nLab
category algebra

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Context

Category theory

</head> <body> <p><strong><a class='existingWikiWord' href='/nlab/show/category+theory'>category theory</a></strong></p> <h2 id='concepts'>Concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/category'>category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/functor'>functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/natural+transformation'>natural transformation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Cat'>Cat</a></p> </li> </ul> <h2 id='universal_constructions'>Universal constructions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/universal+construction'>universal construction</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/representable+functor'>representable functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor'>adjoint functor</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/limit'>limit</a>/<a class='existingWikiWord' href='/nlab/show/colimit'>colimit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/weighted+limit'>weighted limit</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/end'>end</a>/<a class='existingWikiWord' href='/nlab/show/coend'>coend</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Kan+extension'>Kan extension</a></p> </li> </ul> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Yoneda+lemma'>Yoneda lemma</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Grothendieck+construction'>Grothendieck construction</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/adjoint+functor+theorem'>adjoint functor theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monadicity+theorem'>monadicity theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/adjoint+lifting+theorem'>adjoint lifting theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Tannaka+duality'>Tannaka duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Gabriel-Ulmer+duality'>Gabriel-Ulmer duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/small+object+argument'>small object argument</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Freyd-Mitchell+embedding+theorem'>Freyd-Mitchell embedding theorem</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/relation+between+type+theory+and+category+theory'>relation between type theory and category theory</a></p> </li> </ul> <h2 id='extensions'>Extensions</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/sheaf+and+topos+theory'>sheaf and topos theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/enriched+category+theory'>enriched category theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/higher+category+theory'>higher category theory</a></p> </li> </ul> <h2 id='applications'>Applications</h2> <ul> <li><a class='existingWikiWord' href='/nlab/show/applications+of+%28higher%29+category+theory'>applications of (higher) category theory</a></li> </ul> <div> <p> <a href='/nlab/edit/category+theory+-+contents'>Edit this sidebar</a> </p> </div></body></html></div> <h4 id="algebra">Algebra</h4> <div class="hide"><html xml:lang='en' xmlns:svg='http://www.w3.org/2000/svg' xmlns='http://www.w3.org/1999/xhtml'> <head><meta content='application/xhtml+xml;charset=utf-8' http-equiv='Content-type' /><title /></head> <body> <p><strong><a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebra</a></strong></p> <p><a class='existingWikiWord' href='/nlab/show/universal+algebra'>universal algebra</a></p> <h2 id='algebraic_theories'>Algebraic theories</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/algebraic+theory'>algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/2-algebraic+theory'>2-algebraic theory</a> / <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-algebraic+theory'>(∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monad'>monad</a> / <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-monad'>(∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/operad'>operad</a> / <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-operad'>(∞,1)-operad</a></p> </li> </ul> <h2 id='algebras_and_modules'>Algebras and modules</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/algebra+over+a+monad'>algebra over a monad</a></p> <p><a class='existingWikiWord' href='/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-monad'>∞-algebra over an (∞,1)-monad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/algebra+over+an+algebraic+theory'>algebra over an algebraic theory</a></p> <p><a class='existingWikiWord' href='/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-algebraic+theory'>∞-algebra over an (∞,1)-algebraic theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/algebra+over+an+operad'>algebra over an operad</a></p> <p><a class='existingWikiWord' href='/nlab/show/%E2%88%9E-algebra+over+an+%28%E2%88%9E%2C1%29-operad'>∞-algebra over an (∞,1)-operad</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/action'>action</a>, <a class='existingWikiWord' href='/nlab/show/%E2%88%9E-action'>∞-action</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/representation'>representation</a>, <a class='existingWikiWord' href='/nlab/show/%E2%88%9E-representation'>∞-representation</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/module'>module</a>, <a class='existingWikiWord' href='/nlab/show/%E2%88%9E-module'>∞-module</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/associated+bundle'>associated bundle</a>, <a class='existingWikiWord' href='/nlab/show/associated+%E2%88%9E-bundle'>associated ∞-bundle</a></p> </li> </ul> <h2 id='higher_algebras'>Higher algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/monoidal+%28%E2%88%9E%2C1%29-category'>monoidal (∞,1)-category</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+%28%E2%88%9E%2C1%29-category'>symmetric monoidal (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoid+in+an+%28%E2%88%9E%2C1%29-category'>monoid in an (∞,1)-category</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/commutative+monoid+in+an+%28%E2%88%9E%2C1%29-category'>commutative monoid in an (∞,1)-category</a></p> </li> </ul> </li> <li> <p>symmetric monoidal (∞,1)-category of spectra</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/smash+product+of+spectra'>smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/symmetric+monoidal+smash+product+of+spectra'>symmetric monoidal smash product of spectra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/ring+spectrum'>ring spectrum</a>, <a class='existingWikiWord' href='/nlab/show/module+spectrum'>module spectrum</a>, <a class='existingWikiWord' href='/nlab/show/algebra+spectrum'>algebra spectrum</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/A-%E2%88%9E+algebra'>A-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/A-%E2%88%9E+ring'>A-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/A-%E2%88%9E+space'>A-∞ space</a></li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/C-%E2%88%9E+algebra'>C-∞ algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/E-%E2%88%9E+ring'>E-∞ ring</a>, <a class='existingWikiWord' href='/nlab/show/E-%E2%88%9E+algebra'>E-∞ algebra</a></p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/%E2%88%9E-module'>∞-module</a>, <a class='existingWikiWord' href='/nlab/show/%28%E2%88%9E%2C1%29-module+bundle'>(∞,1)-module bundle</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/multiplicative+cohomology+theory'>multiplicative cohomology theory</a></p> </li> </ul> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/L-%E2%88%9E+algebra'>L-∞ algebra</a></p> <ul> <li><a class='existingWikiWord' href='/nlab/show/deformation+theory'>deformation theory</a></li> </ul> </li> </ul> <h2 id='model_category_presentations'>Model category presentations</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+simplicial+T-algebras'>model structure on simplicial T-algebras</a> / <a class='existingWikiWord' href='/nlab/show/homotopy+T-algebra'>homotopy T-algebra</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+operads'>model structure on operads</a></p> <p><a class='existingWikiWord' href='/nlab/show/model+structure+on+algebras+over+an+operad'>model structure on algebras over an operad</a></p> </li> </ul> <h2 id='geometry_on_formal_duals_of_algebras'>Geometry on formal duals of algebras</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Isbell+duality'>Isbell duality</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/derived+geometry'>derived geometry</a></p> </li> </ul> <h2 id='theorems'>Theorems</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Deligne+conjecture'>Deligne conjecture</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/delooping+hypothesis'>delooping hypothesis</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/monoidal+Dold-Kan+correspondence'>monoidal Dold-Kan correspondence</a></p> </li> </ul> <div> <p><a href='/nlab/edit/higher+algebra+-+contents'>Edit this sidebar</a></p> </div></body></html></div> </div> </div> <h1 id="contents">Contents</h1> <div class="maruku_toc"><ul><li><a href="#Idea">Idea</a></li><li><a href="#definition">Definition</a><ul><li><a href="#for_bare_categories_discrete_geometry">For bare categories (discrete geometry)</a></li><li><a href="#ForLieGroupoids">For Lie groupoids</a></li></ul></li><li><a href="#equivalent_characterizations">Equivalent characterizations</a><ul><li><a href="#WeakColimit">As a weak colimit over a constant <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Vect</mi></mrow><annotation encoding='application/x-tex'>2Vect</annotation></semantics></math>-valued functor</a></li><li><a href="#InTermsOfCompositionOfSpans">In terms of composition of spans</a></li></ul></li><li><a href="#examples">Examples</a><ul><li><a href="#basic_examples">Basic examples</a></li><li><a href="#incidence_algebras_poset_convolution_algebras">Incidence algebras (poset convolution algebras)</a></li><li><a href="#HigherGroupoidConvolutionAlgebras">Higher groupoid convolution algebras and n-vector spaces/n-modules</a></li></ul></li><li><a href="#properties">Properties</a><ul><li><a href="#RelationToTwistedKTheory">Relation to (twisted) K-theory</a></li></ul></li><li><a href="#related_concepts">Related concepts</a></li><li><a href="#references">References</a><ul><li><a href="#for_partial_orders">For partial orders</a></li><li><a href="#for_discrete_geometry">For discrete geometry</a></li><li><a href="#ReferencesForSmoothGeometry">For continuous/smooth geometry</a><ul><li><a href="#convolution_algebras">Convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</a></li><li><a href="#ReferencesConvolutionHopfAlgebroids">Convolution Hopf algebroids</a></li><li><a href="#ReferencesModulesOverConvolutionAlgebra">Modules over Lie groupoid convolution algebras and K-theory</a></li></ul></li></ul></li></ul></div> <h2 id="Idea">Idea</h2> <p>The <a class='existingWikiWord' href='/nlab/show/space'>space</a> of <a class='existingWikiWord' href='/nlab/show/functions'>functions</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}_1 \to R</annotation></semantics></math> on the <a class='existingWikiWord' href='/nlab/show/space'>space</a> of <a class='existingWikiWord' href='/nlab/show/morphisms'>morphisms</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_1</annotation></semantics></math> of a <a class='existingWikiWord' href='/nlab/show/small+category'>small category</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_\bullet</annotation></semantics></math> (with <a class='existingWikiWord' href='/nlab/show/coefficients'>coefficients</a> in some <a class='existingWikiWord' href='/nlab/show/ring'>ring</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>) naturally inherits a <a class='existingWikiWord' href='/nlab/show/convolution+algebra'>convolution algebra</a> structure from the <a class='existingWikiWord' href='/nlab/show/composition'>composition</a> operation on morphisms. This is called the <em>category convolution algebra</em> or just <em>category algebra</em> for short.</p> <p>Often this is considered specifically for <a class='existingWikiWord' href='/nlab/show/groupoids'>groupoids</a> and hence accordingly called <em>groupoid convolution algebra</em> or just <em>groupoid algebra</em> for short. (For one-object <a class='existingWikiWord' href='/nlab/show/delooping'>delooping</a> groupoids of <a class='existingWikiWord' href='/nlab/show/groups'>groups</a>, groupoid algebras reduce to <a class='existingWikiWord' href='/nlab/show/group+algebras'>group algebras</a>.) The <a class='existingWikiWord' href='/nlab/show/inverse'>inversion operation</a> of the groupoid naturally makes its groupoid algebra into a <a class='existingWikiWord' href='/nlab/show/star-algebra'>star-algebra</a> (this is generally the case for category algebras of <a class='existingWikiWord' href='/nlab/show/dagger+categories'>dagger categories</a>) and accordingly groupoid algebras play a role in <a class='existingWikiWord' href='/nlab/show/C-star-algebra'>C-star-algebra</a> theory.</p> <p>More generally, if the groupoid carries a <a class='existingWikiWord' href='/nlab/show/line+2-bundle'>line 2-bundle</a> then (in its incarnation as a <a class='existingWikiWord' href='/nlab/show/bundle+gerbe'>bundle gerbe</a>-like transition bundle) the space of morphisms carries a <a class='existingWikiWord' href='/nlab/show/line+bundle'>line bundle</a> (satisfying some compatibility conditions) and one can consider convolution algebras not just of functions, but of <a class='existingWikiWord' href='/nlab/show/sections'>sections</a> of this line bundle. The resulting algebra is called the <em>twisted groupoid convolution algebra</em>, twisted by the <a class='existingWikiWord' href='/nlab/show/characteristic+class'>characteristic class</a> of the <a class='existingWikiWord' href='/nlab/show/line+2-bundle'>line 2-bundle</a> (<a href="#TXLG">TXLG</a>).</p> <p>For “bare” categories/groupoids (i.e.: <a class='existingWikiWord' href='/nlab/show/internal+category'>internal</a> to <a class='existingWikiWord' href='/nlab/show/Set'>Set</a>) these constructions are canonical. But under mild conditions or else when equipped some suitable extra <a class='existingWikiWord' href='/nlab/show/stuff%2C+structure%2C+property'>structure</a>, it generalizes to <a class='existingWikiWord' href='/nlab/show/internal+categories'>internal categories</a>/<a class='existingWikiWord' href='/nlab/show/internal+groupoids'>internal groupoids</a> in <a class='existingWikiWord' href='/nlab/show/geometry'>geometric</a> contexts, notably in <a class='existingWikiWord' href='/nlab/show/topology'>topology</a> (<a class='existingWikiWord' href='/nlab/show/topological+groupoids'>topological groupoids</a>), <a class='existingWikiWord' href='/nlab/show/differential+geometry'>differential geometry</a> (<a class='existingWikiWord' href='/nlab/show/Lie+groupoids'>Lie groupoids</a>), and <a class='existingWikiWord' href='/nlab/show/algebraic+geometry'>algebraic geometry</a>. In such geometric situations a groupoid convolution algebra equipped with its canonical <a class='existingWikiWord' href='/nlab/show/coalgebra'>coalgebra</a> structure over the functions on its canonical <a class='existingWikiWord' href='/nlab/show/atlas'>atlas</a> is also called a <em><a class='existingWikiWord' href='/nlab/show/Hopf+algebroid'>Hopf algebroid</a></em> and may be used to <em>characterize</em> the geometric groupoid.</p> <p>Therefore to some extent one may think of the relation between groupoids/categories and their groupoid/category algebras as an incarnation of the <a class='existingWikiWord' href='/nlab/show/Isbell+duality'>general duality</a> between <a class='existingWikiWord' href='/nlab/show/geometry'>geometry</a> and <a class='existingWikiWord' href='/nlab/show/algebra'>algebra</a>. Since category/groupoid algebras are generically non-commutative, this relation identifies groupoids/categories as certain spaces in <a class='existingWikiWord' href='/nlab/show/noncommutative+geometry'>noncommutative geometry</a>. From this point of view groupoid convolution algebras have been highlighted and developed notably in (<a href="#Connes094">Connes 94</a>). Due to this relation the groupoid convolution product is also referred to as a <a class='existingWikiWord' href='/nlab/show/star+product'>star product</a> and denoted “<math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⋆</mo></mrow><annotation encoding='application/x-tex'>\star</annotation></semantics></math>”. Groupoid <a class='existingWikiWord' href='/nlab/show/C%2A-algebras'>C*-algebras</a> form a rich sub-class of all <a class='existingWikiWord' href='/nlab/show/C%2A-algebras'>C*-algebras</a>, including <a class='existingWikiWord' href='/nlab/show/crossed+product+C%2A-algebras'>crossed product C*-algebras</a>, <a class='existingWikiWord' href='/nlab/show/Cuntz+algebras'>Cuntz algebras</a>.</p> <p>Groupoid convolution algebras may also be understood as generalizations of <a class='existingWikiWord' href='/nlab/show/matrix+algebras'>matrix algebras</a>, to which they reduce for the case of the <a class='existingWikiWord' href='/nlab/show/pair+groupoid'>pair groupoid</a>. In (<a href="#Connes094">Connes 94, 1.1</a>) it was famously argued that when <a class='existingWikiWord' href='/nlab/show/Werner+Heisenberg'>Werner Heisenberg</a> (re-)discovered (infinite-dimensional) <a class='existingWikiWord' href='/nlab/show/matrix+algebras'>matrix algebras</a> as <a class='existingWikiWord' href='/nlab/show/algebras+of+observables'>algebras of observables</a> in <a class='existingWikiWord' href='/nlab/show/quantum+mechanics'>quantum mechanics</a>, conceptually he rather considered groupoid convolution algebras. This perspective has since been fully developed: in (<a href="#EH">EH 06</a>) <a class='existingWikiWord' href='/nlab/show/strict+deformation+quantization'>strict deformation quantization</a> is given fairly generally by twisted groupoid convolution algebras. See at <em><a class='existingWikiWord' href='/nlab/show/geometric+quantization+of+symplectic+groupoids'>geometric quantization of symplectic groupoids</a></em> for more on this. For the <a class='existingWikiWord' href='/nlab/show/discrete+groupoid'>discrete</a> but <a class='existingWikiWord' href='/nlab/show/higher+geometry'>higher geometry</a> of <a class='existingWikiWord' href='/nlab/show/infinity-Dijkgraaf-Witten+theory'>infinity-Dijkgraaf-Witten theory</a> quantization by higher groupoid convolution algebras is indicated in (<a href="#FHLT">FHLT 09</a>).</p> <h2 id="definition">Definition</h2> <h3 id="for_bare_categories_discrete_geometry">For bare categories (discrete geometry)</h3> <div id="CategoryAlgebra" class="num_defn"> <h6 id="definition_2">Definition</h6> <p>Let <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/small+category'>small category</a> and let <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/ring'>ring</a>.</p> <p>The <strong>category algebra</strong> or <strong><a class='existingWikiWord' href='/nlab/show/convolution+algebra'>convolution algebra</a></strong> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><mi>𝒞</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R[\mathcal{C}]</annotation></semantics></math> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> over <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/associative+algebra'>algebra</a></p> <ul> <li> <p>whose underlying <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi></mrow><annotation encoding='application/x-tex'>R</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/module'>module</a> is the <a class='existingWikiWord' href='/nlab/show/free+module'>free module</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R[\mathcal{C}_1]</annotation></semantics></math> over the set of <a class='existingWikiWord' href='/nlab/show/morphisms'>morphisms</a> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>;</p> </li> <li> <p>whose product operation is defined on <a class='existingWikiWord' href='/nlab/show/basis'>basis</a>-elements <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>↪</mo><mi>R</mi><mo stretchy='false'>[</mo><mi>𝒞</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>f,g \in \mathcal{C}_1 \hookrightarrow R[\mathcal{C}]</annotation></semantics></math> to be their <a class='existingWikiWord' href='/nlab/show/composition'>composition</a> if they are composable and zero otherwise:</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>⋅</mo><mi>g</mi><mo>:</mo><mo>=</mo><mrow><mo>{</mo><mrow><mtable><mtr><mtd><mi>g</mi><mo>∘</mo><mi>f</mi></mtd> <mtd><mi>if</mi><mspace width='thickmathspace' /><mi>composable</mi></mtd></mtr> <mtr><mtd><mn>0</mn></mtd> <mtd><mi>otherwise</mi></mtd></mtr></mtable></mrow></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> f \cdot g := \left\lbrace \array{ g \circ f & if\;composable \\ 0 & otherwise } \right. \,. </annotation></semantics></math></div></li> </ul> </div> <div id="AsConvolutionAlgebra" class="num_remark"> <h6 id="remark">Remark</h6> <p>We may identify elements in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R[\mathcal{C}_1]</annotation></semantics></math> with <a class='existingWikiWord' href='/nlab/show/functions'>functions</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><mi>R</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}_1 \to R</annotation></semantics></math> with the property that they are non-vanishing only on finitely many elements. Under this identification for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ϕ</mi> <mn>2</mn></msub></mrow><annotation encoding='application/x-tex'>\phi_1, \phi_2</annotation></semantics></math> two such functions, their product in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><mi>𝒞</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>R[\mathcal{C}]</annotation></semantics></math> is given by the formula</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>f</mi><mo>↦</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><msub><mi>f</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>=</mo><mi>f</mi></mrow></munder><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>⋅</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><msub><mi>f</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>,</mo></mrow><annotation encoding='application/x-tex'> \phi_1 \cdot \phi_2 \; \colon \; f \mapsto \sum_{f_2 \circ f_1 = f} \phi_2(f_2) \cdot \phi_1(f_1) \,, </annotation></semantics></math></div> <p>where <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><msub><mi>f</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>f</mi> <mn>2</mn></msub><mo>∈</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>f,f_1,f_2 \in \mathcal{C}_1</annotation></semantics></math>. In particular if <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoid</a> so that every <a class='existingWikiWord' href='/nlab/show/morphisms'>morphisms</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math> has an <a class='existingWikiWord' href='/nlab/show/inverse'>inverse</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>f</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding='application/x-tex'>f^{-1}</annotation></semantics></math> then this is equivalently</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>ϕ</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>ϕ</mi> <mn>2</mn></msub><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>f</mi><mo>↦</mo><mo>=</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>g</mi><mo>∈</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow></munder><msub><mi>ϕ</mi> <mn>2</mn></msub><mo stretchy='false'>(</mo><mi>f</mi><mo>∘</mo><msup><mi>g</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>)</mo><mo>⋅</mo><msub><mi>ϕ</mi> <mn>1</mn></msub><mo stretchy='false'>(</mo><mi>g</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \phi_1 \cdot \phi_2 \; \colon \; f \mapsto = \sum_{g \in \mathcal{C}_1} \phi_2(f \circ g^{-1}) \cdot \phi_1(g) \,. </annotation></semantics></math></div> <p>This expresses <a class='existingWikiWord' href='/nlab/show/convolution'>convolution</a> of functions.</p> </div> <div id="InvolutionByPullbackAlongInversion" class="num_defn"> <h6 id="remark_2">Remark</h6> <p>If the small category <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_\bullet</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoid</a>, hence equipped with an <a class='existingWikiWord' href='/nlab/show/inverse'>inversion</a> map</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>inv</mi><mo lspace='verythinmathspace'>:</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'> inv \colon \mathcal{C}_1 \to \mathcal{C}_1 </annotation></semantics></math></div> <p>then <a class='existingWikiWord' href='/nlab/show/pullback+of+functions'>pullback of functions</a> along this map makes is an <a class='existingWikiWord' href='/nlab/show/involution'>involution</a> of the convolution algebra of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> and hence makes it into a <a class='existingWikiWord' href='/nlab/show/star-algebra'>star-algebra</a>.</p> <p>More generally for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> equipped with the structure of a <a class='existingWikiWord' href='/nlab/show/dagger-category'>dagger-category</a>, pullback along the dagger-functor</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>†</mo><mo lspace='verythinmathspace'>:</mo><msub><mi>𝒞</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'> \dagger \colon \mathcal{C}_1 \to \mathcal{C}_1 </annotation></semantics></math></div> <p>makes the convolution algebra a star-algebra.</p> </div> <div id="catalgebraIsWeakHopfalgebra" class="num_defn"> <h6 id="remark_3">Remark</h6> <p>If the category has finitely many objects and finitely many morphisms, then the category algebra is also a coalgebra via the unique comultiplication where every morphism in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><mo>⊂</mo><mi>R</mi><mo stretchy='false'>[</mo><msub><mi>C</mi> <mn>1</mn></msub><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>C_1\subset R[C_1]</annotation></semantics></math> is group like; the structure of the coalgebra and that of an algebra are compatible in the sense that they form a <a class='existingWikiWord' href='/nlab/show/weak+Hopf+algebra'>weak Hopf algebra</a>.</p> </div> <h3 id="ForLieGroupoids">For Lie groupoids</h3> <p>We discuss groupoid convolution <a class='existingWikiWord' href='/nlab/show/C%2A-algebras'>C*-algebras</a> for <a class='existingWikiWord' href='/nlab/show/Lie+groupoids'>Lie groupoids</a>/<a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a> (def. <a class='maruku-ref' href='#ContinuousCategoryAlgebraWithHalfDensities' />, prop. <a class='maruku-ref' href='#GroupoidConvolutionIs2Functor' /> below).</p> <div id="ForLieGroupoids" class="num_remark"> <h6 id="remark_4">Remark</h6> <p>If <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is a groupoid with extra <a class='existingWikiWord' href='/nlab/show/geometry'>geometric</a> structure, then there are natural variants of the above definition.</p> <p>Notably if <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> is a <a class='existingWikiWord' href='/nlab/show/Lie+groupoid'>Lie groupoid</a> then there is a variant where the functions in remark <a class='maruku-ref' href='#AsConvolutionAlgebra' /> are taken to be <a class='existingWikiWord' href='/nlab/show/smooth+functions'>smooth functions</a> and where the convolution <a class='existingWikiWord' href='/nlab/show/sum'>sum</a> is replaced by an <a class='existingWikiWord' href='/nlab/show/integration'>integration</a>. In order for this to make sense one needs to consider in fact functions with values in half-<a class='existingWikiWord' href='/nlab/show/densities'>densities</a> over the <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_1</annotation></semantics></math>.</p> <p>More generally, for a <a class='existingWikiWord' href='/nlab/show/bundle+gerbe'>bundle gerbe</a> over a Lie groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math>, hence a multiplicative <a class='existingWikiWord' href='/nlab/show/line+bundle'>line bundle</a> over <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒞</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{C}_1</annotation></semantics></math>, one can consider a convolution product on <a class='existingWikiWord' href='/nlab/show/sections'>sections</a> of this line bundle tensored with half-densities.</p> </div> <p>See the <a href="#ReferencesForSmoothGeometry">References – For continuous/smooth geometry</a>.</p> <div id="ContinuousCategoryAlgebraWithHalfDensities" class="num_defn"> <h6 id="definition_3">Definition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/Lie+groupoid+convolution+algebra'>Lie groupoid convolution algebra</a>)</strong></p> <p>Let <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/Lie+groupoid'>Lie groupoid</a>. Write <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>C</mi> <mi>c</mi> <mn>∞</mn></msubsup><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C^\infty_c(\mathcal{G}_1, \sqrt{\Omega})</annotation></semantics></math> for the space of smooth <a class='existingWikiWord' href='/nlab/show/half-densities'>half-densities</a> in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>T</mi> <mrow><mi>d</mi><mi>s</mi><mo>=</mo><mn>0</mn></mrow></msub><msub><mi>Γ</mi> <mn>1</mn></msub><mo>⊕</mo><msub><mi>T</mi> <mrow><mi>d</mi><mi>t</mi><mo>=</mo><mn>0</mn></mrow></msub><msub><mi>Γ</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>T_{d s = 0}\Gamma_1 \oplus T_{d t = 0}\Gamma_1</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/compact+support'>compact support</a> on its <a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_1</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/morphisms'>morphisms</a>. Let the <a class='existingWikiWord' href='/nlab/show/convolution+product'>convolution product</a></p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo>⋆</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo><mo>×</mo><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo><mo>→</mo><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \star \;\colon\; C_c(\mathcal{G}_1, \sqrt{\Omega}) \times C_c(\mathcal{G}_1, \sqrt{\Omega}) \to C_c(\mathcal{G}_1, \sqrt{\Omega}) </annotation></semantics></math></div> <p>be given on elements <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>,</mo><mi>g</mi><mo>∈</mo><msub><mi>C</mi> <mi>c</mi></msub><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f,g \in C_c(\mathcal{G}_1, \sqrt{\Omega})</annotation></semantics></math> over any element <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>γ</mi><mo>∈</mo><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\gamma \in \mathcal{G}_1</annotation></semantics></math> by the <a class='existingWikiWord' href='/nlab/show/integral'>integral</a></p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>⋆</mo><mi>g</mi><mo stretchy='false'>)</mo><mo lspace='verythinmathspace'>:</mo><mi>γ</mi><mo>↦</mo><msub><mo>∫</mo> <mrow><msub><mi>γ</mi> <mn>2</mn></msub><mo>∘</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo>=</mo><mi>γ</mi></mrow></msub><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>⋅</mo><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>γ</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (f \star g) \colon \gamma \mapsto \int_{\gamma_2\circ \gamma_1 = \gamma} f(\gamma_1) \cdot f(\gamma_2) \,. </annotation></semantics></math></div> <p>(Here we regard the integrand naturally as taking values in actual <a class='existingWikiWord' href='/nlab/show/densities'>densities</a> tensored with the pullback of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msqrt><mi>Ω</mi></msqrt></mrow><annotation encoding='application/x-tex'>\sqrt{\Omega}</annotation></semantics></math> along the composition map. This defines the <a class='existingWikiWord' href='/nlab/show/integration'>integration</a> of density-factor which then takes values in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msqrt><mi>Ω</mi></msqrt></mrow><annotation encoding='application/x-tex'>\sqrt{\Omega}</annotation></semantics></math>.)</p> <p>The algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>C</mi> <mi>c</mi> <mn>∞</mn></msubsup><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo><mo>,</mo><mo>⋆</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star)</annotation></semantics></math> is the <strong>groupoid convolution algebra</strong> of smooth compactly supported functions. As in remark <a class='maruku-ref' href='#AsConvolutionAlgebra' />, this is naturally a <a class='existingWikiWord' href='/nlab/show/star-algebra'>star-algebra</a> with <a class='existingWikiWord' href='/nlab/show/involution'>involution</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>inv</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>inv^\ast</annotation></semantics></math>.</p> </div> <p>This construction originates around (<a href="#Connes82">Connes 82</a>).</p> <div class="num_prop"> <h6 id="propositiondefinition">Proposition/Definition</h6> <p>For <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/Lie+groupoid'>Lie groupoid</a> and for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi><mo>∈</mo><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>x \in \mathcal{G}_0</annotation></semantics></math> any point in the manifold of <a class='existingWikiWord' href='/nlab/show/objects'>objects</a> there is an involutive <a class='existingWikiWord' href='/nlab/show/representation'>representation</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>x</mi></msub></mrow><annotation encoding='application/x-tex'>\pi_x</annotation></semantics></math> of the convolution algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>C</mi> <mi>c</mi> <mn>∞</mn></msubsup><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo><mo>,</mo><mo>⋆</mo><mo>,</mo><msup><mi>inv</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star, inv^\ast)</annotation></semantics></math> of def. <a class='maruku-ref' href='#ContinuousCategoryAlgebraWithHalfDensities' /> on the <a class='existingWikiWord' href='/nlab/show/canonical+Hilbert+space+of+half-densities'>canonical Hilbert space of half-densities</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy='false'>(</mo><msup><mi>s</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>L^2(s^{-1}(x))</annotation></semantics></math> of the source fiber of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math> given on any <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ξ</mi><mo>∈</mo><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy='false'>(</mo><msup><mi>s</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\xi \in L^2(s^{-1}(x))</annotation></semantics></math> by</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mi>x</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mi>ξ</mi><mo lspace='verythinmathspace'>:</mo><mi>γ</mi><mo>↦</mo><msub><mo>∫</mo> <mrow><msub><mi>γ</mi> <mn>1</mn></msub><mo>∈</mo><msup><mi>t</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>γ</mi><mo stretchy='false'>)</mo></mrow></msub><mi>f</mi><mo stretchy='false'>(</mo><msub><mi>γ</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mi>ξ</mi><mo stretchy='false'>(</mo><msubsup><mi>γ</mi> <mn>1</mn> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msubsup><mi>γ</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \pi_x(f) \xi \colon \gamma \mapsto \int_{\gamma_1 \in t^{-1}(\gamma)} f(\gamma_1) \xi(\gamma_1^{-1}\gamma) \,. </annotation></semantics></math></div> <p>This defines a <a class='existingWikiWord' href='/nlab/show/norm'>norm</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>|</mo><mrow><mo stretchy='false'>‖</mo><mo stretchy='false'>‖</mo></mrow></mrow><annotation encoding='application/x-tex'>|{\Vert \Vert}</annotation></semantics></math> on the <a class='existingWikiWord' href='/nlab/show/vector+space'>vector space</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msubsup><mi>C</mi> <mi>c</mi> <mn>∞</mn></msubsup><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><mi>Ω</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C_c^\infty(\mathcal{G}_1, \Omega)</annotation></semantics></math> given by the <a class='existingWikiWord' href='/nlab/show/supremum'>supremum</a> of the norms in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mn>2</mn></msup><mo stretchy='false'>(</mo><msup><mi>s</mi> <mrow><mo lspace='verythinmathspace' rspace='0em'>−</mo><mn>1</mn></mrow></msup><mo stretchy='false'>(</mo><mi>x</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>L^2(s^{-1}(x))</annotation></semantics></math> over all points <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>x</mi></mrow><annotation encoding='application/x-tex'>x</annotation></semantics></math>:</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>‖</mo><mi>f</mi><mo stretchy='false'>‖</mo></mrow><mo>≔</mo><msub><mi>Sup</mi> <mrow><mi>x</mi><mo>∈</mo><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></msub><mrow><mo stretchy='false'>‖</mo><msub><mi>π</mi> <mi>x</mi></msub><mo stretchy='false'>(</mo><mi>f</mi><mo stretchy='false'>)</mo><mo stretchy='false'>‖</mo></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> {\Vert f\Vert} \coloneqq Sup_{x \in \mathcal{G}_0} {\Vert \pi_x(f)\Vert} \,. </annotation></semantics></math></div> <p>The <a class='existingWikiWord' href='/nlab/show/Cauchy+completion'>Cauchy completion</a> of the <a class='existingWikiWord' href='/nlab/show/star+algebra'>star algebra</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><msubsup><mi>C</mi> <mi>c</mi> <mn>∞</mn></msubsup><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>,</mo><msqrt><mi>Ω</mi></msqrt><mo stretchy='false'>)</mo><mo>,</mo><mo>⋆</mo><mo>,</mo><msup><mi>inv</mi> <mo>*</mo></msup><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C_c^\infty(\mathcal{G}_1, \sqrt{\Omega}), \star, inv^\ast)</annotation></semantics></math> with respect to this <a class='existingWikiWord' href='/nlab/show/norm'>norm</a> is a <a class='existingWikiWord' href='/nlab/show/C-star-algebra'>C-star-algebra</a>, the <strong>convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebra</strong> of the Lie groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math>.</p> </div> <p>This is recalled at (<a href="#Connes94">Connes 94, prop. 3 on p. 106</a>).</p> <div id="ExtensionToBibundlesAndBimodules" class="num_remark"> <h6 id="remark_5">Remark</h6> <p>With suitable definitions, this construction constitutes something at least close to a <a class='existingWikiWord' href='/nlab/show/2-functor'>2-functor</a> from <a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a> to <a class='existingWikiWord' href='/nlab/show/C-star-algebras'>C-star-algebras</a> and <a class='existingWikiWord' href='/nlab/show/Hilbert+bimodules'>Hilbert bimodules</a> between them:</p> <p>In (<a href="#Mrcun99">Mrčun 99</a>) the convolution algebra construction for <a class='existingWikiWord' href='/nlab/show/%C3%A9tale+Lie+groupoids'>étale Lie groupoids</a> is extended to groupoid <a class='existingWikiWord' href='/nlab/show/bibundles'>bibundles</a> and shown to produce a <a class='existingWikiWord' href='/nlab/show/functor'>functor</a> to <a class='existingWikiWord' href='/nlab/show/C-star-algebras'>C-star-algebras</a> with (isomorphism classes of) <a class='existingWikiWord' href='/nlab/show/bimodules'>bimodules</a> between them. In (<a href="#KalisnikMrcun">Kališnik-Mrčun 07</a>) it is shown that if one remembers the additional <a class='existingWikiWord' href='/nlab/show/Hopf+algebroid'>Hopf algebroid</a> structure on the convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-alegras (the algebraic analog of remebering the <a class='existingWikiWord' href='/nlab/show/atlas'>atlas</a> of the <a class='existingWikiWord' href='/nlab/show/differentiable+stack'>differentiable stack</a> of a Lie groupoid) then this construction becomes a <a class='existingWikiWord' href='/nlab/show/full+subcategory'>full subcategory</a> inclusion of étale Lie groupoids into their convolution Hopf algebroids.</p> <p>In (<a href="#MuhleReaultWilliams87">Muhly-Renault-Williams 87</a>, <a href="#Landsman00">Landsman 00</a>) the generalization of the construction of a <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-bimodule from a groupoid <a class='existingWikiWord' href='/nlab/show/bibundle'>bibundle</a> to general <a class='existingWikiWord' href='/nlab/show/Lie+groupoids'>Lie groupoids</a> is discussed (not necessarily <a class='existingWikiWord' href='/nlab/show/%C3%A9tale+Lie+groupoid'>étale</a>), but only equivalence-bibundles are considered and are shown to yield <a class='existingWikiWord' href='/nlab/show/Morita+equivalence'>Morita equivalence</a> bimodules (no discussion of composition and functoriality here).</p> </div> <p>A precise form of this statement is the following <a href="#Nuiten13">Nuiten 13, theorem 3.3.1</a></p> <div id="GroupoidConvolutionIs2Functor" class="num_prop"> <h6 id="proposition">Proposition</h6> <p><strong>(<a class='existingWikiWord' href='/nlab/show/groupoid+convolution+algebra'>groupoid convolution algebra</a> is <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-functor'>(2,1)-functor</a> from <a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a> to <a class='existingWikiWord' href='/nlab/show/C%2A-algebras'>C*-algebras</a>)</strong></p> <p>The above construction of groupoid convolution algebras (def. <a class='maruku-ref' href='#ContinuousCategoryAlgebraWithHalfDensities' />) extends to a <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-functor'>(2,1)-functor</a></p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><msup><mi>DiffStack</mi> <mi>prop</mi></msup><mover><mo>⟶</mo><mphantom><mi>AAA</mi></mphantom></mover><msup><mi>C</mi> <mo>*</mo></msup><msubsup><mi>Alg</mi> <mi>bim</mi> <mi>op</mi></msubsup></mrow><annotation encoding='application/x-tex'> C^\ast(-) \;\colon\; DiffStack^{prop} \overset{\phantom{AAA}}{\longrightarrow} C^\ast Alg^{op}_{bim} </annotation></semantics></math></div> <p>between</p> <ol> <li> <p>the <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-category'>(2,1)-category</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>DiffStack</mi> <mi>prop</mi></msub></mrow><annotation encoding='application/x-tex'>DiffStack_{prop}</annotation></semantics></math> of <a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a> (i.e. <a class='existingWikiWord' href='/nlab/show/Lie+groupoids'>Lie groupoids</a> regarded as <a class='existingWikiWord' href='/nlab/show/smooth+stacks'>smooth stacks</a>) with proper morphisms between them (<a href="#Nuiten13">Nuiten 13, def. 2.2.35</a>)</p> </li> <li> <p>the <a class='existingWikiWord' href='/nlab/show/opposite+2-category'>opposite</a> <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-category'>(2,1)-category</a> of <a class='existingWikiWord' href='/nlab/show/C%2A-algebras'>C*-algebras</a> with <a class='existingWikiWord' href='/nlab/show/Hilbert+bimodules'>Hilbert bimodules</a> between them and <a class='existingWikiWord' href='/nlab/show/intertwiners'>intertwiners</a> between those.</p> </li> </ol> </div> <p>(<a href="#Nuiten13">Nuiten 13, theorem 3.3.1</a>)</p> <p>This way, much of <a class='existingWikiWord' href='/nlab/show/noncommutative+geometry'>noncommutative geometry</a> is exhibited as actually being the <a class='existingWikiWord' href='/nlab/show/higher+geometry'>higher geometry</a> of <a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a> inside all <a class='existingWikiWord' href='/nlab/show/smooth+stacks'>smooth stacks</a>. In particular, <a class='existingWikiWord' href='/nlab/show/groupoid+K-theory'>groupoid K-theory</a> is realized as the <a class='existingWikiWord' href='/nlab/show/KK-theory'>KK-theory</a> of convolution C<em>-algebras.</em></p> <h2 id="equivalent_characterizations">Equivalent characterizations</h2> <p>We discuss equivalent characterizations of category algebras/groupoid algebras that are useful in certain context</p> <ul> <li> <p><a href="#WeakColimit">As a weak colimit over a constant 2Vect-valued functor</a></p> </li> <li> <p><a href="#InTermsOfCompositionOfSpans">In terms of composition of spans</a></p> </li> </ul> <h3 id="WeakColimit">As a weak colimit over a constant <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mn>2</mn><mi>Vect</mi></mrow><annotation encoding='application/x-tex'>2Vect</annotation></semantics></math>-valued functor</h3> <p>Apparently for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoid</a> the category algebra of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/weak+limit'>weak colimit</a> over <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C}</annotation></semantics></math> of the functor <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>→</mo><mi>Vect</mi><mtext>-</mtext><mi>Mod</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C} \to Vect\text{-}Mod</annotation></semantics></math> constant on the ground field algebra.</p> <p>This statement is for instance in (<a href="#FHLT">FHLT, section 8.4</a>).</p> <p>The 2-cell in the universal co-cone corresponding to the morphism <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding='application/x-tex'>f \in C</annotation></semantics></math> is the <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mtext>-</mtext><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k\text{-}k[C]</annotation></semantics></math>-bimodule homomorphism <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>⋅</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo><mo>:</mo><mi>A</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding='application/x-tex'>f \cdot (-) : A \to A</annotation></semantics></math> that multiplies by <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo>∈</mo><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>f \in k[C]</annotation></semantics></math> from the left.</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd><mi>x</mi></mtd> <mtd /> <mtd><mover><mo>→</mo><mi>f</mi></mover></mtd> <mtd /> <mtd><mi>y</mi></mtd></mtr> <mtr><mtd><mi>k</mi></mtd> <mtd /> <mtd><mover><mo>→</mo><mi>k</mi></mover></mtd> <mtd /> <mtd><mi>k</mi></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mrow> <mpadded lspace='-100%width' width='0'><mrow /></mpadded></msub><mo>↘</mo></mtd> <mtd><msub><mo>⇙</mo> <mrow><mi>f</mi><mo>⋅</mo><mo stretchy='false'>(</mo><mo lspace='verythinmathspace' rspace='0em'>−</mo><mo stretchy='false'>)</mo></mrow></msub></mtd> <mtd><msub><mo>↙</mo> <mpadded width='0'><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mrow></mpadded></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ x &&\stackrel{f}{\to}&& y \\ k &&\stackrel{k}{\to}&& k \\ & {k[C]}_{\mathllap{}}\searrow &\swArrow_{f \cdot (-)}& \swarrow_{\mathrlap{k[C]}} \\ && k[C] } </annotation></semantics></math></div> <p>This description should be compared with the analogous description of the <a class='existingWikiWord' href='/nlab/show/action+groupoid'>action groupoid</a> by a weak colimit. One sees that the groupoid algebra is a linear incarnation of the action groupoid in some sense.</p> <h3 id="InTermsOfCompositionOfSpans">In terms of composition of spans</h3> <p>The category algebra of a category <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> is a special case of a general construction of <a class='existingWikiWord' href='/nlab/show/spans'>spans</a> (see also at <em><a class='existingWikiWord' href='/nlab/show/bi-brane'>bi-brane</a></em>).</p> <p>In order not to get distracted by inessential technicalities, consider the case of a finite <a class='existingWikiWord' href='/nlab/show/category'>category</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, i.e. an <a class='existingWikiWord' href='/nlab/show/internal+category'>internal category</a> in <a class='existingWikiWord' href='/nlab/show/FinSet'>FinSet</a>. This is a <a class='existingWikiWord' href='/nlab/show/span'>span</a></p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&&& C_0 } </annotation></semantics></math></div> <p>equipped with a composition operation: a morphism of spans from the composite span</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd> <mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ &&&& C_1 \times_{t,s} C_1 \\ &&& \swarrow && \searrow \\ && C_1 &&&& C_1 \\ & {}^{s}\swarrow && \searrow^{t} && {}^{s}\swarrow && \searrow^{t} \\ C_0 &&&& C_0 &&&& C_0 } </annotation></semantics></math></div> <p>to the original one, i.e. a morphism</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>comp</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'> comp : C_1 \times_{t,s} C_1 \to C_1 </annotation></semantics></math></div> <p>which respects source and target morphisms.</p> <p>Given this, consider the trivial vector bundle on the set of objects <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>C_0</annotation></semantics></math>. This is nothing but an assignment</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>I</mi><mo>:</mo><msub><mi>C</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>Vect</mi> <mi>k</mi></msub></mrow><annotation encoding='application/x-tex'> I : C_0 \to Vect_k </annotation></semantics></math></div> <p>of the <a class='existingWikiWord' href='/nlab/show/ground+field'>ground field</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math> to each element of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>0</mn></msub></mrow><annotation encoding='application/x-tex'>C_0</annotation></semantics></math>. There are two different ways to pull this vector bundle on objects back to a vector bundle on morphisms, once along the source, once along the target map.</p> <p>Then notice that the set of natural transformations between these two vector bundles</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy='false'>[</mo><mi>Sets</mi><mo>,</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy='false'>]</mo></mrow></msub><mo stretchy='false'>(</mo><msup><mi>s</mi> <mo>*</mo></msup><mi>I</mi><mo>,</mo><msup><mi>t</mi> <mo>*</mo></msup><mi>I</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> Hom_{[Sets,Vect_k]}(s^* I , t^* I) </annotation></semantics></math></div> <p>whose elements are 2-arrows of the form</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd><mover><mo>⇒</mo><mi>f</mi></mover></mtd> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mi>I</mi></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mi>I</mi></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><mi>Vect</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } </annotation></semantics></math></div> <p>are canonically in bijection with <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-calued functions on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>C_1</annotation></semantics></math>, hence with the vector space spanned by <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>C_1</annotation></semantics></math>, hence with the vector space underlying the category algebra</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>Hom</mi> <mrow><mo stretchy='false'>[</mo><mi>Sets</mi><mo>,</mo><msub><mi>Vect</mi> <mi>k</mi></msub><mo stretchy='false'>]</mo></mrow></msub><mo stretchy='false'>(</mo><msup><mi>s</mi> <mo>*</mo></msup><mi>I</mi><mo>,</mo><msup><mi>t</mi> <mo>*</mo></msup><mi>I</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> Hom_{[Sets,Vect_k]}(s^* I , t^* I) \simeq k[C] \,. </annotation></semantics></math></div> <p>The algebra structure on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><mo stretchy='false'>[</mo><mi>C</mi><mo stretchy='false'>]</mo></mrow><annotation encoding='application/x-tex'>k[C]</annotation></semantics></math> is similarly encoded in the diagrammatics: given two elements</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd><mover><mo>⇒</mo><mi>f</mi></mover></mtd> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mi>I</mi></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mi>I</mi></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><mi>Vect</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mi>and</mi><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mspace width='thickmathspace' /><mrow><mtable><mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd><mover><mo>⇒</mo><mi>g</mi></mover></mtd> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msub><mrow /> <mi>I</mi></msub><mo>↘</mo></mtd> <mtd /> <mtd><msub><mo>↙</mo> <mi>I</mi></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><mi>Vect</mi> <mi>k</mi></msub></mtd></mtr></mtable></mrow></mrow><annotation encoding='application/x-tex'> \array{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } \;\;\;\; and \;\;\;\; \array{ && C_1 \\ & {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{g}{\Rightarrow}&& C_0 \\ & {}_I \searrow && \swarrow_I \\ && Vect_k } </annotation></semantics></math></div> <p>their pre-composite is the diagram</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mtable><mtr><mtd /> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd /> <mtd /> <mtd><mo>↙</mo></mtd> <mtd /> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd> <mtd /> <mtd /> <mtd /> <mtd><msub><mi>C</mi> <mn>1</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd> <mtd /> <mtd><msup><mrow /> <mi>s</mi></msup><mo>↙</mo></mtd> <mtd /> <mtd><msup><mo>↘</mo> <mi>t</mi></msup></mtd></mtr> <mtr><mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd><mover><mo>⇒</mo><mi>f</mi></mover></mtd> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd> <mtd /> <mtd><mover><mo>⇒</mo><mi>g</mi></mover></mtd> <mtd /> <mtd><msub><mi>C</mi> <mn>0</mn></msub></mtd></mtr> <mtr><mtd /> <mtd><mo>↘</mo></mtd> <mtd /> <mtd /> <mtd><mo stretchy='false'>↓</mo></mtd> <mtd /> <mtd /> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd /> <mtd /> <mtd><mo>→</mo></mtd> <mtd /> <mtd><msub><mi>Vect</mi> <mi>k</mi></msub></mtd> <mtd /> <mtd><mo>←</mo></mtd></mtr></mtable></mrow><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \array{ &&&& C_1 \times_{t,s} C_1 \\ &&& \swarrow && \searrow \\ && C_1 &&&& C_1 \\ & {}^{s}\swarrow && \searrow^{t} && {}^{s}\swarrow && \searrow^{t} \\ C_0 &&\stackrel{f}{\Rightarrow}&& C_0 &&\stackrel{g}{\Rightarrow}&& C_0 \\ & \searrow &&& \downarrow &&& \swarrow \\ && \to &&Vect_k&& \leftarrow } \,. </annotation></semantics></math></div> <p>This is a composite transformation between three trivial vector bundles on the set <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>C_1 \times_{t,s} C_1</annotation></semantics></math> of composable morphisms in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>. As such, it is a function, which on the element consisting of the composable pair <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mover><mo>→</mo><mi>r</mi></mover><mover><mo>→</mo><mi>s</mi></mover></mrow><annotation encoding='application/x-tex'>\stackrel{r}{\to}\stackrel{s}{\to}</annotation></semantics></math> takes the value <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo stretchy='false'>(</mo><mi>r</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>f(r)\cdot g(s)</annotation></semantics></math>.</p> <p>In order to get back a transformation between vector bundles on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>C_1</annotation></semantics></math>, hence a transformation between vector bundles on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>C_1</annotation></semantics></math>, we <em>push forward</em> along the composition map <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>comp</mi><mo>:</mo><msub><mi>C</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>C</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>C</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>comp: C_1 \times_{t,s} C_1 \to C_1</annotation></semantics></math>. This just means that we add up the values on the fibers of this map.</p> <p>The result is the <a class='existingWikiWord' href='/nlab/show/convolution+product'>convolution product</a></p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>f</mi><mo>⋆</mo><mi>g</mi><mo stretchy='false'>)</mo><mo>:</mo><mi>t</mi><mo>↦</mo><munder><mo lspace='thinmathspace' rspace='thinmathspace'>∑</mo> <mrow><mi>s</mi><mo>∘</mo><mi>r</mi><mo>=</mo><mi>t</mi></mrow></munder><mi>f</mi><mo stretchy='false'>(</mo><mi>r</mi><mo stretchy='false'>)</mo><mo>⋅</mo><mi>g</mi><mo stretchy='false'>(</mo><mi>s</mi><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> (f\star g) : t \mapsto \sum_{s\circ r = t} f(r)\cdot g(s) \,. </annotation></semantics></math></div> <p>This is indeed the product in the category algebra.</p> <p>Looking at category algebras realizes them as a puny special case of a bigger story which involves <a class='existingWikiWord' href='/nlab/show/bi-brane'>bi-brane</a>s as morphisms between <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>n</mi></mrow><annotation encoding='application/x-tex'>n</annotation></semantics></math>-bundles/<math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>n</mi><mo>−</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(n-1)</annotation></semantics></math>-gerbes which live on spaces connected by correspondence spaces. This is related to a bunch of things, such as T-duality, Fourier-Mukai transformations and other issues of quantization. A description of this perspective is in</p> <ul> <li><a class='existingWikiWord' href='/schreiber/show/Nonabelian+cocycles+and+their+quantum+symmetries'>Nonabelian cocycles and their quantum symmetries</a>.</li> </ul> <p>This is related to observations such as described here:</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/John+Baez'>John Baez</a>, <a href="http://golem.ph.utexas.edu/category/2007/03/quantization_and_cohomology_we_16.html"><em>Quantization and Cohomology (Week 17)</em></a></p> </li> <li> <p>Urs Schreiber, <a href="http://golem.ph.utexas.edu/category/2007/02/qft_of_charged_nparticle_tdual.html"><em>QFT of Charged n-Particle: T-Duality</em></a></p> </li> </ul> <h2 id="examples">Examples</h2> <h3 id="basic_examples">Basic examples</h3> <div class="num_example"> <h6 id="example">Example</h6> <p>The convolution algebra of a <a class='existingWikiWord' href='/nlab/show/set'>set</a>/<a class='existingWikiWord' href='/nlab/show/manifold'>manifold</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> regarded as a <a class='existingWikiWord' href='/nlab/show/discrete+groupoid'>discrete groupoid</a>/<a class='existingWikiWord' href='/nlab/show/Lie+groupoid'>Lie groupoid</a> with only <a class='existingWikiWord' href='/nlab/show/identity'>identity</a> <a class='existingWikiWord' href='/nlab/show/morphisms'>morphisms</a> is the ordinary <a class='existingWikiWord' href='/nlab/show/function+algebra'>function algebra</a> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_2">Example</h6> <p>For <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/set'>set</a> the convolution algebra of the <a class='existingWikiWord' href='/nlab/show/pair+groupoid'>pair groupoid</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pair</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>Pair(X)_\bullet</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/matrix+algebra'>matrix algebra</a> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mrow><mo stretchy='false'>|</mo><mi>X</mi><mo stretchy='false'>|</mo></mrow><mo>×</mo><mrow><mo stretchy='false'>|</mo><mi>X</mi><mo stretchy='false'>|</mo></mrow></mrow><annotation encoding='application/x-tex'>{\vert X\vert} \times {\vert X\vert}</annotation></semantics></math> matrices.</p> </div> <div class="num_example"> <h6 id="example_3">Example</h6> <p>For <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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 class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Pair</mi><mo stretchy='false'>(</mo><mi>X</mi><msub><mo stretchy='false'>)</mo> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>Pair(X)_\bullet</annotation></semantics></math> its <a class='existingWikiWord' href='/nlab/show/pair+groupoid'>pair groupoid</a> regarded as a <a class='existingWikiWord' href='/nlab/show/Lie+groupoid'>Lie groupoid</a> its smooth convolution algebra is the algebra of <span class='newWikiWord'>smoothing kernels<a href='/nlab/new/smoothing+kernels'>?</a></span> on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>X</mi></mrow><annotation encoding='application/x-tex'>X</annotation></semantics></math>.</p> </div> <div class="num_example"> <h6 id="example_4">Example</h6> <p><math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒞</mi><mo>=</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\mathcal{C} = \mathbf{B}G</annotation></semantics></math> is the <a class='existingWikiWord' href='/nlab/show/delooping'>delooping</a> <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoid</a> of a <a class='existingWikiWord' href='/nlab/show/discrete+group'>discrete group</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> (the groupoid with a single object and <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> as its set of morphisms), then def. <a class='maruku-ref' href='#CategoryAlgebra' /> reduces to that of the <a class='existingWikiWord' href='/nlab/show/group+algebra'>group algebra</a> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>:</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>R</mi><mo stretchy='false'>[</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>]</mo><mo>≃</mo><mi>R</mi><mo stretchy='false'>[</mo><mi>G</mi><mo stretchy='false'>]</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> R[\mathbf{B}G] \simeq R[G] \,. </annotation></semantics></math></div></div> <h3 id="incidence_algebras_poset_convolution_algebras">Incidence algebras (poset convolution algebras)</h3> <p>See at <a class='existingWikiWord' href='/nlab/show/M%C3%B6bius+inversion#IncidenceAlgebrasZetaFn'>Möbius inversion#IncidenceAlgebrasZetaFn</a>.</p> <h3 id="HigherGroupoidConvolutionAlgebras">Higher groupoid convolution algebras and n-vector spaces/n-modules</h3> <blockquote> <p>under construction</p> </blockquote> <p>We discuss here a natural generalization of the notion of <a class='existingWikiWord' href='/nlab/show/groupoid+convolution+algebras'>groupoid convolution algebras</a> to <a class='existingWikiWord' href='/nlab/show/higher+algebra'>higher algebras</a> for <a class='existingWikiWord' href='/nlab/show/n-groupoid'>higher groupoids</a>.</p> <p>There may be several sensible such generalizations. The one discussed now follows the principle of iterated <a class='existingWikiWord' href='/nlab/show/internalization'>internalization</a> and naturally matches to the concept of <a class='existingWikiWord' href='/nlab/show/n-modules'>n-modules</a> (<a class='existingWikiWord' href='/nlab/show/n-vector+spaces'>n-vector spaces</a>) as they appear in <a class='existingWikiWord' href='/nlab/show/extended+prequantum+field+theory'>extended prequantum field theory</a>.</p> <p>In order to disentangle conceptual from technical aspects, we first discuss <a class='existingWikiWord' href='/nlab/show/discrete+%E2%88%9E-groupoid'>geometrically discrete higher groupoids</a>. The results of this discussion then in particular help to suggest what the right definition of “higher Lie groupoid” in the context of higher convolution algebras should be in the first place.</p> <p>The considerations are based on the following</p> <div class="num_remark"> <h6 id="remark_6">Remark</h6> <p>By the discussion at <em><a class='existingWikiWord' href='/nlab/show/2-module'>2-module</a></em> we may think of the <a class='existingWikiWord' href='/nlab/show/2-category'>2-category</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><msub><mi>Alg</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>k Alg_b</annotation></semantics></math> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/associative+algebras'>associative algebras</a> and <a class='existingWikiWord' href='/nlab/show/bimodules'>bimodules</a> between them as a model for the 2-category <a class='existingWikiWord' href='/nlab/show/2Mod'>2Mod</a> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi></mrow><annotation encoding='application/x-tex'>k</annotation></semantics></math>-<a class='existingWikiWord' href='/nlab/show/2-modules'>2-modules</a> that admit a 2-<a class='existingWikiWord' href='/nlab/show/basis'>basis</a> (<a class='existingWikiWord' href='/nlab/show/2-vector+spaces'>2-vector spaces</a>). Hence the groupoid convolution algebra constructiuon is a 2-functor</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mspace width='thickmathspace' /><mo lspace='verythinmathspace'>:</mo><mspace width='thickmathspace' /><mi>Grpd</mi><mo>→</mo><mn>2</mn><mi>Mod</mi><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> C \;\colon\; Grpd \to 2 Mod \,. </annotation></semantics></math></div> <p>There is then the following systematic refinement of this to higher groupoids and higher algebra: by the discussion at <em><a class='existingWikiWord' href='/nlab/show/n-module'>n-module</a></em>, 3-modules are algebra objects in <a class='existingWikiWord' href='/nlab/show/2Mod'>2Mod</a> and maps between them are <a class='existingWikiWord' href='/nlab/show/bimodule'>bimodule</a> objects in there. An algebra object in <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>k</mi><msub><mi>Alg</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>k Alg_b</annotation></semantics></math> is equivalently a <a class='existingWikiWord' href='/nlab/show/sesquialgebra'>sesquialgebra</a>, an algebra equipped with a second algebra structure up to coherent homotopy, that is exhibited by structure bimodules.</p> <p>Special cases of this are <a class='existingWikiWord' href='/nlab/show/bialgebras'>bialgebras</a>, for which these structure bimodules come from actual algebra homomorphisms. Examples of these in turn are <a class='existingWikiWord' href='/nlab/show/Hopf+algebras'>Hopf algebras</a>. These we naturally re-discover as special higher groupoid convolution higher algebras in example <a class='maruku-ref' href='#DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup' /> below.</p> </div> <p>This iterated internalization on the codomain of the groupoid convolution algebra functor has a natural analog on its domain: a 2-groupoid we may present by a <a class='existingWikiWord' href='/nlab/show/double+groupoid'>double groupoid</a>, namely a <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category'>groupoid object in an (∞,1)-category</a> in <a class='existingWikiWord' href='/nlab/show/Grpd'>Grpd</a> which is 3-<a class='existingWikiWord' href='/nlab/show/coskeleton'>coskeletal</a> as a <a class='existingWikiWord' href='/nlab/show/simplicial+object'>simplicial object</a> in <a class='existingWikiWord' href='/nlab/show/Grpd'>Grpd</a>.</p> <div class="num_remark"> <h6 id="remark_7">Remark</h6> <p>Given a <a class='existingWikiWord' href='/nlab/show/groupoid+object+in+an+%28%E2%88%9E%2C1%29-category'>groupoid object</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math> in the <a class='existingWikiWord' href='/nlab/show/%282%2C1%29-topos'>(2,1)-topos</a> <a class='existingWikiWord' href='/nlab/show/Grpd'>Grpd</a> hence a <a class='existingWikiWord' href='/nlab/show/double+groupoid'>double groupoid</a>, applying the groupoid convolution algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(2,1)</annotation></semantics></math>-functor <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math> to the corresponding <a class='existingWikiWord' href='/nlab/show/simplicial+object'>simplicial object</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub><mo>∈</mo><msup><mi>Grpd</mi> <mrow><msup><mi>Δ</mi> <mi>op</mi></msup></mrow></msup></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet \in Grpd^{\Delta^{op}}</annotation></semantics></math> yields:</p> <ul> <li> <p>groupoid convolution algebras <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathcal{G}_0)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathcal{G}_1)</annotation></semantics></math>,</p> </li> <li> <p>a <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><msub><mo>⊗</mo> <mrow><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow></msub><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><mo stretchy='false'>)</mo><mo>−</mo><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mn>0</mn></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathcal{G}_1) \otimes_{C(\mathcal{G}_{0,1})} C(\mathcal{G}_1)-C(\mathcal{G}_{0})</annotation></semantics></math>-bimodule, assigned to the <a class='existingWikiWord' href='/nlab/show/composition'>composition</a> functor <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mo>∂</mo> <mn>1</mn></msub><mo lspace='verythinmathspace'>:</mo><msub><mi>𝒢</mi> <mn>1</mn></msub><munder><mo>×</mo><mrow><msub><mi>𝒢</mi> <mn>0</mn></msub></mrow></munder><msub><mi>𝒢</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>𝒢</mi> <mn>1</mn></msub></mrow><annotation encoding='application/x-tex'>\partial_1 \colon \mathcal{G}_1 \underset{\mathcal{G}_0}{\times} \mathcal{G}_1 \to \mathcal{G}_1</annotation></semantics></math>.</p> </li> </ul> <p>Under the 2-functoriality of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi></mrow><annotation encoding='application/x-tex'>C</annotation></semantics></math>, the <a class='existingWikiWord' href='/nlab/show/Segal+conditions'>Segal conditions</a> satisfied by <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math> make this bimodule exhibi a <a class='existingWikiWord' href='/nlab/show/sesquialgebra'>sesquialgebra</a> structure over <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><msub><mi>𝒢</mi> <mrow><mn>0</mn><mo>,</mo><mn>1</mn></mrow></msub><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathcal{G}_{0,1})</annotation></semantics></math>.</p> <p>This sesquialgebra we call the the <strong>double groupoid convolution 2-algebra</strong> of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒢</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{G}_\bullet</annotation></semantics></math>.</p> <p>(Here we make invariant sense of the <a class='existingWikiWord' href='/nlab/show/tensor+product'>tensor product</a> by evaluating on a <a class='existingWikiWord' href='/nlab/show/Reedy+model+structure'>Reedy fibrant</a> representative.)</p> </div> <div id="DoubleGroupoid2AlgebraOfDeloopingOfFiniteGroup" class="num_example"> <h6 id="example_5">Example</h6> <p>Let <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> be a <a class='existingWikiWord' href='/nlab/show/finite+group'>finite group</a>. Write <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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> for its <a class='existingWikiWord' href='/nlab/show/delooping'>delooping</a> <a class='existingWikiWord' href='/nlab/show/groupoid'>groupoid</a> (the connected groupoid with <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>π</mi> <mn>1</mn></msub><mo>=</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>\pi_1 = G</annotation></semantics></math>). Since this is just a <a class='existingWikiWord' href='/nlab/show/1-groupoid'>1-groupoid</a>, there are two natural ways to present <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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> as a <a class='existingWikiWord' href='/nlab/show/double+groupoid'>double groupoid</a>:</p> <ol> <li> <p><math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mo stretchy='false'>(</mo><mi>⋯</mi><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mover><munder><mo>→</mo><mi>id</mi></munder><mover><mo>→</mo><mi>id</mi></mover></mover><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo><mo>≃</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\underset{\longrightarrow}{\lim}(\cdots \mathbf{B}G\stackrel{\to}{\stackrel{\to}{\to}} \mathbf{B}G \stackrel{\overset{id}{\to}}{\underset{id}{\to}} \mathbf{B}G) \simeq \mathbf{B}G</annotation></semantics></math>;</p> </li> <li> <p><math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><munder><mi>lim</mi><mo>⟶</mo></munder><mo stretchy='false'>(</mo><mi>⋯</mi><mi>G</mi><mo>×</mo><mi>G</mi><mover><mover><mo>→</mo><mo>→</mo></mover><mo>→</mo></mover><mi>G</mi><mover><mo>→</mo><mo>→</mo></mover><mo>*</mo><mo stretchy='false'>)</mo><mo>≃</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding='application/x-tex'>\underset{\longrightarrow}{\lim}(\cdots G \times G \stackrel{\to}{\stackrel{\to}{\to}} G \stackrel{\to}{\to} *) \simeq \mathbf{B}G</annotation></semantics></math>.</p> </li> </ol> <p>(The first is “vertically constant”, the second is “horizontally constant”).</p> <p>Applying the <a class='existingWikiWord' href='/nlab/show/groupoid+convolution+algebra'>groupoid convolution algebra</a> functor to the first presentation yields the groupoid convolution algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathbf{B}G)</annotation></semantics></math> equipped with a trivial multiplication bimodule, hence just the group convolution algebra <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mstyle mathvariant='bold'><mi>B</mi></mstyle><mi>G</mi><mo stretchy='false'>)</mo><mo>≃</mo><msub><mi>C</mi> <mi>conv</mi></msub><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(\mathbf{B}G) \simeq C_{conv}(G)</annotation></semantics></math>.</p> <p>Applying however the groupoid convolution algebra functor to the second presentation yields the <em>commutative</em> algebra of functions <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(G)</annotation></semantics></math> equipped with the multiplication bimodule which is <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>C(G \times G)</annotation></semantics></math> regarded as a <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy='false'>)</mo><mo>,</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(C(G\times G), C(G))</annotation></semantics></math>-bimdodule, where the right action is induced by pullback along the group product map <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi><mo>×</mo><mi>G</mi><mo>→</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>G \times G \to G</annotation></semantics></math>.</p> <p>This bimodule is in the image of the functor <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Alg</mi><mo>→</mo><msub><mi>Alg</mi> <mi>b</mi></msub></mrow><annotation encoding='application/x-tex'>Alg \to Alg_b</annotation></semantics></math> that sends algebra homomorphisms to their induced bimodules, by sending <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi><mo lspace='verythinmathspace'>:</mo><mi>A</mi><mo>→</mo><mi>B</mi></mrow><annotation encoding='application/x-tex'>f \colon A \to B</annotation></semantics></math> to <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>A</mi></mrow><annotation encoding='application/x-tex'>A</annotation></semantics></math> regarded as an <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mo stretchy='false'>(</mo><mi>A</mi><mo>,</mo><mi>B</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>(A,B)</annotation></semantics></math>-bimodule with the canonical left action on itself and the right action induced by <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>f</mi></mrow><annotation encoding='application/x-tex'>f</annotation></semantics></math>. Namely this bimdoule correspondonds to the map</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mo lspace='verythinmathspace'>:</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo><mo>→</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo>×</mo><mi>G</mi><mo stretchy='false'>)</mo><mo>≃</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo><mo>⊗</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'> \Delta \colon C(G) \to C(G \times G) \simeq C(G) \otimes C(G) </annotation></semantics></math></div> <p>given on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>ϕ</mi><mo>∈</mo><mi>C</mi><mo stretchy='false'>(</mo><mi>G</mi><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\phi \in C(G)</annotation></semantics></math> and <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo>∈</mo><mi>G</mi></mrow><annotation encoding='application/x-tex'>g_1, g_2 \in G</annotation></semantics></math> by</p> <div class="maruku-equation"><math class='maruku-mathml' display='block' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>Δ</mi><mi>ϕ</mi><mo lspace='verythinmathspace'>:</mo><mo stretchy='false'>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mo>↦</mo><mi>ϕ</mi><mo stretchy='false'>(</mo><msub><mi>g</mi> <mn>1</mn></msub><mo>⋅</mo><msub><mi>g</mi> <mn>2</mn></msub><mo stretchy='false'>)</mo><mspace width='thinmathspace' /><mo>.</mo></mrow><annotation encoding='application/x-tex'> \Delta \phi \colon (g_1, g_2) \mapsto \phi(g_1 \cdot g_2) \,. </annotation></semantics></math></div> <p>This is that standard <a class='existingWikiWord' href='/nlab/show/coalgebra'>coproduct</a> on the standard dual <a class='existingWikiWord' href='/nlab/show/Hopf+algebra'>Hopf algebra</a> associated with <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> <p>In summary this means that (for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math> a finite group):</p> <ol> <li> <p>If we regard <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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> as presented as a double groupoid constant on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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>, then the corresponding groupoid convolution <a class='existingWikiWord' href='/nlab/show/sesquialgebra'>sesquialgebra</a> (basis for a <a class='existingWikiWord' href='/nlab/show/n-module'>3-module</a>) is the convolution algebra of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>;</p> </li> <li> <p>If instead we regard <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/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> as presented as the double groupoid which is degreewise constant as a groupoid, then the corresponding groupoid convolution sesquialgebra is the standard (“dual”) <a class='existingWikiWord' href='/nlab/show/Hopf+algebra'>Hopf algebra</a> structure on the commutative pointwise product algebra of functions on <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>G</mi></mrow><annotation encoding='application/x-tex'>G</annotation></semantics></math>.</p> </li> </ol> </div> <h2 id="properties">Properties</h2> <h3 id="RelationToTwistedKTheory">Relation to (twisted) K-theory</h3> <p>The <a class='existingWikiWord' href='/nlab/show/operator+K-theory'>operator K-theory</a> of the convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebra of a <a class='existingWikiWord' href='/nlab/show/topological+groupoid'>topological groupoid</a> <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msub><mi>𝒳</mi> <mo>•</mo></msub></mrow><annotation encoding='application/x-tex'>\mathcal{X}_\bullet</annotation></semantics></math> may be thought of as the <a class='existingWikiWord' href='/nlab/show/topological+K-theory'>topological K-theory</a> of the corresponding <a class='existingWikiWord' href='/nlab/show/topological+stack'>topological stack</a>. More generally, for <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><mi>𝒳</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><mo stretchy='false'>)</mo></mrow><annotation encoding='application/x-tex'>\mathcal{X} \to \mathbf{B}^2 U(1)</annotation></semantics></math> a <a class='existingWikiWord' href='/nlab/show/principal+2-bundle'>principal 2-bundle</a> (<a class='existingWikiWord' href='/nlab/show/bundle+gerbe'>bundle gerbe</a>) on the groupoid/stack, the <a class='existingWikiWord' href='/nlab/show/operator+K-theory'>operator K-theory</a> of the corresponding twisted convolution algebra is the <a class='existingWikiWord' href='/nlab/show/twisted+K-theory'>twisted K-theory</a> of the stack.</p> <p>(<a href="#TXLG">Tu, Xu, Laurent-Gengoux 04</a>)</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/groupoid+quantale'>groupoid quantale</a> - an analogue of groupoid ring/algebra where free abelian groups/vector spaces are replaced by free <a class='existingWikiWord' href='/nlab/show/sup-lattice'>sup-lattice</a>s</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/noncommutative+topology'>noncommutative topology</a>, <a class='existingWikiWord' href='/nlab/show/noncommutative+geometry'>noncommutative geometry</a></p> <p><a class='existingWikiWord' href='/nlab/show/KK-theory'>KK-theory</a></p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/bibundle'>bibundle</a>, <a class='existingWikiWord' href='/nlab/show/bimodule'>bimodule</a></p> </li> </ul> <h2 id="references">References</h2> <h3 id="for_partial_orders">For partial orders</h3> <ul id="Rota64"> <li><a class='existingWikiWord' href='/nlab/show/Gian-Carlo+Rota'>Gian-Carlo Rota</a>, <em>On the foundations of combinatorial theory I: theory of Möbius functions</em> , Z. Wahrscheinlichkeitstheorie und Verw. Gebiete 2 (1964), 340–368.</li> </ul> <h3 id="for_discrete_geometry">For discrete geometry</h3> <p>The <a class='existingWikiWord' href='/nlab/show/homotopy+colimit'>homotopy colimit</a>-interpretation of category algebras over discrete categories is discussed in</p> <ul id="FHLT"> <li><a class='existingWikiWord' href='/nlab/show/Dan+Freed'>Dan Freed</a>, <a class='existingWikiWord' href='/nlab/show/Mike+Hopkins'>Mike Hopkins</a>, <a class='existingWikiWord' href='/nlab/show/Jacob+Lurie'>Jacob Lurie</a>, <a class='existingWikiWord' href='/nlab/show/Constantin+Teleman'>Constantin Teleman</a>, <em><a class='existingWikiWord' href='/nlab/show/Topological+Quantum+Field+Theories+from+Compact+Lie+Groups'>Topological Quantum Field Theories from Compact Lie Groups</a></em> (<a href="http://arxiv.org/abs/0905.0731">arXiv</a>)</li> </ul> <p>Groupoid algebras of geometrically discrete groupoids twisted by <a class='existingWikiWord' href='/nlab/show/principal+2-bundles'>principal 2-bundles</a>/<a class='existingWikiWord' href='/nlab/show/bundle+gerbes'>bundle gerbes</a>/<a class='existingWikiWord' href='/nlab/show/central+extension+of+groupoids'>groupoid central extension</a> is reviwed in</p> <ul> <li> <p>Eitan Angel, <em>A Geometric Construction of Cyclic Cocycles on Twisted Convolution Algebras</em>, PhD thesis (2010)</p> <p>Cyclic cocycles on twisted convolution algebras, (<a href="http://arxiv.org/abs/1103.0578">arXiv.1103.0578</a>)</p> </li> </ul> <h3 id="ReferencesForSmoothGeometry">For continuous/smooth geometry</h3> <h4 id="convolution_algebras">Convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</h4> <p>The study of convolution <a class='existingWikiWord' href='/nlab/show/C-star+algebras'>C-star algebras</a> of <a class='existingWikiWord' href='/nlab/show/Lie+groupoids'>Lie groupoids</a> goes back to</p> <ul> <li id="Renault80"><a class='existingWikiWord' href='/nlab/show/Jean+Renault'>Jean Renault</a>, <em>A groupoid approach to <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math> algebras</em>, <p>Springer Lecture Notes in Mathematics, 793, Springer-Verlag, New York, 1980.</p> </li> </ul> <p>Where the <a class='existingWikiWord' href='/nlab/show/integration'>integration</a> is performed against a fixed <a class='existingWikiWord' href='/nlab/show/Haar+measure'>Haar measure</a>. Surveys are for instance in</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Nigel+Higson'>Nigel Higson</a>, <em>Groupoids, <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras and Index theory</em> (<a href="http://folk.uio.no/rognes/higson/zurich.pdf">pdf</a>)</p> </li> <li> <p>PlanetMath, <em><a href="http://planetmath.org/groupoidcconvolutionalgebras">groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-convolution algebras</a></em>.</p> </li> </ul> <p>The construction via <a class='existingWikiWord' href='/nlab/show/sections'>sections</a> of <a class='existingWikiWord' href='/nlab/show/bundles'>bundles</a> of <a class='existingWikiWord' href='/nlab/show/half-densities'>half-densities</a> (avoiding a choice of Haar measure) is due to</p> <ul> <li id="Connes82"><a class='existingWikiWord' href='/nlab/show/Alain+Connes'>Alain Connes</a>, <em>A survey of foliations and operator algebras</em>, Proc. Sympos. Pure Math., AMS Providence, 32 (1982), 521–628</li> </ul> <p>A review is on page 106 of</p> <ul> <li id="Connes94"><a class='existingWikiWord' href='/nlab/show/Alain+Connes'>Alain Connes</a>, <em><a class='existingWikiWord' href='/nlab/show/Noncommutative+Geometry'>Noncommutative Geometry</a></em>, Academic Press, San Diego, CA, (1994)</li> </ul> <p>See also</p> <ul> <li id="Nuiten13"><a class='existingWikiWord' href='/nlab/show/Joost+Nuiten'>Joost Nuiten</a>, <em><a class='existingWikiWord' href='/schreiber/show/master+thesis+Nuiten'>Cohomological quantization of local prequantum boundary field theory</a></em>, master thesis, August 2013</li> </ul> <p>More along these lines is in</p> <ul> <li> <p><a class='existingWikiWord' href='/nlab/show/Paul+Muhly'>Paul Muhly</a>, Dana P. Williams, <em>Continuous trace groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras II</em> Math. Scand. 70 (1992), no. 1, 127–145; MR 93i:46117). (<a href="http://www.math.dartmouth.edu/~dana/dpwpdfs/muhwil-ms90.pdf">pdf</a>)</p> </li> <li> <p><a class='existingWikiWord' href='/nlab/show/Paul+Muhly'>Paul Muhly</a>, <a class='existingWikiWord' href='/nlab/show/Jean+Renault'>Jean Renault</a>, Dana P. Williams, <em>Continuous trace groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras III</em> , Transactions of the AMS, vol 348, Number 9 (1996) (<a href="http://www.jstor.org/stable/2155247">jstor</a>)</p> </li> <li> <p>Mădălina Roxana Buneci, <em>Groupoid Representations,</em> Ed. Mirton: Timishoara (2003).</p> </li> <li> <p>Mădălina Roxana Buneci, <em>Groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-Algebras</em>, Surveys in Mathematics and its Applications, Volume 1: 71–98. (<a href="http://www.utgjiu.ro/math/mbuneci/preprint/p0024.pdf">pdf</a>)</p> </li> <li> <p>Mădălina Roxana Buneci, <em>Convolution algebras for topological groupoids with locally compact fibers</em> (2011) (<a href="http://journals.bg.agh.edu.pl/OPUSCULA/31-2/31-2-02.pdf">pdf</a>)</p> </li> </ul> <p>A review in the context of <a class='existingWikiWord' href='/nlab/show/geometric+quantization'>geometric quantization</a> is in section 4.3 of</p> <ul id="Bos"> <li><a class='existingWikiWord' href='/nlab/show/Rogier+Bos'>Rogier Bos</a>, <em>Groupoids in geometric quantization</em> PhD Thesis (2007) <a href="http://www.math.ist.utl.pt/~rbos/ProefschriftA4.pdf">pdf</a></li> </ul> <p>Specifically the convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras of <a class='existingWikiWord' href='/nlab/show/bundle+gerbes'>bundle gerbes</a> regarded as <a class='existingWikiWord' href='/nlab/show/centrally+extended+groupoids'>centrally extended groupoids</a> (algebras whose <a class='existingWikiWord' href='/nlab/show/modules'>modules</a> (see <a href="#ReferencesModulesOverConvolutionAlgebra">below</a>) are <a class='existingWikiWord' href='/nlab/show/gerbe+modules'>gerbe modules</a>/<a class='existingWikiWord' href='/nlab/show/twisted+bundle'>twisted bundle</a>) is discussed in section 5 of</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Alan+Carey'>Alan Carey</a>, Stuart Johnson, <a class='existingWikiWord' href='/nlab/show/Michael+Murray'>Michael Murray</a>, <em>Holonomy on D-Branes</em>, J. Geom. Phys. 52 (2004) 186-216 (<a href="http://arxiv.org/abs/hep-th/0204199">arXiv:hep-th/0204199</a>)</li> </ul> <p>A discussion of convolution algebras of <a class='existingWikiWord' href='/nlab/show/symplectic+groupoids'>symplectic groupoids</a> (in the context of <a class='existingWikiWord' href='/nlab/show/geometric+quantization+of+symplectic+groupoids'>geometric quantization of symplectic groupoids</a>) is in</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Eli+Hawkins'>Eli Hawkins</a>, <em>A groupoid approach to quantization</em> (<a href="http://arxiv.org/abs/math.SG/0612363">arXiv:math.SG/0612363</a>)</li> </ul> <p>Functoriality of the construction of <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-convolution algebras (its extension to groupoid-<a class='existingWikiWord' href='/nlab/show/bibundles'>bibundles</a>) is discussed in</p> <ul id="MuhleReaultWilliams87"> <li><a class='existingWikiWord' href='/nlab/show/Paul+Muhly'>Paul Muhly</a>, <a class='existingWikiWord' href='/nlab/show/Jean+Renault'>Jean Renault</a>, D. Williams, <em>Equivalence and isomorphism for groupoid <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras</em>, J. Operator Theory 17 (1987), no. 1, 3–22.</li> </ul> <ul id="Mrcun99"> <li><span class='newWikiWord'>Janez Mr?un<a href='/nlab/new/Janez+Mr%3Fun'>?</a></span>, <em>Functoriality of the bimodule associated to a Hilsum-Skandalis map</em>. K-Theory 18 (1999) 235–253.</li> </ul> <ul id="Landsman00"> <li><a class='existingWikiWord' href='/nlab/show/Klaas+Landsman'>Klaas Landsman</a>, <em>The Muhly-Renault-Williams theorem for Lie groupoids and its classical counterpart</em>, Lett. Math. Phys. 54 (2000), no. 1, 43–59. (<a href="http://arxiv.org/abs/math-ph/0008005">arXiv:math-ph/0008005</a>)</li> </ul> <ul> <li><a class='existingWikiWord' href='/nlab/show/Klaas+Landsman'>Klaas Landsman</a>, <em>Operator Algebras and Poisson Manifolds Associated to Groupoids</em>, Commun. Math. Phys. 222, 97 – 116 (2001) (<a href="http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.7.2716">web</a>)</li> </ul> <h4 id="ReferencesConvolutionHopfAlgebroids">Convolution Hopf algebroids</h4> <p>A characterization of the convolution algebras of <a class='existingWikiWord' href='/nlab/show/%C3%A9tale+groupoids'>étale groupoids</a> with their <a class='existingWikiWord' href='/nlab/show/Hopf+algebroid'>Hopf algebroid</a> structure is in</p> <ul id="KalisnikMrcun"> <li> <p><span class='newWikiWord'>Janez Mr?un<a href='/nlab/new/Janez+Mr%3Fun'>?</a></span>, <em>On spectral representation of coalgebras and Hopf algebroids</em> (<a href="http://arxiv.org/abs/math/0208199">arXiv:math/0208199</a>)</p> </li> <li> <p><span class='newWikiWord'>Jure Kali?nik<a href='/nlab/new/Jure+Kali%3Fnik'>?</a></span>, <span class='newWikiWord'>Janez Mr?un<a href='/nlab/new/Janez+Mr%3Fun'>?</a></span>, <em>Equivalence between the Morita categories of etale Lie groupoids and of locally grouplike Hopf algebroids</em> (<a href="http://arxiv.org/abs/math/0703374">arXiv:math/0703374</a>)</p> </li> </ul> <h4 id="ReferencesModulesOverConvolutionAlgebra">Modules over Lie groupoid convolution algebras and K-theory</h4> <p>Discussion of <a class='existingWikiWord' href='/nlab/show/modules'>modules</a> over Lie groupoid convolution algebras is in the following articles.</p> <p>In (<a href="#Renault80">Renault80</a>) <a class='existingWikiWord' href='/nlab/show/measurable+space'>measurable</a> representations of topological groupoids are related to modules over their <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>L</mi> <mn>1</mn></msup></mrow><annotation encoding='application/x-tex'>L^1</annotation></semantics></math> convolution <a class='existingWikiWord' href='/nlab/show/star+algebra'>star algebra</a> <a class='existingWikiWord' href='/nlab/show/Banach+algebras'>Banach algebras</a> hence over their envoloping <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebras.</p> <p>In (<a href="#Bos">Bos, chapter 7</a>) is discussion refining this to continuous representations and representation of a convolution <math class='maruku-mathml' display='inline' xmlns='http://www.w3.org/1998/Math/MathML'><semantics><mrow><msup><mi>C</mi> <mo>*</mo></msup></mrow><annotation encoding='application/x-tex'>C^\ast</annotation></semantics></math>-algebra, also in section 4 of:</p> <ul> <li><a class='existingWikiWord' href='/nlab/show/Rogier+Bos'>Rogier Bos</a>, <em>Continuous representations of groupoids</em> (<a href="http://arxiv.org/abs/math/0612639">arXiv:math/0612639</a>)</li> </ul> <p>Representation of convolution algebras of <a class='existingWikiWord' href='/nlab/show/%C3%A9tale+groupoids'>étale groupoids</a> is in</p> <ul> <li><span class='newWikiWord'>Jure Kali?nik<a href='/nlab/new/Jure+Kali%3Fnik'>?</a></span>, <em>Groupoid representations and modules over the convolution algebras</em> (<a href="http://arxiv.org/abs/0806.1832">arXiv:0806.1832</a>)</li> </ul> <p>The <a class='existingWikiWord' href='/nlab/show/operator+K-theory'>operator K-theory</a> of groupoid convolution algebras (the <a class='existingWikiWord' href='/nlab/show/topological+K-theory'>topological K-theory</a> of the corresponding <a class='existingWikiWord' href='/nlab/show/differentiable+stacks'>differentiable stacks</a>) is discussed in</p> <ul id="TXLG"> <li><a class='existingWikiWord' href='/nlab/show/Jean-Louis+Tu'>Jean-Louis Tu</a>, <a class='existingWikiWord' href='/nlab/show/Ping+Xu'>Ping Xu</a>, <a class='existingWikiWord' href='/nlab/show/Camille+Laurent-Gengoux'>Camille Laurent-Gengoux</a>, <em>Twisted K-theory of differentiable stacks</em>, Annales scientifiques de l’École Normale Supérieure (2004) Volume: 37, Issue: 6, page 841-910 (<a href="http://arxiv.org/abs/math/0306138">arXiv:math/0306138</a>)</li> </ul> <p>Construction of cocycles in <a class='existingWikiWord' href='/nlab/show/KK-theory'>KK-theory</a> and <a class='existingWikiWord' href='/nlab/show/spectral+triples'>spectral triples</a> from groupoid convolution is in</p> <ul id="Meland"> <li>Bram Mesland, <em>Groupoid cocycles and K-theory</em> (<a href="http://arxiv.org/abs/1005.3677">arXiv:1005.3677</a>)</li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on August 30, 2018 at 10:35:21. 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