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href="/nlab/show/linear+algebra">linear algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebra">algebra</a>, <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebra</a></li> </ul> <h2 id="sidebar_definitions">Definitions</h2> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/group">group</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/group+algebra">group algebra</a>, <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a>, <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>, <a class="existingWikiWord" href="/nlab/show/n-vector+space">n-vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/affine+space">affine space</a>, <a class="existingWikiWord" href="/nlab/show/symplectic+vector+space">symplectic vector space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/equivariant+object">equivariant object</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bimodule">bimodule</a>, <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a>, <a class="existingWikiWord" href="/nlab/show/Frobenius+reciprocity">Frobenius reciprocity</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Hilbert+space">Hilbert space</a>, <a class="existingWikiWord" href="/nlab/show/Banach+space">Banach space</a>, <a class="existingWikiWord" href="/nlab/show/Fourier+transform">Fourier transform</a>, <a class="existingWikiWord" href="/nlab/show/functional+analysis">functional analysis</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbit">orbit</a>, <a class="existingWikiWord" href="/nlab/show/coadjoint+orbit">coadjoint orbit</a>, <a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+representation">unitary representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+quantization">geometric quantization</a>, <a class="existingWikiWord" href="/nlab/show/coherent+state">coherent state</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/socle">socle</a>, <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module+algebra">module algebra</a>, <a class="existingWikiWord" href="/nlab/show/comodule+algebra">comodule algebra</a>, <a class="existingWikiWord" href="/nlab/show/Hopf+action">Hopf action</a>, <a class="existingWikiWord" href="/nlab/show/measuring">measuring</a></p> </li> </ul> <h2 id="geometric_representation_theory">Geometric representation theory</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>, <a class="existingWikiWord" href="/nlab/show/perverse+sheaf">perverse sheaf</a>,</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Grothendieck+group">Grothendieck group</a>, <a class="existingWikiWord" href="/nlab/show/lambda-ring">lambda-ring</a>, <a class="existingWikiWord" href="/nlab/show/symmetric+function">symmetric function</a>, <a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a>, <a class="existingWikiWord" href="/nlab/show/torsor">torsor</a>, <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>, <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/geometric+function+theory">geometric function theory</a>, <a class="existingWikiWord" href="/nlab/show/groupoidification">groupoidification</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Eilenberg-Moore+category">Eilenberg-Moore category</a>, <a class="existingWikiWord" href="/nlab/show/algebra+over+an+operad">algebra over an operad</a>, <a class="existingWikiWord" href="/nlab/show/actegory">actegory</a>, <a class="existingWikiWord" href="/nlab/show/crossed+module">crossed module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/reconstruction+theorems">reconstruction theorems</a></p> </li> </ul> <h2 id="sidebar_theorems">Theorems</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Borel-Weil-Bott+theorem">Borel-Weil-Bott theorem</a></p> </li> <li> <p><span class="newWikiWord">Be?linson-Bernstein localization<a href="/nlab/new/Be%3Flinson-Bernstein+localization">?</a></span></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kazhdan-Lusztig+theory">Kazhdan-Lusztig theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/BBDG+decomposition+theorem">BBDG decomposition theorem</a></p> </li> </ul> <div> <p> <a href="/nlab/edit/representation+theory+-+contents">Edit this sidebar</a> </p> </div></div></div> <h4 id="higher_algebra">Higher algebra</h4> <div class="hide"><div> <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></div></div> <h4 id="category_theory"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-Category theory</h4> <div class="hide"><div> <p><strong><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a></strong></p> <p><strong>Background</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+theory">category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(n,r)-category</a></p> </li> </ul> <p><strong>Basic concepts</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/hom-object+in+a+quasi-category">hom-objects</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/equivalence+in+a+quasi-category">equivalences in</a>/<a class="existingWikiWord" href="/nlab/show/equivalence+of+quasi-categories">of</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/sub-quasi-category">sub-(∞,1)-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/reflective+sub-%28%E2%88%9E%2C1%29-category">reflective sub-(∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/localization+of+an+%28%E2%88%9E%2C1%29-category">reflective localization</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/opposite+quasi-category">opposite (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/over+quasi-category">over (∞,1)-category</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/join+of+quasi-categories">join of quasi-categories</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/exact+%28%E2%88%9E%2C1%29-functor">exact (∞,1)-functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-functors">(∞,1)-category of (∞,1)-functors</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+of+%28%E2%88%9E%2C1%29-presheaves">(∞,1)-category of (∞,1)-presheaves</a></p> </li> </ul> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/fibrations+of+quasi-categories">fibrations</a></strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/inner+fibration">inner fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/left+fibration">left/right fibration</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Cartesian+fibration">Cartesian fibration</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cartesian+morphism">Cartesian morphism</a></li> </ul> </li> </ul> </li> </ul> <p><strong>Universal constructions</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/limit+in+quasi-categories">limit</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/terminal+object+in+a+quasi-category">terminal object</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor">adjoint functors</a></p> </li> </ul> <p><strong>Local presentation</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+presentable+%28%E2%88%9E%2C1%29-category">locally presentable</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/essentially+small+%28%E2%88%9E%2C1%29-category">essentially small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/locally+small+%28%E2%88%9E%2C1%29-category">locally small</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/accessible+%28%E2%88%9E%2C1%29-category">accessible</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/idempotent-complete+%28%E2%88%9E%2C1%29-category">idempotent-complete</a></p> </li> </ul> </li> </ul> <p><strong>Theorems</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Yoneda+lemma">(∞,1)-Yoneda lemma</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-Grothendieck+construction">(∞,1)-Grothendieck construction</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/adjoint+%28%E2%88%9E%2C1%29-functor+theorem">adjoint (∞,1)-functor theorem</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/monadicity+theorem">(∞,1)-monadicity theorem</a></p> </li> </ul> <p><strong>Extra stuff, structure, properties</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/stable+%28%E2%88%9E%2C1%29-category">stable (∞,1)-category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a></p> </li> </ul> <p><strong>Models</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/category+with+weak+equivalences">category with weak equivalences</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+category">model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/derivator">derivator</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/quasi-category">quasi-category</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+quasi-categories">model structure for quasi-categories</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+Cartesian+fibrations">model structure for Cartesian fibrations</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/relation+between+quasi-categories+and+simplicial+categories">relation to simplicial categories</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+coherent+nerve">homotopy coherent nerve</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/simplicial+model+category">simplicial model category</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presentable quasi-category</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kan+complex">Kan complex</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+sets">model structure for Kan complexes</a></li> </ul> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <ul> <li><a href='#representations_as_functors'>Representations as functors</a></li> <li><a href='#structured_representations_as_morphisms_in_2toposes'>Structured representations as morphisms in 2-toposes</a></li> <li><a href='#IdeaInfRepresentations'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Representations</a></li> <li><a href='#representations_2'><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Representations</a></li> </ul> <li><a href='#Definition'>Definition</a></li> <ul> <li><a href='#InHomotopyTypeTheory'>In homotopy type theory</a></li> </ul> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>In full generality, an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation</strong> is an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a> of some <a class="existingWikiWord" href="/nlab/show/higher+algebra">higher algebraic</a> structure on some object in a <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category</a> up to <a class="existingWikiWord" href="/nlab/show/coherent">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a>. One also speaks of <em>representation up to homotopy</em> or maybe <em>sh-representations</em> .</p> <p>We motivate the general notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation by recalling first some <a class="existingWikiWord" href="/nlab/show/category+theory">category theoretic</a> aspects of the ordinary notion of representation, and then leading over to the analogous <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theoretic</a> notions.</p> <h3 id="representations_as_functors">Representations as functors</h3> <p>Recall that for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a> with <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>=</mo><mo stretchy="false">(</mo><mi>G</mi><mover><mo>→</mo><mo>→</mo></mover><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G = (G \stackrel{\to}{\to})</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math><a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> the category of <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a>s (over some base <a class="existingWikiWord" href="/nlab/show/field">field</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>), ordinary linear <a class="existingWikiWord" href="/nlab/show/representation">representation</a>s of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> are equivalently <a class="existingWikiWord" href="/nlab/show/functor">functor</a>s</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>k</mi><mi>Vect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to k Vect \,. </annotation></semantics></math></div> <p>Such a functor takes the single object of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> to some vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi></mrow><annotation encoding="application/x-tex">V</annotation></semantics></math> and takes every morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mo>*</mo><mover><mo>→</mo><mi>g</mi></mover><mo>*</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(* \stackrel{g}{\to} * )</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> labeled by an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> to a linear <a class="existingWikiWord" href="/nlab/show/automorphism">automorphism</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\rho(g) : V \to V</annotation></semantics></math> such that composition and the identity is respected. We have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+categories">equivalence of categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Func</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>,</mo><mi>k</mi><mi>Vect</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mi>Rep</mi> <mi>k</mi></msub><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> Func(\mathbf{B}G, k Vect) \simeq Rep_k(G) \,. </annotation></semantics></math></div> <p>Here the category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Vect</mi></mrow><annotation encoding="application/x-tex">Vect</annotation></semantics></math> could be replaced by other categories and they need not be <a class="existingWikiWord" href="/nlab/show/abelian+categories">abelian categories</a> or otherwise linear for the above to make sense. For instance if we take instead the category <a class="existingWikiWord" href="/nlab/show/Set">Set</a> itself, then a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to Set </annotation></semantics></math></div> <p>is what is called a <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>. In <a class="existingWikiWord" href="/nlab/show/topology">topology</a> one is interested in representations in <a class="existingWikiWord" href="/nlab/show/Top">Top</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Top</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to Top \,. </annotation></semantics></math></div> <p>(However, rarely is it sufficient to regard these just as functors to the 1-category <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Top</mi></mrow><annotation encoding="application/x-tex">Top</annotation></semantics></math>. Instead, in order to speak about topological <a class="existingWikiWord" href="/nlab/show/fiber+bundle">fiber bundle</a>s and fibrations, one needs to regard <a class="existingWikiWord" href="/nlab/show/Top">Top</a> here as an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> and regard <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex">\rho : \mathbf{B}G \to Top</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a>. This we come to below in <a href="#IdeaInfRepresentations">∞-Representations</a>)</p> <p>Moreover, we may replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> by a more general <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> any groupoid, a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>k</mi><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> \rho : K \to k Vect </annotation></semantics></math></div> <p>is called a linear representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>. This now picks not just a single vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>V</mi><mo>∈</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex">V \in Vect</annotation></semantics></math>, but one vector space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mi>x</mi></msub></mrow><annotation encoding="application/x-tex">V_x</annotation></semantics></math> for each <a class="existingWikiWord" href="/nlab/show/object">object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>x</mi><mo>∈</mo><mi>K</mi></mrow><annotation encoding="application/x-tex">x \in K</annotation></semantics></math>. And to each morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mover><mo>→</mo><mi>g</mi></mover><mi>y</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x \stackrel{g}{\to} y)</annotation></semantics></math> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> is assigns a <a class="existingWikiWord" href="/nlab/show/linear+map">linear map</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>V</mi> <mi>x</mi></msub><mo>→</mo><msub><mi>V</mi> <mi>y</mi></msub></mrow><annotation encoding="application/x-tex">\rho(g) : V_x \to V_y</annotation></semantics></math>.</p> <p>For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>=</mo><msub><mi>Π</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">K = \Pi_1(X)</annotation></semantics></math> is the <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> of a <a class="existingWikiWord" href="/nlab/show/manifold">manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, then a representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>Π</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>vect</mi></mrow><annotation encoding="application/x-tex"> \rho : \Pi(X) \to vect </annotation></semantics></math></div> <p>is a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/curvature">flat</a> <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>.</p> <p>And we do not even need to assume that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> here is a groupoid. For instance if <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/directed+graph">directed graph</a> (or <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>) and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">F(D)</annotation></semantics></math> its <a class="existingWikiWord" href="/nlab/show/path+category">path category</a>, then a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Vect</mi></mrow><annotation encoding="application/x-tex"> F(D) \to Vect </annotation></semantics></math></div> <p>is called a <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a>-representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>D</mi></mrow><annotation encoding="application/x-tex">D</annotation></semantics></math>.</p> <p>One could in principle therefore speak of a functor</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>F</mi><mo stretchy="false">(</mo><mi>D</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Set</mi></mrow><annotation encoding="application/x-tex"> F(D) \to Set </annotation></semantics></math></div> <p>as a “quiver permutation representation”, but there does not seem to be much use of this terminology in practice. The examples do show, however, that there is considerable overlap between the notion of <em><a class="existingWikiWord" href="/nlab/show/representation">representation</a></em> and of <em><a class="existingWikiWord" href="/nlab/show/functor">functor</a></em> .</p> <h3 id="structured_representations_as_morphisms_in_2toposes">Structured representations as morphisms in 2-toposes</h3> <p>Still a bit more generally, we can speak of representations that preserve extra structure, such as <a class="existingWikiWord" href="/nlab/show/smooth+structure">smooth structure</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> we have that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>: an object in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> of <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-sheaves">(2,1)-sheaves</a> over the <a class="existingWikiWord" href="/nlab/show/site">site</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C = </annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> or <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">C =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>.</p> <p>We may also promote the category <a class="existingWikiWord" href="/nlab/show/Vect">Vect</a> to this <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-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>-topos, by replacing it by the <a class="existingWikiWord" href="/nlab/show/stack">stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>VectBund</mi></mrow><annotation encoding="application/x-tex">VectBund</annotation></semantics></math>, which assigns to each test manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">U \in C</annotation></semantics></math> the groupoid of smooth <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a>s over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>U</mi></mrow><annotation encoding="application/x-tex">U</annotation></semantics></math>. Then a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>VectBund</mi></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to VectBund </annotation></semantics></math></div> <p>in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(C)</annotation></semantics></math> is a <em>smooth representation</em> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, in that the linear automorphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><mi>V</mi><mo>→</mo><mi>V</mi></mrow><annotation encoding="application/x-tex">\rho(g) : V \to V</annotation></semantics></math> depend smoothly on the point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math>.</p> <p>We can recover the underlying ordinary representation by applying the <a class="existingWikiWord" href="/nlab/show/global+section">global section</a> functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo>:</mo><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo><mo>→</mo><mi>Grpd</mi></mrow><annotation encoding="application/x-tex">\Gamma : Sh_{(2,1)}(C) \to Grpd</annotation></semantics></math>. This is given by evaluating every thing on the <a class="existingWikiWord" href="/nlab/show/terminal+object">terminal object</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo>*</mo><mo>∈</mo><mi>C</mi></mrow><annotation encoding="application/x-tex">{*} \in C</annotation></semantics></math>, which here is just the ordinary <a class="existingWikiWord" href="/nlab/show/point">point</a>, regarded as a smmoth manifold.</p> <p>This yields the underlying bare representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><mi>ρ</mi><mo stretchy="false">)</mo><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>Vect</mi><mspace width="thinmathspace"></mspace><mo>.</mo></mrow><annotation encoding="application/x-tex"> \Gamma(\rho) : \mathbf{B}G \to Vect \,. </annotation></semantics></math></div> <p>Conversely, one finds that extending such a bare representation from the point to all test spaces in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> amounts to equipping it with smooth structure.</p> <p>As before, this is not restricted to <a class="existingWikiWord" href="/nlab/show/connected">connected</a> objects: we may replace <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> here with any <a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a>. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{P}_1(X)</annotation></semantics></math> is smooth <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a> a representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mi>VectBund</mi></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{P}_1(X) \to VectBund </annotation></semantics></math></div> <p>as a morphism in the 2-topos is a <a class="existingWikiWord" href="/nlab/show/vector+bundle">vector bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a>. Or if we consider non-linear representations, a representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{P}_1(X) \to \mathbf{B}G </annotation></semantics></math></div> <p>is a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> with <a class="existingWikiWord" href="/nlab/show/connection+on+a+bundle">connection on a bundle</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. See <a class="existingWikiWord" href="/nlab/show/parallel+transport">parallel transport</a> for more details and references.</p> <p>By just changing the <a class="existingWikiWord" href="/nlab/show/site">site</a> here, we can implement other geometric structures. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/algebraic+group">algebraic group</a> we may think of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> as an <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a> over something like the <a class="existingWikiWord" href="/nlab/show/fppf-site">fppf-site</a> structure <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><msubsup><mi>CAlg</mi> <mi>k</mi> <mi>op</mi></msubsup></mrow><annotation encoding="application/x-tex">C = CAlg_k^{op}</annotation></semantics></math> on formal duals of commutative <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-algebras or similar.</p> <p>In this case there is a well-known good generalization of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>VectBund</mi></mrow><annotation encoding="application/x-tex">VectBund</annotation></semantics></math>: instead of just vector bundles we can consider their completion to <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaves">quasicoherent sheaves</a>. The stack of these is the object in the <a class="existingWikiWord" href="/nlab/show/%282%2C1%29-topos">(2,1)-topos</a> given by</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo>:</mo><mi>Spec</mi><mi>A</mi><mo>↦</mo><mi>A</mi><mi>Mod</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> QC : Spec A \mapsto A Mod \,, </annotation></semantics></math></div> <p>where on the right we have the groupoid of <a class="existingWikiWord" href="/nlab/show/module">module</a>s over <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>. Over the point this is again just a <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi></mrow><annotation encoding="application/x-tex">k</annotation></semantics></math>-module hence a <a class="existingWikiWord" href="/nlab/show/vector+space">vector space</a> and hence a representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mi>QC</mi></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to QC </annotation></semantics></math></div> <p>of an algebraic group is a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> on a vector space.</p> <p>But also here we may allow the represented structure to have more than one object. For instance for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/scheme">scheme</a> regarded as an object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Sh</mi> <mrow><mo stretchy="false">(</mo><mn>2</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow></msub><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(C)</annotation></semantics></math> a representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> in the context of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>QC</mi></mrow><annotation encoding="application/x-tex">QC</annotation></semantics></math> is a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>QC</mi><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> \rho : X \to QC \,, </annotation></semantics></math></div> <p>which is equivalently a <a class="existingWikiWord" href="/nlab/show/quasicoherent+sheaf">quasicoherent sheaf</a> of modules on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. As before, we may think of this as assigning to each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> one representation space, only that in a scheme <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> there are no morphisms that would act on these.</p> <p>But more generally for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/algebraic+stack">algebraic stack</a>, a representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>QC</mi></mrow><annotation encoding="application/x-tex"> \rho : K \to QC </annotation></semantics></math></div> <p>assigns to each point of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>K</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">K_0</annotation></semantics></math> a representation space, such that these glue together to a quasicoherent sheaf of modules, and to each <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> a morphism between the corresponding representation spaces, as before.</p> <p>Analogous constructions are available for more general sites, effectively we can take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi></mrow><annotation encoding="application/x-tex">C</annotation></semantics></math> to be the <a class="existingWikiWord" href="/nlab/show/opposite+category">opposite category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/algebras+over+a+Lawvere+theory">algebras over a Lawvere theory</a> for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi></mrow><annotation encoding="application/x-tex">T</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/algebraic+theory">algebraic theory</a> that contains the theory of <a class="existingWikiWord" href="/nlab/show/abelian+group">abelian group</a>s. If for instance we take <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>T</mi><mo>=</mo></mrow><annotation encoding="application/x-tex">T =</annotation></semantics></math> <a class="existingWikiWord" href="/nlab/show/CartSp">CartSp</a> we are back to the smooth case discussed before.</p> <p>Also notice that if we take the site to be the point, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo>=</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">C = *</annotation></semantics></math>, then sheaves over it are just sets and stacks over it are just bare groupoids, so that we recover the discussion at the very beginning.</p> <h3 id="IdeaInfRepresentations"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Representations</h3> <p>We have found above the term “representation” is, to a large extent, congruent with the term “morphism in a (2,1)-topos with codomain a stack of modules”.</p> <p>This way of thinking of representations has an immediate generalization to <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a> and in particular to <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category+theory">(∞,1)-category theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>To start with the simple discussion over the point again, a model for a notion of a category of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-modules that is useful is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Ch_\bullet(k)</annotation></semantics></math> that is <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/model+structure+on+chain+complexes">model structure on chain complexes</a>.</p> <p>If again <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/discrete+group">discrete group</a>, then an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a> (equivalently: a “strong homotopy-functor” or “homotopy coherent functor”, see there for details)</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><msub><mi>Ch</mi> <mo>•</mo></msub><mo stretchy="false">(</mo><mi>k</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex"> \rho : \mathbf{B}G \to Ch_\bullet(k) </annotation></semantics></math></div> <p>assigns</p> <ul> <li> <p>to the single obect of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math> a chain complex <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">V_\bullet</annotation></semantics></math>;</p> </li> <li> <p>to a group element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>∈</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">g \in G</annotation></semantics></math> a chain map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>:</mo><msub><mi>V</mi> <mo>•</mo></msub><mo>→</mo><msub><mi>V</mi> <mo>•</mo></msub></mrow><annotation encoding="application/x-tex">\rho(g) : V_\bullet \to V_\bullet</annotation></semantics></math>;</p> </li> <li> <p>to a pair of group elements <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo></mrow><annotation encoding="application/x-tex">g, g'</annotation></semantics></math> a chain homotopy</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo>:</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo>∘</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⇒</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mi>g</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(g,g') : \rho(g')\circ\rho(g) \Rightarrow \rho(g' g)</annotation></semantics></math>;</p> </li> <li> <p>to a triple of group elements a homotopy of homotopies between composites of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>′</mo><mo stretchy="false">)</mo><mo>,</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>,</mo><mi>g</mi><mo>″</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(g,g'), \rho(g,g'')</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>g</mi><mo>′</mo><mo>,</mo><mi>g</mi><mo>″</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(g',g'')</annotation></semantics></math> and so on</p> <p>(see the diagrams at <a class="existingWikiWord" href="/nlab/show/group+cohomology">group cohomology</a> for more details in low degree).</p> </li> </ul> <p>In other words, this is much like representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> as before on an ordinary vector space, only that now the action property of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi></mrow><annotation encoding="application/x-tex">\rho</annotation></semantics></math> holds only <em>up to <a class="existingWikiWord" href="/nlab/show/coherent">coherent</a> <a class="existingWikiWord" href="/nlab/show/homotopy">homotopy</a></em> . Therefore people also speak of <em>representations up to homotopy</em> (<a href="#AbadCrainic">AbadCrainic</a>) as well as <em>strong homotopy representations</em> and many other variants.</p> <p>As before, there is in principle no reason to restrict oneself to representations of groupoids here. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> or even <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> (recall the <a class="existingWikiWord" href="/nlab/show/quiver">quiver</a> representations) and for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Mod</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-modules (for instance as <a class="existingWikiWord" href="/nlab/show/presentable+%28%E2%88%9E%2C1%29-category">presented</a> by a <a class="existingWikiWord" href="/nlab/show/model+structure+on+modules+over+an+algebra+over+an+operad">model structure on modules over an algebra over an operad</a>) we may call an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>K</mi><mo>→</mo><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> \rho : K \to Mod </annotation></semantics></math></div> <p>an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math>.</p> <p>If we wish to consider <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-generalizations of <a class="existingWikiWord" href="/nlab/show/permutation+representation">permutation representation</a>s we can also consider more general codomain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories here. For instance if we take <a class="existingWikiWord" href="/nlab/show/%E2%88%9EGrpd">∞Grpd</a> itself, then an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-permutation representation</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mn>∞</mn><mi>Grpd</mi></mrow><annotation encoding="application/x-tex"> K \to \infty Grpd </annotation></semantics></math></div> <p>is known as an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-presheaf">(∞,1)-presheaf</a>. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>K</mi></mrow><annotation encoding="application/x-tex">K</annotation></semantics></math> the <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> of an ordinary group or the <a class="existingWikiWord" href="/nlab/show/orbit+category">orbit category</a> of a <a class="existingWikiWord" href="/nlab/show/topological+group">topological group</a>, such appear genuinely from the point of view of representations for instance in <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> and <a class="existingWikiWord" href="/nlab/show/orbit+category">equivariant homotopy</a> theory. Notice that by the <a class="existingWikiWord" href="/nlab/show/homotopy+hypothesis">homotopy hypothesis</a>-theorem we have an <a class="existingWikiWord" href="/nlab/show/equivalence+of+%28%E2%88%9E%2C1%29-categories">equivalence of (∞,1)-categories</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mn>∞</mn><mi>Grpd</mi><mo>≃</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> \infty Grpd \simeq Top </annotation></semantics></math></div> <p>so that the above is equivalently an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-functor">(∞,1)-functor</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>K</mi><mo>→</mo><mi>Top</mi></mrow><annotation encoding="application/x-tex"> K \to Top </annotation></semantics></math></div> <p>hence literally a <em>representation up to homotopy</em> in the classical sense of <a class="existingWikiWord" href="/nlab/show/homotopy+theory">homotopy theory</a>.</p> <p>As before, all this may be lifted from the point into large classes of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>es to equip the notion of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation with geometric structure (algebraic structure, smooth structure, etc.)</p> <p>There are then analogs of the above relation between representations of path groupoids and connections on bundles. For more on this see <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>. Quite generally, in every <a class="existingWikiWord" href="/nlab/show/locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">locally ∞-connected (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> there is a notion of <a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid+in+a+locally+%E2%88%9E-connected+%28%E2%88%9E%2C1%29-topos">fundamental ∞-groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>P</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{P}(X)</annotation></semantics></math> of any object <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>. Representation of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>Π</mi></mstyle><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\mathbf{\Pi}(X)</annotation></semantics></math> define general abstract <a href="http://nlab.mathforge.org/nlab/show/cohesive+(infinity,1)-topos#FlatDifferentialCohomology">flat differential cohomology</a> and <a class="existingWikiWord" href="/nlab/show/local+system">local system</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, generally also in <a class="existingWikiWord" href="/nlab/show/nonabelian+cohomology">nonabelian cohomology</a> (see there for some more properties and examples).</p> <p>For instance <a class="existingWikiWord" href="/nlab/show/dg-geometry">dg-geometry</a> is the study of the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a> over an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-site">(∞,1)-site</a> of formal duals of <a class="existingWikiWord" href="/nlab/show/dg-algebra">dg-algebra</a>s. Again there is the canonical <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>QC</mi><mo>:</mo><mi>Spec</mi><mi>A</mi><mo>↦</mo><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex"> QC : Spec A \mapsto A Mod </annotation></semantics></math></div> <p>on this site, where now however <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mi>Mod</mi></mrow><annotation encoding="application/x-tex">A Mod</annotation></semantics></math> denotes the <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a> (or <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> if we do a more comprehensive discussion) of <a class="existingWikiWord" href="/nlab/show/chain+complex">chain complex</a>es equipped with the structure of a <a class="existingWikiWord" href="/nlab/show/module">module</a> ove the dg-algebra <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi></mrow><annotation encoding="application/x-tex">A</annotation></semantics></math>.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> any <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-stack">∞-stack</a> then a morphism</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>X</mi><mo>→</mo><mi>QC</mi></mrow><annotation encoding="application/x-tex"> \rho : X \to QC </annotation></semantics></math></div> <p>is a equivalently a <a class="existingWikiWord" href="/nlab/show/quasicoherent+%E2%88%9E-stack">quasicoherent ∞-stack</a> of modules on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>, or an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation with “dg-algebraic structure”.</p> <p>If one replaces <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> here by its <a class="existingWikiWord" href="/nlab/show/de+Rham+space">de Rham stack</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>dR</mi></msub></mrow><annotation encoding="application/x-tex">X_{dR}</annotation></semantics></math> then dg-algebraic <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representations</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><msub><mi>X</mi> <mi>dR</mi></msub><mo>→</mo><mi>QC</mi></mrow><annotation encoding="application/x-tex"> \rho : X_{dR} \to QC </annotation></semantics></math></div> <p>are <a class="existingWikiWord" href="/nlab/show/D-module">D-module</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math>.</p> <p>A discussion of these higher categorical structure in <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> is in (<a href="#Ben-ZviNadler">Ben-ZviNadler</a>).</p> <h3 id="representations_2"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-Representations</h3> <p>If the codomain <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Mod</mi></mrow><annotation encoding="application/x-tex">Mod</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> that is just a <a class="existingWikiWord" href="/nlab/show/%28n%2C1%29-category">(n,1)-category</a> (all <a class="existingWikiWord" href="/nlab/show/k-morphisms">k-morphisms</a>s for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>k</mi><mo>></mo><mi>n</mi></mrow><annotation encoding="application/x-tex">k \gt n</annotation></semantics></math> are effectively identities) then an <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation is called an <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-representation</strong>. These are <em>representations up to homotopy</em> where from degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> on all homotopies are actually identities: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> representations up to homotopy.</p> <p>As always in <a class="existingWikiWord" href="/nlab/show/higher+category+theory">higher category theory</a>, the cases for low <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math> are more restrictive but typically admit a more tractable detailed analysis and construction.</p> <p>2-representations of <a class="existingWikiWord" href="/nlab/show/2-groups">2-groups</a> and <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>s on various variants of <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a>s have been considered for instance in (<a href="#Schreiber">Schreiber</a>, <a href="#BaezBaratinFreidelWise">BaezBaratinFreidelWise</a>, and other places).</p> <p>In analogy to the case for <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">n=1</annotation></semantics></math>, 2-Representations <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>P</mi></mstyle> <mn>2</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo>→</mo><mn>2</mn><mi>Vect</mi></mrow><annotation encoding="application/x-tex">\mathbf{P}_2(X) \to 2 Vect</annotation></semantics></math> of the smooth <a class="existingWikiWord" href="/nlab/show/path+2-groupoid">path 2-groupoid</a> of a <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> describe <a class="existingWikiWord" href="/nlab/show/connections+on+a+2-bundle">connections on a 2-bundle</a>. See there for more details.</p> <h2 id="Definition">Definition</h2> <h3 id="InHomotopyTypeTheory">In homotopy type theory</h3> <p>In a general abstract context of <a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a> we may define <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representations as follows.</p> <p>For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math> an <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos">(∞,1)-topos</a>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>∈</mo><mi>Grp</mi><mo stretchy="false">(</mo><mstyle mathvariant="bold"><mi>H</mi></mstyle><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">G \in Grp(\mathbf{H})</annotation></semantics></math> be a <a class="existingWikiWord" href="/nlab/show/group+object+in+an+%28%E2%88%9E%2C1%29-category">group object</a> in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>H</mi></mstyle></mrow><annotation encoding="application/x-tex">\mathbf{H}</annotation></semantics></math>, hence an <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a>. Then the <a class="existingWikiWord" href="/nlab/show/slice+%28%E2%88%9E%2C1%29-topos">slice (∞,1)-topos</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub></mrow><annotation encoding="application/x-tex">\mathbf{H}_{/\mathbf{B}G}</annotation></semantics></math> over its <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> is the <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-category">(∞,1)-category</a> of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-actions">∞-actions</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>Act</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>≃</mo><msub><mstyle mathvariant="bold"><mi>H</mi></mstyle> <mrow><mo stretchy="false">/</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow></msub><mspace width="thinmathspace"></mspace><mo>,</mo></mrow><annotation encoding="application/x-tex"> Act(G) \simeq \mathbf{H}_{/\mathbf{B}G} \,, </annotation></semantics></math></div> <p>hence of possibly “non-linear” <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representations. (See at <em><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></em> for details). A genuine (linear) <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation is then an <a class="existingWikiWord" href="/nlab/show/abelian+%E2%88%9E-group">abelian ∞-group</a> object in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Act</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Act(G)</annotation></semantics></math>.</p> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><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/action">action</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/module">module</a>, <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-module">∞-module</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/representation">representation</a>, <strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-representation</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/2-representation">2-representation</a>,</li> </ul> </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> <div> <p><strong><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> and <a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a> in terms of <a class="existingWikiWord" href="/nlab/show/%28%E2%88%9E%2C1%29-topos+theory">(∞,1)-topos theory</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></strong> (<a class="existingWikiWord" href="/schreiber/show/Principal+%E2%88%9E-bundles+--+theory%2C+presentations+and+applications">FSS 12 I, exmp. 4.4</a>):</p> <table><thead><tr><th><a class="existingWikiWord" href="/nlab/show/homotopy+type+theory">homotopy type theory</a></th><th><a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a></th></tr></thead><tbody><tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/pointed+type">pointed</a> <a class="existingWikiWord" href="/nlab/show/connected+homotopy+type">connected</a> <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+type">dependent type</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-action">∞-action</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-representation">∞-representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinvariants">coinvariants</a>/<a class="existingWikiWord" href="/nlab/show/homotopy+quotient">homotopy quotient</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/trivial+representation">trivial representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/homotopy+invariants">homotopy invariants</a>/<a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group+cohomology">∞-group cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> of <a class="existingWikiWord" href="/nlab/show/internal+hom">internal hom</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mo>*</mo></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \ast</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/equivariant+cohomology">equivariant cohomology</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+sum">dependent sum</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/induced+representation">induced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/context+extension">context extension</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/restricted+representation">restricted representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/dependent+product">dependent product</a> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>H</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G \to \mathbf{B}H</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/coinduced+representation">coinduced representation</a></td></tr> <tr><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+object">spectrum object</a> in <a class="existingWikiWord" href="/nlab/show/context">context</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{B}G</annotation></semantics></math></td><td style="text-align: left;"><a class="existingWikiWord" href="/nlab/show/spectrum+with+G-action">spectrum with G-action</a> (<a class="existingWikiWord" href="/nlab/show/naive+G-spectrum">naive G-spectrum</a>)</td></tr> </tbody></table> </div> <h2 id="references">References</h2> <p>2-representations of <a class="existingWikiWord" href="/nlab/show/2-groups">2-groups</a> and <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a>s such as the <a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a> on <a class="existingWikiWord" href="/nlab/show/2-vector+space">2-vector space</a>s are disscussed in</p> <ul> <li id="Schreiber"> <p><a class="existingWikiWord" href="/nlab/show/Urs+Schreiber">Urs Schreiber</a>, <em>AQFT from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>n</mi></mrow><annotation encoding="application/x-tex">n</annotation></semantics></math>-functorial QFT</em></p> <p>(<a href="http://arxiv.org/abs/0806.1079">arXiv:0806.1079</a>) (appendix)</p> </li> <li id="BaezBaratinFreidelWise"> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a>, Aristide Baratin, Laurent Freidel, Derek Wise, <em>Infinite-Dimensional Representations of 2-Groups</em> (<a href="http://arxiv.org/abs/0812.4969">arXiv:0812.4969</a>)</p> </li> </ul> <p>References for 2- and 3-representations of <a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a>s are at <a class="existingWikiWord" href="/nlab/show/higher+parallel+transport">higher parallel transport</a>.</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-Representations of <a class="existingWikiWord" href="/nlab/show/groupoids">groupoids</a> and <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>s on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mn>∞</mn><mo>,</mo><mn>1</mn><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\infty,1)</annotation></semantics></math>-categories of chain complexes are discussed under the term <em>representations up to homotopy</em> in</p> <ul> <li id="AbadCrainic"><a class="existingWikiWord" href="/nlab/show/Camilo+Arias+Abad">Camilo Arias Abad</a>, <a class="existingWikiWord" href="/nlab/show/Marius+Crainic">Marius Crainic</a> , <em>Representations up to homotopy and Bott’s spectral sequence for Lie groupoids</em> (<a href="http://arxiv.org/abs/0911.2859">arXiv</a>)</li> </ul> <p>A discussion of quasicoherent <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>∞</mn></mrow><annotation encoding="application/x-tex">\infty</annotation></semantics></math>-stacks and D-modules in the context of <a class="existingWikiWord" href="/nlab/show/representation+theory">representation theory</a> is for instance in</p> <ul> <li id="Ben-ZviNadler"> <p><a class="existingWikiWord" href="/nlab/show/David+Ben-Zvi">David Ben-Zvi</a>, <a class="existingWikiWord" href="/nlab/show/David+Nadler">David Nadler</a>,</p> <p><em>The character theory of a complex group</em> (<a href="http://arxiv.org/abs/0904.1247">arXiv:0904.1247</a>)</p> <p><em>Loop Spaces and Representations</em> (<a href="http://arxiv.org/abs/1004.5120">arXiv:1004.5120</a>)</p> </li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on February 4, 2024 at 07:22:04. 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