><annotation encoding="application/x-tex">\dashv</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><mo>♯</mo></mrow><annotation encoding="application/x-tex">\sharp</annotation></semantics></math></td></tr> </tbody></table> <p>This implies (strictly speaking we need <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a> for that, coming from another adjoint triple of (co)monads) that for every <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \in Grp(\mathbf{H})</annotation></semantics></math> there is a canonical object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> which modulats <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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a>.</p> <h4 id="higher_atiyah_groupoids">Higher Atiyah groupoids</h4> <p>Looking at the <a href="#OrdinaryPrequantumGeometryInTermsofAutomorphismsInSlices">above</a> table and noticing the <a href="#TheNeedForHigherPrequantumBundles">above</a> need for higher prequantum bundles, we should try to find an analogous table of concepts in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a>, something like this:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice</a>-<a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-groups">automorphism ∞-groups</a> in <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heisenberg+L-%E2%88%9E+algebra">Heisenberg L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+L-%E2%88%9E+algebra">Poisson L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+L-%E2%88%9E+algebra">Courant L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;">twisted vector fields</td></tr> </tbody></table> </div> <p>(…)</p> <p>The way all these notions and theorems work is by considering <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-groups">automorphism ∞-groups</a> of the classifying (or rather: modulating) maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/prequantum+%E2%88%9E-bundle">prequantum ∞-bundle</a> in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over the domain. For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{QuantMorph}(\nabla) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \,. </annotation></semantics></math></div> <p>The others are obtained by succesively forgetting connection data. For instance</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo lspace="0em" rspace="thinmathspace">BiSect</mo><mo stretchy="false">(</mo><mi>Cou</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \BiSect(Cou(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_1}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_1}} \\ && \mathbf{B}(\mathbf{B}\mathbb{G}_{conn}) } \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 lspace="0em" rspace="thinmathspace">BiSect</mo><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \BiSect(At(\nabla)) = \left\{ \array{ X && \underoverset{\simeq}{\phi}{\to} && X \\ & {}_{\mathllap{\nabla_0}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla_0}} \\ && \mathbf{B}\mathbb{G} } \right\} \,. </annotation></semantics></math></div> <p>The extension sequence is then schematically simply the following</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>{</mo><mrow><mtable><mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mo>∇</mo></mpadded></msub><mo>↘</mo></mtd> <mtd><mo>⇙</mo></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mo>∇</mo></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mtd></mtr></mtable></mrow><mo>}</mo></mrow><mspace width="thickmathspace"></mspace><mo>→</mo><mspace width="thickmathspace"></mspace><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mo>→</mo><mo>≃</mo></mover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow><mo>}</mo></mrow></mrow><annotation encoding="application/x-tex"> \left\{ \array{ && X \\ & \swarrow & & \searrow \\ & \searrow &\swArrow& \swarrow \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X &&\stackrel{\simeq}{\to}&& X \\ & {}_{\mathllap{\nabla}}\searrow &\swArrow& \swarrow_{\mathrlap{\nabla}} \\ && \mathbf{B}\mathbb{G}_{conn} } \right\} \; \to \; \left\{ \array{ X && \stackrel{\simeq}{\to} && X } \right\} </annotation></semantics></math></div> <p>in this generality this now includes various other notions, too:</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a>:</th><th>standard <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th>groupoid version of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>:</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_1"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_2"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_3"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbb{G}_{conn}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> without top-degree connection form</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a></td></tr> </tbody></table> </div> <h4 id="the_central_theorem_quantomorphism_group_extensions">The central theorem: Quantomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group extensions</h4> <div class="num_theorem"> <h6 id="theorem">Theorem</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">\mathbb{G}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> a higher prequantum geometry with respect to <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">\mathbb{G}</annotation></semantics></math> there is a long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mo>(</mo><mi>Ω</mi><mi>𝔾</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mrow><mo>(</mo><mo>∇</mo><mo>)</mo></mrow><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mrow><mo>(</mo><mrow><mo>(</mo><mi>Ω</mi><mi>𝔾</mi><mo>)</mo></mrow><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mrow><mo>(</mo><mo>∇</mo><mo>)</mo></mrow><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}\left(\left(\Omega \mathbb{G}\right)\mathbf{FlatConn}\left(\nabla\right) \right) \,. </annotation></semantics></math></div> <p>Similarly there is the <a class="existingWikiWord" href="/nlab/show/Heisenberg+infinity-group">Heisenberg infinity-group</a> extension</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{Heis}(\nabla) \to G </annotation></semantics></math></div></div> <div class="num_theorem"> <h6 id="theorem_2">Theorem</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> sequence of prop. <a class="maruku-ref" href="#QuantomorphismExtension"></a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>𝒳</mi> <mi>Ham</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mi>ω</mi></mrow></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \to \mathfrak{Poisson}(X,\omega) \to \mathcal{X}_{Ham}(X,\omega) \stackrel{\iota_{(-)\omega}}{\to} \mathbf{B}\mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \,, </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><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Poisson}(X,\omega)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a> as defined in (<a href="#Rogers11">Rogers 11</a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒳</mi> <mi>Ham</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{X}_{Ham}</annotation></semantics></math> is the Lie algebra of <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a> restricted to the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \flat (\mathbf{B}^{n-1})\mathbb{R})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> for flat <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> in the given degree, regarded as an abelian <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>.</p> </li> </ul> </div> <p>The following table shows what this sequence reduces to when one chooses <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = \mathbf{B}^{n-1}U(1)</annotation></semantics></math>.</p> <div> <p><strong>higher and integrated <a class="existingWikiWord" href="/nlab/show/Kostant-Souriau+extensions">Kostant-Souriau extensions</a></strong>:</p> <p>(<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a> 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla) </annotation></semantics></math></div> <table><thead><tr><th><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><th>geometry</th><th>structure</th><th>unextended structure</th><th>extension by</th><th>quantum extension</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><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega \mathbb{G})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a> under <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+group">Hamiltonian symplectomorphism group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphism">Hamiltonian 2-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism 2-group</a></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>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a></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>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphisms">Hamiltonian n-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></td></tr> </tbody></table> <p>(extension are listed for sufficiently connected <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> <h4 id="examples__and__as_heisenberg_groups">Examples: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi></mrow><annotation encoding="application/x-tex">String</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Fivebrane</mi></mrow><annotation encoding="application/x-tex">Fivebrane</annotation></semantics></math> as Heisenberg <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</h4> <div class="num_example"> <h6 id="example">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 simply connected semisimple compact Lie group such as the <a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo>≔</mo><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">]</mo><mo>)</mo></mrow><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>G</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex"> \nabla \coloneqq \exp\left(2 \pi i \int_{S^1} [S^1, \tfrac{1}{2}\hat \mathbf{p}_1]\right) \;\colon\; G \to \mathbf{B}^2 U(1)_{conn} </annotation></semantics></math></div> <p>be the canonical <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> over it. Then the <a class="existingWikiWord" href="/nlab/show/Heisenberg+infinity-group">Heisenberg 2-group</a> <a class="existingWikiWord" href="/nlab/show/infinity-group+extension">extension</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><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex"> U(1)\mathbf{FlatConn}(G) \to \mathbf{Heis}(\nabla) \to G </annotation></semantics></math></div> <p>is the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> extension</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1) \to String(G) \to G \,. </annotation></semantics></math></div></div> <p>(by classification of extensions by cohomology… by Lie 2-algebra computation…)</p> <p>(and analogously for <a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a>…)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>6</mn></msup><mi>U</mi><mrow><mo>(</mo><mn>1</mn><mo>)</mo></mrow><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mrow><mo>(</mo><mi>exp</mi><mrow><mo>(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mrow><mo>[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo>]</mo></mrow><mo>)</mo></mrow><mo>)</mo></mrow><mo>→</mo><mi>String</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}^6 U\left(1\right) \to \mathbf{Heis}\left(\exp\left(2 \pi i \int_{S^1} \left[S^1, \tfrac{1}{2}\hat \mathbf{p}_2\right] \right)\right) \to String </annotation></semantics></math></div> <h2 id="Definition">Definition</h2> <p>We now turn to the formal definition of higher Atiyah groupoids and the basic constructions on them.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>.</p> <p>We define now for every <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/principal+%E2%88%9E-bundle">principal ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> a <em><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a></em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">At(P) \in Grpd(\mathbf{H})</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. In order to do so we invoke two basic facts.</p> <div class="num_prop" id="EffectiveEpisAreEquivalentlyGroupoids"> <h6 id="proposition">Proposition</h6> <p>The construction of forming the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> consitutes an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a> from that of <a class="existingWikiWord" href="/nlab/show/1-epimorphisms">1-epimorphisms</a> to that of <a class="existingWikiWord" href="/nlab/show/groupoid+objects+in+an+%28%E2%88%9E%2C1%29-category">groupoid objects</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{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><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msup><msub><mo stretchy="false">)</mo> <mrow><mn>1</mn><mi>epi</mi></mrow></msub><mover><mo>→</mo><mo>≃</mo></mover><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\mathbf{H}^{\Delta^1})_{1epi} \stackrel{\simeq}{\to} Grpd(\mathbf{H}) \,. </annotation></semantics></math></div></div> <p>This is a refined version of one of the <a class="existingWikiWord" href="/nlab/show/Giraud-Rezk-Lurie+axioms">Giraud-Rezk-Lurie axioms</a> characterizing <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, discussed at <em><a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object in an (∞,1)-category</a></em>.</p> <div class="num_remark"> <h6 id="remark">Remark</h6> <p>In terms of traditional terminology in the literature on <a class="existingWikiWord" href="/nlab/show/topological+stacks">topological stacks</a>/<a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a> etc, this says that a groupoid object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> is equivalently an object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X \in \mathbf{H}</annotation></semantics></math> which is equipped with an <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">X_0 \to X</annotation></semantics></math>.</p> </div> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}G \in \mathbf{H}</annotation></semantics></math> for the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> (the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</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/principal+%E2%88%9E-bundles">principal ∞-bundles</a>, as the following proposition asserts:</p> <div class="num_prop" id="ClassificationOfGPrincipalBundles"> <h6 id="proposition_2">Proposition</h6> <p>The operation of forming <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-fibers">(∞,1)-fibers</a> (<a class="existingWikiWord" href="/nlab/show/homotopy+fibers">homotopy fibers</a>) constitutes an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%E2%88%9E-groupoids">equivalence of ∞-groupoids</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>fib</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mi>Bund</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> fib \;\colon\; \mathbf{H}(X, \mathbf{B}G) \to G Bund(X) \,. </annotation></semantics></math></div></div> <p>This is discussed at <em><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></em>.</p> <p>Using these two facts we now set:</p> <div class="num_defn" id="HigherAtiyahGroupoid"> <h6 id="definition_2">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> 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/principal+%E2%88%9E-bundle">principal ∞-bundle</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, its <strong>Atiyah groupoid</strong> is the <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo>∈</mo><mi mathvariant="normal">Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mo>≃</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><msup><mi>Δ</mi> <mn>1</mn></msup></mrow></msup><msub><mo stretchy="false">)</mo> <mrow><mn>1</mn><mi>epi</mi></mrow></msub></mrow><annotation encoding="application/x-tex">At \in \mathrm{Grpd}(\mathbf{H}) \simeq (\mathbf{H}^{\Delta^1})_{1epi}</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/1-image">1-image</a> projection of the classifying map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g \colon X \to \mathbf{B}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>g</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mrow></mrow></mover><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>↪</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> g \;\colon\; X \stackrel{}{\to} At(P) \coloneqq im_1(g) \hookrightarrow \mathbf{B}G \,. </annotation></semantics></math></div></div> <div class="num_remark" id="AtiyahGroupoidIsCechNerve"> <h6 id="remark_2">Remark</h6> <p>By the discussion at <em><a class="existingWikiWord" href="/nlab/show/1-image">1-image</a></em>, the 1-image projection of any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> is equivalently given as the canonical map given by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-colimit">(∞,1)-colimit</a> over the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><msub><mi>im</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≃</mo><munder><mi>lim</mi><mo>→</mo></munder><mo stretchy="false">(</mo><msup><mi>X</mi> <mrow><msubsup><mo>×</mo> <mi>Y</mi> <mrow><mo>•</mo><mo lspace="verythinmathspace" rspace="0em">+</mo><mn>1</mn></mrow></msubsup></mrow></msup><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \to im_1(f) \simeq \underset{\rightarrow}{\lim} (X^{\times^{\bullet+1}_Y}) \,. </annotation></semantics></math></div> <p>This means that regarded as an object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd(\mathbf{H})</annotation></semantics></math>, the Atiyah groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">At(P)</annotation></semantics></math> <em>is</em> simply the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of the classifying map. This means that the definition of Atiyah groupoids in <a class="existingWikiWord" href="/nlab/show/higher+geometry">higher geometry</a> is much more fundamental than in traditional <a class="existingWikiWord" href="/nlab/show/geometry">geometry</a>.</p> </div> <h2 id="properties">Properties</h2> <h3 id="equivalence_of_atiyahgroupoid_bisections_to_slice_automorphisms">Equivalence of Atiyah-groupoid bisections to slice automorphisms</h3> <p>We discuss how the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> of a higher Atiyah groupoid is canonically equivalent to the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> of the modulating map that gave rise to it, regarded as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> over its <a class="existingWikiWord" href="/nlab/show/codomain">codomain</a>.</p> <p>To this end we need the following two definitions</p> <div class="num_defn" id="HValuedAutomorphismGroup"> <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>f</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, its <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Aut}_{\mathbf{H}}(f)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> over <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> over the <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> 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> regarded as an object in the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/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><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mi>Y</mi></munder><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mrow><mo stretchy="false">/</mo><mi>Y</mi></mrow></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Aut}_{\mathbf{H}}(f) \coloneqq \underset{Y}{\prod} \mathbf{Aut}_{/Y}(f) \,. </annotation></semantics></math></div></div> <div class="num_remark" id="Concretification"> <h6 id="remark_3">Remark</h6> <p>For <a class="existingWikiWord" href="/nlab/show/concrete+object">non-concrete</a> codomains <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> one is usually interested in the <a class="existingWikiWord" href="/nlab/show/concretification">concretification</a> of this group. To be discussed… For an example see at <em><a href="#TheQuantomorphismNGroups">The quantomorphism <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>-group</a></em> below.</p> </div> <div class="num_prop" id="HValuedAutomorphismAsFiber"> <h6 id="proposition_3">Proposition</h6> <p>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>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">f \colon X \to Y</annotation></semantics></math> a morphism in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, its <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued slice automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group according to prop. <a class="maruku-ref" href="#HValuedAutomorphismGroup"></a> sits in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> <a class="existingWikiWord" href="/nlab/show/diagram">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><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>f</mi><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⊢</mo><mi>f</mi></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{Aut}_{\mathbf{H}}(f) &\to& \mathbf{Aut}(X) \\ \downarrow^{\mathrlap{}} && \downarrow^{\mathrlap{f \circ (-)}} \\ {*} &\stackrel{\vdash f}{\to}& [X,Y] } \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>↪</mo><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>X</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">\mathbf{Aut}(X) \hookrightarrow [X,X]</annotation></semantics></math> is the ordinary <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> 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> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>.</p> </div> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝒢</mi><mo>∈</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathcal{G} \in Grpd(\mathbf{H})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, its <strong><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo><mo>∈</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{BiSect}(\mathcal{G}) \in Grpd(\mathbf{H})</annotation></semantics></math> is the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group, def. <a class="maruku-ref" href="#HValuedAutomorphismGroup"></a>, of the <a class="existingWikiWord" href="/nlab/show/atlas">atlas</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mi>𝒢</mi></msub><mo lspace="verythinmathspace">:</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo>→</mo><mi>𝒢</mi></mrow><annotation encoding="application/x-tex">\phi_{\mathcal{G}} \colon \mathcal{G}_0 \to \mathcal{G}</annotation></semantics></math> 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">\mathcal{G}</annotation></semantics></math> under prop. <a class="maruku-ref" href="#EffectiveEpisAreEquivalentlyGroupoids"></a>:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>𝒢</mi><mo stretchy="false">)</mo><mo>≔</mo><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><msub><mi>ϕ</mi> <mi>𝒢</mi></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(\mathcal{G}) \coloneqq \mathbf{Aut}_{\mathbf{H}}(\phi_{\mathcal{G}}) \,. </annotation></semantics></math></div></div> <p>With this the following proposition is immediate, but important for the interpretation of higher Atiyah groupoids:</p> <div class="num_prop"> <h6 id="propostition">Propostition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>c</mi><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>F</mi></mstyle></mrow><annotation encoding="application/x-tex">c \;\colon\; X \to \mathbf{F}</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, modulating an <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>, there is an canonical <a class="existingWikiWord" href="/nlab/show/equivalence+in+an+%28infinity%2C1%29-category">equivalence</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mi>c</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(At(P)) \simeq \mathbf{Aut}_{\mathbf{H}}(c) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_4">Remark</h6> <p>We may read this as saying that the higher Atiyah groupoid of an <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a> is the <a class="existingWikiWord" href="/nlab/show/universal+property">universal</a> solution to the problem of finding a <a class="existingWikiWord" href="/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category">groupoid object</a> whose <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> reproduces a given slice <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a>. In many applications, this is indeed the crucial property that drives the interest in higher Atiyah groupoids, see the <em><a href="#Examples">Examples</a></em> below.</p> </div> <h3 id="SequencesOfInclusionsOfGroupsOfBisections">Sequences of inclusions of Atiyah-bisection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> which is <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive</a>. As discussed there, this implies that there is an internal notion of <a class="existingWikiWord" href="/nlab/show/differential+cohomology">differential cohomology</a> and in particular of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>. We note here how the canonical forgetful maps between <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a> of <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundles">principal ∞-bundles</a> equipped with differing degree of differential refinement induce canonical inclusions of the corresponding higher Atiyah groupoids.</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo><mi>be</mi><mi>a</mi></mrow><annotation encoding="application/x-tex">\mathbb{G} \in Grp(\mathbf{H}) be a </annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a>. Then there exists, by <a class="existingWikiWord" href="/nlab/show/cohesion">cohesion</a>, a canonical notion 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a>, whose <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> we denote <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_{\mathrm{conn}}</annotation></semantics></math>. This is equipped with a canonical map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex"> \mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G} </annotation></semantics></math></div> <p>which “forgets the connection”. Then for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> we write</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>0</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mi>X</mi><mover><mo>→</mo><mo>∇</mo></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex"> \nabla_0 \;\colon\; X \stackrel{\nabla}{\to} \mathbf{B}\mathbb{G}_{conn} \to \mathbf{B}\mathbb{G} </annotation></semantics></math></div> <p>for the corresponding underlying map.</p> <div class="num_prop" id="InclusionOfNableBisectionsIntoNable0Bisections"> <h6 id="proposition_4">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along this map induces a canonical map of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(At(\nabla)) \to \mathbf{BiSect}(At(\nabla_0)) \,. </annotation></semantics></math></div></div> <p>If we regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo></mrow><annotation encoding="application/x-tex">\nabla</annotation></semantics></math> as a <a class="existingWikiWord" href="/nlab/show/prequantum+n-bundle">prequantum n-bundle</a> then this is a canonical inclusion of the <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a> into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_0</annotation></semantics></math>-twisted diffeomorphisms”.</p> <p>If <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">\mathbb{G}</annotation></semantics></math> is even a <a class="existingWikiWord" href="/nlab/show/sylleptic+%E2%88%9E-group">sylleptic ∞-group</a>, then the above moduli <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks have a further <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> and we obtain a 2-step sequence of forgetful maps</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>𝔾</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 \mathbb{G}_{conn} \to \mathbf{B}(\mathbf{B}\mathbb{G}_{conn}) \to \mathbf{B}^2 \mathbb{G} \,. </annotation></semantics></math></div> <p>Accordingly:</p> <div class="num_prop" id="InclusionOfNableBisectionsIntoNable1AndNabla0Bisections"> <h6 id="proposition_5">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along these maps induces inclusions of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-groups</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>BiSect</mi></mstyle><mo stretchy="false">(</mo><mi>At</mi><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>0</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{BiSect}(At(\nabla)) \to \mathbf{BiSect}(At(\nabla_1)) \to \mathbf{BiSect}(At(\nabla_0)) \,. </annotation></semantics></math></div></div> <div class="num_remark"> <h6 id="remark_5">Remark</h6> <p>This now interprets as the inclusion</p> <ol> <li> <p>of the <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a> into the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group of bisections of the <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group of bisections of the <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a> into that of “<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_0</annotation></semantics></math>-twisted diffeomorphisms”.</p> </li> </ol> </div> <p>In summary we have the following table of inclusions</p> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice</a>-<a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-groups">automorphism ∞-groups</a> in <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</a>:</th><th><a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></th><th><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-bisections">∞-bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Heisenberg+L-%E2%88%9E+algebra">Heisenberg L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+L-%E2%88%9E+algebra">Poisson L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Courant+L-%E2%88%9E+algebra">Courant L-∞ algebra</a></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>↪</mo></mrow><annotation encoding="application/x-tex">\hookrightarrow</annotation></semantics></math></td><td style="text-align: left;">twisted vector fields</td></tr> </tbody></table> </div> <p>See below at <em><a href="#TheTraditionalCourantLie2Algebroid">Examples– The traditional Courant Lie 2-algebroid</a></em> for more on this.</p> <h2 id="Examples">Examples</h2> <p>We first show how the general notion of <em>higher Atiyah groupoid</em> reproduces various traditonal structures.</p> <ul> <li> <p><em><a href="#TheTraditionalAtiyagLieGroupoid">The traditonal Atiyah Lie groupoid</a></em></p> </li> <li> <p><em><a href="#TheTraditionalCourantLie2Algebroid">The traditional Courant Lie 2-algebroid</a></em></p> </li> <li> <p><em><a href="#TheTraditionalQuantomorphismGroup">The traditional quantomorphism group</a></em></p> </li> <li> <p><em><a href="#TheQuantomorphismNGroups">The quantomorphism n-groups</a></em></p> </li> <li> <p><em><a href="#TheTraditionalHeisenbergGroup">The traditional Heisenberg group</a></em></p> </li> <li> <p><em><a href="#TheHeisenbergnGroup">The Heisenberg n-group</a></em></p> </li> </ul> <h3 id="TheTraditionalAtiyagLieGroupoid">The traditional Atiyah Lie groupoid</h3> <p>We discuss how the traditional notion of <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoids">Atiyah Lie groupoids</a> in traditional <a class="existingWikiWord" href="/nlab/show/differential+geometry">differential geometry</a> is a special case of higher Atiyah groupoids of def. <a class="maruku-ref" href="#HigherAtiyahGroupoid"></a>.</p> <p>To set this up we take the ambient <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> to be</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>≔</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} \coloneqq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and make use of the canonical embeddings</p> <p><a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/diffeological+space">DiffeologicalSpace</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/smooth+spaces">SmoothSpace</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>,</p> <p>and</p> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">LieGrpd</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">\simeq</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a></p> <p>which are understood in the following.</p> <div class="num_prop"> <h6 id="proposition_6">Proposition</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/Lie+group">Lie group</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> 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>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> 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/principal+bundle">principal bundle</a>, the traditional <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> of <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> is equivalent to that of def. <a class="maruku-ref" href="#HigherAtiyahGroupoid"></a>.</p> </div> <div class="proof"> <h6 id="proof">Proof</h6> <p>Write <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">g \colon X \to \mathbf{B}G</annotation></semantics></math> for the classifying map of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math>, by prop. <a class="maruku-ref" href="#ClassificationOfGPrincipalBundles"></a>.</p> <p>By remark <a class="maruku-ref" href="#AtiyahGroupoidIsCechNerve"></a> the higher Atiyah groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">At(P)</annotation></semantics></math> is simply the <a class="existingWikiWord" href="/nlab/show/Cech+nerve">Cech nerve</a> of this map. Since <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> and <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 hence <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> are all <a class="existingWikiWord" href="/nlab/show/truncated+object+in+an+%28infinity%2C1%29-category">0-truncated objects</a>, hence <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/1-truncated">1-truncated</a> object, this Cech nerve is <a class="existingWikiWord" href="/nlab/show/coskeleton">2-coskeletal</a> and hence is sufficient to consider the first three degrees. By definition these are</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo><mo>≃</mo><mrow><mo>(</mo><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><mover><mo>→</mo><mo>→</mo></mover><mi>X</mi><mo>)</mo></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> At(P) \simeq \left( X \underset{\mathbf{B}G}{\times}X \underset{\mathbf{B}G}{\times}X \stackrel{\to}{\stackrel{\to}{\to}} X \underset{\mathbf{B}G}{\times}X \stackrel{\to}{\to} X \right) \,, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi></mrow><annotation encoding="application/x-tex">X \underset{\mathbf{B}G}{\times}X </annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+product">homotopy fiber product</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> with itself, and so forth. To see what this object is, pick any <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a>, and observe that</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><mo stretchy="false">)</mo><mo>≃</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo stretchy="false">)</mo></mrow></munder><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(U, X\underset{\mathbf{B}G}{\times}X ) \simeq \mathbf{H}(U,X) \underset{\mathbf{H}(U,\mathbf{B}G)}{\times} \mathbf{H}(U,X) </annotation></semantics></math></div> <p>(using that the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-categorical+hom">(∞,1)-categorical hom</a>-<a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>U</mi><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(U,-)</annotation></semantics></math> preserves <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-limits">(∞,1)-limits</a>) is equivalently the <a class="existingWikiWord" href="/nlab/show/set">set</a> of <a class="existingWikiWord" href="/nlab/show/triples">triples</a> consisting of two <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><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi_1, \phi_2 \colon X</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> between the <a class="existingWikiWord" href="/nlab/show/pullback">pulled-back bundles</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo lspace="verythinmathspace">:</mo><msubsup><mi>ϕ</mi> <mn>1</mn> <mo>*</mo></msubsup><mi>P</mi><mo>→</mo><msubsup><mi>ϕ</mi> <mn>2</mn> <mo>*</mo></msubsup><mi>P</mi></mrow><annotation encoding="application/x-tex">\eta \colon \phi_1^* P \to \phi_2^* P</annotation></semantics></math> on <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>.</p> <p>Since <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> is topologically <a class="existingWikiWord" href="/nlab/show/contractible+topological+space">contractible</a>, and hence every <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/principal+bundle">principal bundle</a> over <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> admits a <a class="existingWikiWord" href="/nlab/show/section">section</a>, every such triple induces a <a class="existingWikiWord" href="/nlab/show/function">function</a>, in fact a <a class="existingWikiWord" href="/nlab/show/bijection">bijection</a>, from the set of lifts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\hat \phi_1 \colon U \to P</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\phi_1</annotation></semantics></math> to the set of lifts <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover> <mn>2</mn></msub><mo lspace="verythinmathspace">:</mo><mi>U</mi><mo>→</mo><mi>P</mi></mrow><annotation encoding="application/x-tex">\hat \phi_2 \colon U \to P</annotation></semantics></math> which are <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>U</mi><mo>,</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(U,G)</annotation></semantics></math>-equivariant. By <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>-equivariant every pair consisting of a single lift <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\hat \phi_1</annotation></semantics></math> and its image <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>η</mi><mo stretchy="false">(</mo><msub><mover><mi>ϕ</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\eta(\hat \phi_1)</annotation></semantics></math> already uniquely determes <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>. Therefore the above set of triples is <a class="existingWikiWord" href="/nlab/show/natural+isomorphism">naturally isomorphic</a> to the set of <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>U</mi><mo>→</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>P</mi><mo>≔</mo><mo stretchy="false">(</mo><mi>P</mi><mo>×</mo><mi>P</mi><mo stretchy="false">)</mo><msub><mo stretchy="false">/</mo> <mi>diag</mi></msub><mi>G</mi></mrow><annotation encoding="application/x-tex">U \to P \times_G P \coloneqq (P \times P)/_{diag} G</annotation></semantics></math>. This is precisely the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/morphisms">morphisms</a> of the traditional <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a>. Since this is true for all <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">U \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> and <a class="existingWikiWord" href="/nlab/show/natural+transformation">naturally</a> so, and since <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> is a <a class="existingWikiWord" href="/nlab/show/site">site</a> of definition of <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> it follows by the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a> (which in the present cases reduces to the ordinary <a class="existingWikiWord" href="/nlab/show/Yoneda+lemma">Yoneda lemma</a>), we have a <a class="existingWikiWord" href="/nlab/show/natural+equivalence">natural equivalence</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>X</mi><munder><mo>×</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></munder><mi>X</mi><mo>≃</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>P</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> X \underset{\mathbf{B}G}{\times} X \simeq P \times_G P \,. </annotation></semantics></math></div> <p>In this manner it is immediate to check that this identification respects all the structure maps, and hence the above Cech nerve is indeed identified as the <a class="existingWikiWord" href="/nlab/show/simplicial+manifold">simplicial manifold</a> which is the <a class="existingWikiWord" href="/nlab/show/nerve">nerve</a> of the traditional <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>P</mi><msub><mo>×</mo> <mi>G</mi></msub><mi>P</mi><mover><mo>→</mo><mo>→</mo></mover><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(P \times_G P \stackrel{\to}{\to} X)</annotation></semantics></math>.</p> </div> <h3 id="TheTraditionalCourantLie2Algebroid">The traditional Courant Lie 2-algebroid</h3> <p>There is a traditional construction which assigns to a <a class="existingWikiWord" href="/nlab/show/bundle+gerbe">bundle gerbe</a> “with connective structure but without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>” a <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a>. We discuss here how this is the <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the corresponding higher Atiyah groupoid.</p> <p>In order to do so, we pick again, as <a href="#TheTraditionalAtiyagLieGroupoid">above</a>, as ambient context <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>.</p> <div class="num_prop"> <h6 id="proposition_7">Proposition</h6> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>≔</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi>LieGrp</mi><mo>↪</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \coloneqq U(1) \in LieGrp \hookrightarrow Grp(\mathbf{H})</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/circle+Lie+group">circle Lie group</a> (which is in particular a <a class="existingWikiWord" href="/nlab/show/sylleptic+%E2%88%9E-group">sylleptic ∞-group</a>), the sequence of maps of <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stacks">moduli ∞-stacks</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}^2 U(1)_{conn} \to \mathbf{B}(\mathbf{B}U(1)_{conn}) \to \mathbf{B}^2 U(1) </annotation></semantics></math></div> <p>of prop. <a class="maruku-ref" href="#InclusionOfNableBisectionsIntoNable1AndNabla0Bisections"></a> is presented under the canonical equivalence <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>≃</mo> <mrow><msub><mi>L</mi> <mi>lwhe</mi></msub></mrow></msub><mi>Func</mi><mo stretchy="false">(</mo><msubsup><mi>CartSp</mi> <mi>smooth</mi> <mi>op</mi></msubsup><mo>,</mo><mi>sSet</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\simeq _{L_{lwhe}} Func(CartSp^{op}_{smooth}, sSet)</annotation></semantics></math> by the image under the <a class="existingWikiWord" href="/nlab/show/Dold-Kan+correspondence">Dold-Kan correspondence</a> of the evident sequence of <a class="existingWikiWord" href="/nlab/show/chain+maps">chain maps</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>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mi>d</mi><mi>log</mi></mrow></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mi>id</mi></mover></mtd> <mtd><msup><mi>Ω</mi> <mn>1</mn></msup></mtd> <mtd><mover><mo>→</mo><mpadded width="0"><mn>0</mn></mpadded></mover></mtd> <mtd><mn>0</mn></mtd></mtr> <mtr><mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mi>d</mi></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mn>0</mn></mpadded></msup></mtd></mtr> <mtr><mtd><msup><mi>Ω</mi> <mn>2</mn></msup></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd> <mtd><mo>→</mo></mtd> <mtd><mn>0</mn></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ U(1) &\stackrel{id}{\to}& U(1) &\stackrel{id}{\to}& U(1) \\ \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{d log}} && \downarrow^{\mathrlap{0}} \\ \Omega^1 &\stackrel{id}{\to}& \Omega^1 &\stackrel{\mathrlap{0}}{\to}& 0 \\ \downarrow^{\mathrlap{d}} && \downarrow^{\mathrlap{0}} && \downarrow^{\mathrlap{0}} \\ \Omega^2 &\to& 0 &\to& 0 } \,, </annotation></semantics></math></div> <p>where on the left we have the <a class="existingWikiWord" href="/nlab/show/Deligne+complex">Deligne complex</a> for degree-3-<a class="existingWikiWord" href="/nlab/show/ordinary+differential+cohomology">ordinary differential cohomology</a>.</p> </div> <div class="proof"> <h6 id="proof_2">Proof</h6> <p>This is a direct consequence of the discussion at <em><a class="existingWikiWord" href="/nlab/show/circle+n-bundle+with+connection">circle n-bundle with connection</a></em>.</p> </div> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>This makes precise how</p> <ul> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1)_{conn}</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/moduli+infinity-stack">moduli 2-stack</a> of <a class="existingWikiWord" href="/nlab/show/circle+2-bundles+with+connection">circle 2-bundles with connection</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}U(1)_{conn})</annotation></semantics></math> is the moduli 2-stack of circle 2-bundle “with connection but without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>”;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}^2 U(1)</annotation></semantics></math> is the moduli 2-stack of <a class="existingWikiWord" href="/nlab/show/circle+2-group">circle 2-group</a>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundles">principal 2-bundles</a>.</p> </li> </ul> </div> <p>Let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>1</mn></msub><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \nabla_1 \colon X \to \mathbf{B}(\mathbf{B}U(1)_{conn}) </annotation></semantics></math></div> <p>be the map modulating <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</a> but “without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>”. Then then higher Atiyah groupoid of th <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathbf{B}U(1)_{conn})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+2-bundle">principal 2-bundle</a> classified by this map has as higher Atiyah groupoid the corresponding <em><a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-groupoid">Courant Lie 2-groupoid</a></em>: the object which is the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the traditional <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a> associated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_1</annotation></semantics></math>.</p> <p>To see we observe that the corresponding <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">2-group of bisections</a> is</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mo>≔</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∏</mo><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow></munder><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><msub><mo>∇</mo> <mn>1</mn></msub><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Aut}_{\mathbf{H}}(\nabla_1) \coloneqq \underset{\mathbf{B}(\mathbf{B}U(1)_{conn})}{\prod} \mathbf{Aut}_{/\mathbf{B}(\mathbf{B}U(1)_{conn})}(\nabla_1) \,. </annotation></semantics></math></div> <p>This has as objects <a class="existingWikiWord" href="/nlab/show/diagrams">diagrams</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</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>X</mi></mtd> <mtd></mtd> <mtd><munderover><mo>→</mo><mo>≃</mo><mi>ϕ</mi></munderover></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><msub><mrow></mrow> <mpadded width="0" lspace="-100%width"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mi>η</mi></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width="0"><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow></mpadded></msub></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \array{ X &&\underoverset{\simeq}{\phi}{\to}&& X \\ & {}_{\mathllap{\nabla_1}}\searrow &\swArrow_\eta& \swarrow_{\mathrlap{\nabla_1}} \\ && \mathbf{B}(\mathbf{B}U(1)_{conn}) } \,, </annotation></semantics></math></div> <p>hece equivalently pairs consisting of a <a class="existingWikiWord" href="/nlab/show/diffeomorphism">diffeomorphism</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>X</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">\phi \colon X \to X</annotation></semantics></math> and a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> (of 2-connections without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>)</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><msup><mi>ϕ</mi> <mo>*</mo></msup><msub><mo>∇</mo> <mn>1</mn></msub><mo>→</mo><msub><mo>∇</mo> <mn>1</mn></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \eta \;\colon \; \phi^* \nabla_1 \to \nabla_1 \,. </annotation></semantics></math></div> <p>The morphisms are accordingly the suitable <a class="existingWikiWord" href="/nlab/show/natural+transformations">natural transformations</a> of these diagrams.</p> <p>This is precisely the 2-group of “bundle gerbe symmetries” of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mo>∇</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\nabla_1</annotation></semantics></math> which is studient in (<a href="#Collier">Collier</a>). With this identification the main result there is the above claim.</p> <p>Moreover, the canonical inclusions of smooth 2-groups of prop. <a class="maruku-ref" href="#InclusionOfNableBisectionsIntoNable1AndNabla0Bisections"></a> reproduces, under <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a>, the inclusion of the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a> into that <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> of sections of the corresponding <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a> observed in (<a href="#Rogers10">Rogers 10</a>).</p> <h3 id="TheTraditionalQuantomorphismGroup">The traditional quantomorphism group</h3> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>∇</mo><mo lspace="verythinmathspace">:</mo><mi>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \colon X \to \mathbf{B}U(1)_{conn}</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/group+of+bisections">group of bisections</a> of the corresponding Atiyah groupoid is the <a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a> 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">\nabla</annotation></semantics></math> regarded as a <a class="existingWikiWord" href="/nlab/show/prequantum+bundle">prequantum bundle</a>.</p> <p>(…)</p> <h3 id="TheQuantomorphismNGroups">The quantomorphism <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>-groups</h3> <p>In (<a href="#Rogers">Rogers 11</a>) is a proposal for the generalization of the notion of <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a> of a <a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a> to a notion of <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a> induced by an <em><a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a></em>. Since the <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a> of the <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a> is traditionally known as the <em><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></em>, the Lie integration of these <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebras">Poisson Lie n-algebras</a> should be called an <em><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></em>.</p> <p>We here discuss a general abstract theory of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-groups">quantomorphism n-groups</a> as <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups+of+bisections">∞-groups of bisections</a> of a higher Atiyah groupoid associated with a <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a>. Then we show that under <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> this reproduces the construction in (<a href="#Rogers11">Rogers 11</a>).</p> <div> <p><strong>higher and integrated <a class="existingWikiWord" href="/nlab/show/Kostant-Souriau+extensions">Kostant-Souriau extensions</a></strong>:</p> <p>(<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+of+bisections">∞-group of bisections</a> of <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a> 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(X,\nabla) \to \mathbf{HamSympl}(X,\nabla) </annotation></semantics></math></div> <table><thead><tr><th><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><th>geometry</th><th>structure</th><th>unextended structure</th><th>extension by</th><th>quantum extension</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><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/cohesive">cohesive</a> <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega \mathbb{G})</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</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></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/symplectic+geometry">symplectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/real+numbers">real numbers</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonians">Hamiltonians</a> under <a class="existingWikiWord" href="/nlab/show/Poisson+bracket">Poisson bracket</a></td></tr> <tr><td style="text-align: left;">1</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+group">Hamiltonian symplectomorphism group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+group">quantomorphism group</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie 2-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a></td></tr> <tr><td style="text-align: left;">2</td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphism">Hamiltonian 2-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle 2-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism 2-group</a></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>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/n-plectic+geometry">n-plectic geometry</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Lie+n-algebra">Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a></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>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math></td><td style="text-align: left;"></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-group">smooth n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/Hamiltonian+n-plectomorphisms">Hamiltonian n-plectomorphisms</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></td></tr> </tbody></table> <p>(extension are listed for sufficiently connected <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>For all of the following, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/cohesive+%28%E2%88%9E%2C1%29-topos">cohesive (∞,1)-topos</a> equipped with <a class="existingWikiWord" href="/nlab/show/differential+cohesion">differential cohesion</a>. Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo lspace="0em" rspace="thinmathspace">on</mo><mi>Grpd</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \on Grpd(\mathbf{H})</annotation></semantics></math> be equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a>. Then there is a canonical object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}_{conn} \in \mathbf{H}</annotation></semantics></math> which is the <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</a>.</p> <p>Fox such a <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> given by a map</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>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn} \,. </annotation></semantics></math></div> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>By prop. <a class="maruku-ref" href="#HValuedAutomorphismAsFiber"></a> the <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>-valued automorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Aut}_{\mathbf{H}}(\nabla)</annotation></semantics></math> according to def. <a class="maruku-ref" href="#HValuedAutomorphismGroup"></a> sits in an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> diagram 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><msub><mstyle mathvariant="bold"><mi>Aut</mi></mstyle> <mstyle mathvariant="bold"><mi>H</mi></mstyle></msub><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⊢</mo><mo>∇</mo></mrow></mover></mtd> <mtd><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{Aut}_{\mathbf{H}}(\nabla) &\to& \mathbf{Aut}(X) \\ \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}} \\ {*} &\stackrel{\vdash \nabla}{\to}& [X, \mathbf{B}\mathbb{G}_{conn}] } \,. </annotation></semantics></math></div> <p>By remark <a class="maruku-ref" href="#Concretification"></a> we want to pass to its <a class="existingWikiWord" href="/nlab/show/concretification">concretification</a>. Indeed, in the above diagram the <a class="existingWikiWord" href="/nlab/show/mapping+%E2%88%9E-stack">mapping ∞-stack</a> <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><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[X, \mathbf{B}\mathbb{G}_{conn}]</annotation></semantics></math> is not quite yet the correct <a class="existingWikiWord" href="/nlab/show/moduli+%E2%88%9E-stack">moduli ∞-stack</a> 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connections">principal ∞-connections</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>, but instead its <a class="existingWikiWord" href="/nlab/show/differential+concretification">differential concretification</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mstyle mathvariant="bold"><mi>Conn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G}\mathbf{Conn}(X)</annotation></semantics></math> is, as defined at <em><a href="concretification#ConcretificationOfDifferentialModuli">concretification - Examples - Of differential moduli</a></em>. Therefore the following definition states the above pullback diagram with that replacement.</p> </div> <div class="num_defn"> <h6 id="definition_5">Definition</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">\mathbb{G}</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/braided+%E2%88%9E-group">braided ∞-group</a> as above and let <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>X</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \;\colon\; X \to \mathbf{B}\mathbb{G}_{conn}</annotation></semantics></math> be 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">\mathbb{G}</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a>.</p> <p>The <strong><a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QuantMorph</mi><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="normal">Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">QuantMorph(\nabla) \in \mathrm{Grp}(\mathbf{H})</annotation></semantics></math> of 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">\nabla</annotation></semantics></math> is the object fitting into the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</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><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mpadded></msup></mtd></mtr> <mtr><mtd><mo>*</mo></mtd> <mtd><mover><mo>→</mo><mrow><mo>⊢</mo><mo>∇</mo></mrow></mover></mtd> <mtd><mi>𝔾</mi><mstyle mathvariant="bold"><mi>Conn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{QuantMorph}(\nabla) &\to& \mathbf{Aut}(X) \\ \downarrow && \downarrow^{\mathrlap{\nabla \circ (-)}} \\ {*} &\stackrel{\vdash \nabla}{\to}& \mathbb{G}\mathbf{Conn}(X) } \,. </annotation></semantics></math></div></div> <div class="num_defn" id="HamiltonianSymplectomorphismInfinityGroup"> <h6 id="definition_6">Definition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{HamSympl}(\nabla)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/1-image">1-image</a> of the canonical map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{QuantMorph}(\nabla) \to \mathbf{Aut}(X)</annotation></semantics></math>.</p> </div> <div class="num_prop" id="QuantomorphismExtension"> <h6 id="proposition_8">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{QuantMorph}(\nabla)</annotation></semantics></math> in an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> of the Hamiltonian symplectomorphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-group of def. <a class="maruku-ref" href="#HamiltonianSymplectomorphismInfinityGroup"></a> by the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega\mathbb{G})\mathbf{FlatConn}(X)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/concretification">concretified</a> <a class="existingWikiWord" href="/nlab/show/flat+%E2%88%9E-connections">flat ∞-connections</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>: we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Omega\mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \,. </annotation></semantics></math></div> <p>Moreover, at least at the level of the underlying objects, this extension is classified by the <a class="existingWikiWord" href="/nlab/show/cocycle">cocycle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}((\Omega\mathbb{G})\mathbf{FlatConn}(X))</annotation></semantics></math> in that we have a long <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>HamSympl</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><mo>∇</mo><mo>∘</mo><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Omega\mathbb{G})\mathbf{FlatConn}(X) \to \mathbf{QuantMorph}(\nabla) \to \mathbf{HamSympl}(\nabla) \stackrel{\nabla \circ (-)}{\to} \mathbf{B}((\Omega \mathbb{G})\mathbf{FlatConn}(X)) \,. </annotation></semantics></math></div></div> <p>We now restrict this to a special case and describe it more in detail:</p> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>∈</mo></mrow><annotation encoding="application/x-tex">X \in </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/SmthMfd">SmthMfd</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">\hookrightarrow</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> and let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>≔</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>∈</mo><mi mathvariant="normal">Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} \coloneqq \mathbf{B}^{n-1}U(1) \in \mathrm{Grp}(\mathbf{H})</annotation></semantics></math> be the <a class="existingWikiWord" href="/nlab/show/circle+n-group">circle n-group</a>. Finally 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>X</mi><mo>→</mo><msup><mi>Ω</mi> <mrow><mi>n</mi><mo>+</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">\omega \colon X \to \Omega^{n+1}</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/n-plectic+form">n-plectic form</a> and <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>X</mi><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mi>n</mi></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\nabla \;\colon\; X \to \mathbf{B}^n U(1)_{conn}</annotation></semantics></math> a prequantization by a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a>.</p> <div class="num_prop"> <h6 id="proposition_9">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a> sequence of prop. <a class="maruku-ref" href="#QuantomorphismExtension"></a> is a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebras">L-∞ algebras</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mo>→</mo><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mo>→</mo><msub><mi>𝒳</mi> <mi>Ham</mi></msub><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo><mover><mo>→</mo><mrow><msub><mi>ι</mi> <mrow><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mi>ω</mi></mrow></msub></mrow></mover><mstyle mathvariant="bold"><mi>B</mi></mstyle><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mi>ℝ</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \to \mathfrak{Poisson}(X,\omega) \to \mathcal{X}_{Ham}(X,\omega) \stackrel{\iota_{(-)\omega}}{\to} \mathbf{B}\mathbf{H}(X, \flat \mathbf{B}^{n-1}\mathbb{R}) \,, </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><mi>𝔓𝔬𝔦𝔰𝔰𝔬𝔫</mi><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mi>ω</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathfrak{Poisson}(X,\omega)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a> as defined in (<a href="#Rogers11">Rogers 11</a>).</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>𝒳</mi> <mi>Ham</mi></msub></mrow><annotation encoding="application/x-tex">\mathcal{X}_{Ham}</annotation></semantics></math> is the Lie algebra of <a class="existingWikiWord" href="/nlab/show/vector+fields">vector fields</a> restricted to the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+vector+fields">Hamiltonian vector fields</a>;</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo>,</mo><mo>♭</mo><mo stretchy="false">(</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mrow><mi>n</mi><mo>−</mo><mn>1</mn></mrow></msup><mo stretchy="false">)</mo><mi>ℝ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{H}(X, \flat (\mathbf{B}^{n-1})\mathbb{R})</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a> for flat <a class="existingWikiWord" href="/nlab/show/de+Rham+cohomology">de Rham cohomology</a> in the given degree, regarded as an abelian <a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E+algebra">L-∞ algebra</a>.</p> </li> </ul> </div> <h3 id="TheTraditionalHeisenbergGroup">The traditional Heisenberg group</h3> <p>(…) <a class="existingWikiWord" href="/nlab/show/Heisenberg+group">Heisenberg group</a> (…)</p> <h3 id="TheHeisenbergnGroup">The Heisenberg <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>-group</h3> <p>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><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">X = G \in \mathbf{H}</annotation></semantics></math> has itself <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> structure, then it is natural to restrict the <a class="existingWikiWord" href="/nlab/show/quantomorphism+%E2%88%9E-group">quantomorphism ∞-group</a> to that subgroup of the <a class="existingWikiWord" href="/nlab/show/Hamiltonian+symplectomorphism+%E2%88%9E-group">Hamiltonian symplectomorphism ∞-group</a> whose elements come from 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>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> on itself. This is the corresponding <em><a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a></em>.</p> <div class="num_defn"> <h6 id="definition_7">Definition</h6> <p>With all assumtions as <a href="#TheQuantomorphismNGroups">above</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> be an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</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>G</mi><mo>↪</mo><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> G \hookrightarrow \mathbf{Aut}(G) </annotation></semantics></math></div> <p>(where on the right we have the <a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-group">automorphism ∞-group</a> of the underlying object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">G \in \mathbf{H}</annotation></semantics></math>) the inclusion that exhibits the left <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/%E2%88%9E-action">∞-action</a> on itself.</p> <p>The the <strong><a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a></strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Heis}(\nabla)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-pullback">(∞,1)-pullback</a> in the <a class="existingWikiWord" href="/nlab/show/diagram">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><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd><mi>G</mi></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>Aut</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \array{ \mathbf{Heis}(\nabla) &\to& \mathbf{QuantMorph}(\nabla) \\ \downarrow && \downarrow \\ G &\to& \mathbf{Aut}(G) } \,. </annotation></semantics></math></div></div> <p>The following is an immediate consequence of the definition</p> <div class="num_prop" id="HeisenbergInfinityGroupExtension"> <h6 id="proposition_10">Proposition</h6> <p>The <a class="existingWikiWord" href="/nlab/show/Heisenberg+%E2%88%9E-group">Heisenberg ∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{Heis}(\nabla)</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</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> by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\Omega \mathbb{G})\mathbf{FlatConn}(G)</annotation></semantics></math>: we have a <a class="existingWikiWord" href="/nlab/show/homotopy+fiber+sequence">homotopy fiber sequence</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groups">∞-groups</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>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>QuantMorph</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> (\Omega \mathbb{G})\mathbf{FlatConn}(G) \to \mathbf{QuantMorph}(\nabla) \to G \,. </annotation></semantics></math></div></div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>In <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo>=</mo></mrow><annotation encoding="application/x-tex">\mathbf{H} = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Smooth%E2%88%9EGrpd">Smooth∞Grpd</a> consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>LieGrp</mi><mo>↪</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in LieGrp \hookrightarrow Grp(\mathbf{H})</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/connected+topological+space">connected</a>, <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a> <a class="existingWikiWord" href="/nlab/show/compact+Lie+group">compact</a> <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a>, say the <a class="existingWikiWord" href="/nlab/show/Spin+group">Spin group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>=</mo><mi>Spin</mi></mrow><annotation encoding="application/x-tex">G = Spin</annotation></semantics></math>. Then the <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a> <a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a> is a pre-3-plectic form on the <a class="existingWikiWord" href="/nlab/show/moduli+stack">moduli stack</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/principal+connections">principal connections</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><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">⟩</mo><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>→</mo><msubsup><mstyle mathvariant="bold"><mi>Ω</mi></mstyle> <mi>cl</mi> <mn>4</mn></msubsup><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \langle -,-\rangle \;\colon\; \mathbf{B}G_{conn} \to \mathbf{\Omega}_{cl}^4 \,. </annotation></semantics></math></div> <p>This has a <a class="existingWikiWord" href="/nlab/show/higher+geometric+prequantization">higher geometric prequantization</a> by the <a class="existingWikiWord" href="/nlab/show/smooth+first+fractional+Pontryagin+class">smooth first fractional Pontryagin class</a>, a map</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle displaystyle="false"><mfrac><mn>1</mn><mn>2</mn></mfrac></mstyle><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mspace width="thickmathspace"></mspace><mo lspace="verythinmathspace">:</mo><mspace width="thickmathspace"></mspace><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>G</mi> <mi>conn</mi></msub><mo>→</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \tfrac{1}{2}\hat\mathbf{p}_1 \;\colon\; \mathbf{B}G_{conn} \to \mathbf{B}^3 U(1)_{conn} \,. </annotation></semantics></math></div> <p>The <a class="existingWikiWord" href="/nlab/show/transgression">transgression</a> of this to maps oout of the <a class="existingWikiWord" href="/nlab/show/circle">circle</a> yields a <a class="existingWikiWord" href="/nlab/show/circle+2-bundle+with+connection">circle 2-bundle with connection</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>G</mi><mo>→</mo><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mi>G</mi><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><mfrac><mn>1</mn><mn>2</mn></mfrac><msub><mover><mstyle mathvariant="bold"><mi>p</mi></mstyle><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><mo stretchy="false">]</mo></mrow></mover><mo stretchy="false">[</mo><msup><mi>S</mi> <mn>1</mn></msup><mo>,</mo><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>3</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><msub><mo stretchy="false">)</mo> <mi>conn</mi></msub><mo stretchy="false">]</mo><mover><mo>→</mo><mrow><mi>exp</mi><mo stretchy="false">(</mo><mn>2</mn><mi>π</mi><mi>i</mi><msub><mo>∫</mo> <mrow><msup><mi>S</mi> <mn>1</mn></msup></mrow></msub><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">)</mo><mo stretchy="false">)</mo></mrow></mover><msup><mstyle mathvariant="bold"><mi>B</mi></mstyle> <mn>2</mn></msup><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \nabla \;\colon\; G \to [S^1, G] \stackrel{[S^1, \frac{1}{2}\hat\mathbf{p}_1]}{\to} [S^1, \mathbf{B}^3 U(1)_{conn}] \stackrel{\exp(2 \pi i \int_{S^1}(-))}{\to} \mathbf{B}^2 U(1) \,. </annotation></semantics></math></div> <p>This is a <a class="existingWikiWord" href="/nlab/show/prequantum+circle+2-bundle">prequantum circle 2-bundle</a> which prequantizes the canonical <a class="existingWikiWord" href="/nlab/show/differential+3-form">differential 3-form</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>, the one which is <a class="existingWikiWord" href="/nlab/show/invariant+differential+form">left invariant</a> and at the neutral element is <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>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">⟩</mo></mrow><annotation encoding="application/x-tex">\langle -,[-,-]\rangle</annotation></semantics></math>.</p> <p>Consider now the <a class="existingWikiWord" href="/nlab/show/higher+prequantum+geometry">higher prequantum geometry</a> of this 2-connection. So now <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>𝔾</mi><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbb{G} = \mathbf{B}U(1)</annotation></semantics></math>.</p> <p>Observe that</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><mo stretchy="false">(</mo><mi>Ω</mi><mi>𝔾</mi><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd> <mtd><mo>≃</mo><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>=</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{aligned} (\Omega \mathbb{G})\mathbf{FlatConn}(G) & \simeq U(1) \mathbf{FlatConn}(G) \\ & = \mathbf{B}U(1) \end{aligned} </annotation></semantics></math></div> <p>because <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 assumed to be <a class="existingWikiWord" href="/nlab/show/simply+connected+topological+space">simply connected</a>. (Notice that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}U(1)</annotation></semantics></math> does appear here with its canonical smooth structure: while a <a class="existingWikiWord" href="/nlab/show/gauge+transformation">gauge transformation</a> from the trivial <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+connection">principal connection</a> to itself is a constant function along <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 smooth structure in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mstyle mathvariant="bold"><mi>FlatConn</mi></mstyle><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">U(1)\mathbf{FlatConn}(G)</annotation></semantics></math> comes from how this may vary in parameterized collections ).</p> <p>Therefore by prop. <a class="maruku-ref" href="#HeisenbergInfinityGroupExtension"></a> we have an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+extension">∞-group extension</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>U</mi><mo stretchy="false">(</mo><mn>1</mn><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>→</mo><mi>G</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{B}U(1) \to \mathbf{Heis}(\nabla) \to G \,. </annotation></semantics></math></div> <p>This exhibits the Heisenberg 2-group here as the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">String(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><mstyle mathvariant="bold"><mi>Heis</mi></mstyle><mo stretchy="false">(</mo><mo>∇</mo><mo stretchy="false">)</mo><mo>≃</mo><mi>String</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \mathbf{Heis}(\nabla) \simeq String(G) \,. </annotation></semantics></math></div></div> <h2 id="related_concepts">Related concepts</h2> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></strong></p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a>:</th><th>standard <a class="existingWikiWord" href="/nlab/show/higher+Atiyah+groupoid">higher Atiyah groupoid</a></th><th><a class="existingWikiWord" href="/nlab/show/higher+Courant+groupoid">higher Courant groupoid</a></th><th>groupoid version of <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coefficient">coefficient</a> for <a class="existingWikiWord" href="/nlab/show/cohomology">cohomology</a>:</td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_1"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>𝔾</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}\mathbb{G}</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_2"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi mathvariant="normal">conn</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}(\mathbf{B}\mathbb{G}_{\mathrm{conn}})</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" class="maruku-mathml" display="inline" id="mathml_25bb27a7b2a5a8fcca6d17f6a351b913ce28a816_3"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><msub><mi>𝔾</mi> <mi>conn</mi></msub></mrow><annotation encoding="application/x-tex">\mathbf{B} \mathbb{G}_{conn}</annotation></semantics></math></td></tr> <tr><td style="text-align: left;">type of <a class="existingWikiWord" href="/nlab/show/fiber+%E2%88%9E-bundle">fiber ∞-bundle</a>:</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a> without top-degree connection form</td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-connection">principal ∞-connection</a></td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>The <a href="#TheTraditionalCourantLie2Algebroid">above</a> identification of higher Atiyah groupoids of “bundle gerbes with connective structure but without <a class="existingWikiWord" href="/nlab/show/curving">curving</a>” with those <a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integrating</a> the corresponding <a class="existingWikiWord" href="/nlab/show/standard+Courant+Lie+2-algebroid">standard Courant Lie 2-algebroid</a> is directly implied (under the above translations) by the main result in</p> <ul> <li id="Collier"><a class="existingWikiWord" href="/nlab/show/Braxton+Collier">Braxton Collier</a>, <em>Infinitesimal Symmetries of Dixmier-Douady Gerbes</em> (<a href="http://arxiv.org/abs/1108.1525">arXiv:1108.1525</a>)</li> </ul> <p>The corresponding inclusion of the <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+2-algebra">Poisson Lie 2-algebra</a> into the <a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a> of bisections of the <a class="existingWikiWord" href="/nlab/show/Courant+Lie+2-algebroid">Courant Lie 2-algebroid</a> was first observed in</p> <ul> <li id="Rogers10"><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>2-plectic geometry, Courant algebroids, and categorified prequantization</em> , <a href="http://arxiv.org/abs/1009.2975">arXiv:1009.2975</a>.</li> </ul> <p>in the context of <em><a class="existingWikiWord" href="/nlab/show/2-plectic+geometry">2-plectic geometry</a></em> over <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>.</p> <p>The <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+n-algebra">Poisson Lie n-algebra</a> over an <a class="existingWikiWord" href="/nlab/show/n-plectic+manifold">n-plectic manifold</a>, which by prop. <a class="maruku-ref" href="#spring"></a> is the <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a> of the <a class="existingWikiWord" href="/nlab/show/quantomorphism+n-group">quantomorphism n-group</a> of any <a class="existingWikiWord" href="/nlab/show/prequantum+circle+n-bundle">prequantum circle n-bundle</a> prequantizing 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>-plectic form, has been proposed in</p> <ul> <li id="Rogers11"><a class="existingWikiWord" href="/nlab/show/Chris+Rogers">Chris Rogers</a>, <em>Higher symplectic geometry</em> PhD thesis (2011) (<a href="http://arxiv.org/abs/1106.4068">arXiv:1106.4068</a>)</li> </ul> <p>The further statements and the discussion above follow:</p> <ul> <li id="FiorenzaRogersSchreiber"><a class="existingWikiWord" href="/nlab/show/Domenico+Fiorenza">Domenico Fiorenza</a>, <a class="existingWikiWord" href="/nlab/show/Chris+L.+Rogers">Chris L. Rogers</a>, <a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>: <em><a class="existingWikiWord" href="/schreiber/show/Higher+geometric+prequantum+theory">Higher <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"> <semantics> <mrow> <mi>U</mi> <mo stretchy="false">(</mo> <mn>1</mn> <mo stretchy="false">)</mo> </mrow> <annotation encoding="application/x-tex">U(1)</annotation> </semantics> </math>-gerbe connections in geometric prequantization</a></em>, Reviews in Mathematical Physics <strong>28</strong> 06 (2016) [<a href="https://arxiv.org/abs/1304.0236">arXiv:1304.0236</a>, <a href="https://doi.org/10.1142/S0129055X16500124">doi:10.1142/S0129055X16500124</a>]</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on January 23, 2025 at 15:30:11. 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