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geometry</a>)</p> <p><strong>Background</strong></p> <p><em>Smooth structure</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/generalized+smooth+space">generalized smooth space</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/diffeological+space">diffeological space</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Fr%C3%B6licher+space">Frölicher space</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+topos">smooth topos</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Cahiers+topos">Cahiers topos</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+%E2%88%9E-groupoid">smooth ∞-groupoid</a>, <a class="existingWikiWord" href="/nlab/show/concrete+smooth+%E2%88%9E-groupoid">concrete smooth ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/synthetic+differential+%E2%88%9E-groupoid">synthetic differential ∞-groupoid</a></p> </li> </ul> <p><em>Higher groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-groupoid">∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/2-groupoid">2-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+%E2%88%9E-groupoid">strict ∞-groupoid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/crossed+complex">crossed complex</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-group">∞-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simplicial+group">simplicial group</a></li> </ul> </li> </ul> <p><em>Lie theory</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory">Lie theory</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+integration">Lie integration</a>, <a class="existingWikiWord" href="/nlab/show/Lie+differentiation">Lie differentiation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie%27s+three+theorems">Lie's three theorems</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+theory+for+stacky+Lie+groupoids">Lie theory for stacky Lie groupoids</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie groupoids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">strict ∞-Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+groupoid">Lie groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+group">∞-Lie group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/simple+Lie+group">simple Lie group</a>, <a class="existingWikiWord" href="/nlab/show/semisimple+Lie+group">semisimple Lie group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a></p> </li> </ul> </li> </ul> <p><strong>∞-Lie algebroids</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+algebroid">∞-Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+%E2%88%9E-algebroid+representation">Lie ∞-algebroid representation</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/L-%E2%88%9E-algebra">L-∞-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/model+structure+for+L-%E2%88%9E+algebras">model structure for L-∞ algebras</a>: <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-Lie+algebras">on dg-Lie algebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+dg-coalgebras">on dg-coalgebras</a>, <a class="existingWikiWord" href="/nlab/show/model+structure+on+simplicial+Lie+algebras">on simplicial Lie algebras</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/semisimple+Lie+algebra">semisimple Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/compact+Lie+algebra">compact Lie algebra</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+2-algebra">Lie 2-algebra</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/strict+Lie+2-algebra">strict Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/differential+crossed+module">differential crossed module</a></p> </li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+3-algebra">Lie 3-algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/differential+2-crossed+module">differential 2-crossed module</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/dg-Lie+algebra">dg-Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/simplicial+Lie+algebra">simplicial Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/super+L-%E2%88%9E+algebra">super L-∞ algebra</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/super+Lie+algebra">super Lie algebra</a></li> </ul> </li> </ul> <p><strong>Formal Lie groupoids</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/formal+group">formal group</a>, <a class="existingWikiWord" href="/nlab/show/formal+groupoid">formal groupoid</a></li> </ul> <p><strong>Cohomology</strong></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Lie+algebra+cohomology">Lie algebra cohomology</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Chevalley-Eilenberg+algebra">Chevalley-Eilenberg algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Weil+algebra">Weil algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/invariant+polynomial">invariant polynomial</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Killing+form">Killing form</a></li> </ul> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/nonabelian+Lie+algebra+cohomology">nonabelian Lie algebra cohomology</a></p> </li> </ul> <p><strong>Homotopy</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></li> </ul> <p><strong>Related topics</strong></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Chern-Weil+theory">∞-Chern-Weil theory</a></li> </ul> <p><strong>Examples</strong></p> <p><em><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>-Lie groupoids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fundamental+%E2%88%9E-groupoid">fundamental ∞-groupoid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/path+n-groupoid">path n-groupoid</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">smooth principal ∞-bundle</a></p> </li> </ul> <p><em><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>-Lie groups</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+group">orthogonal group</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/special+orthogonal+group">special orthogonal group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/spin+group">spin group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+2-group">string 2-group</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+6-group">fivebrane 6-group</a></p> </li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/unitary+group">unitary group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/special+unitary+group">special unitary group</a></li> </ul> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">circle Lie n-group</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/circle+group">circle group</a></li> </ul> </li> </ul> <p><em><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>-Lie algebroids</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+Lie+n-algebroid">symplectic Lie n-algebroid</a></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/symplectic+manifold">symplectic manifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Courant+Lie+algebroid">Courant Lie algebroid</a></p> <ul> <li><a class="existingWikiWord" href="/nlab/show/generalized+complex+geometry">generalized complex geometry</a></li> </ul> </li> </ul> </li> </ul> <p><em><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>-Lie algebras</em></p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/general+linear+Lie+algebra">general linear Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orthogonal+Lie+algebra">orthogonal Lie algebra</a>, <a class="existingWikiWord" href="/nlab/show/special+orthogonal+Lie+algebra">special orthogonal Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/endomorphism+L-%E2%88%9E+algebra">endomorphism L-∞ algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/automorphism+%E2%88%9E-Lie+algebra">automorphism ∞-Lie algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/string+Lie+2-algebra">string Lie 2-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/fivebrane+Lie+6-algebra">fivebrane Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+3-algebra">supergravity Lie 3-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/supergravity+Lie+6-algebra">supergravity Lie 6-algebra</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/line+Lie+n-algebra">line Lie n-algebra</a></p> </li> </ul> </div></div> </div> </div> <h1 id="contents">Contents</h1> <div class='maruku_toc'> <ul> <li><a href='#idea'>Idea</a></li> <li><a href='#definition'>Definition</a></li> <ul> <li><a href='#examples_for_lie_groupoids'>Examples for Lie groupoids</a></li> <li><a href='#terminology'>Terminology</a></li> <li><a href='#specialisations'>Specialisations</a></li> <li><a href='#2CatOfGrpds'>The (2,1)-category of Lie groupoids</a></li> </ul> <li><a href='#lie_algebroid'>Lie algebroid</a></li> <ul> <li><a href='#examples_of_lie_algebroids'>Examples of Lie algebroids</a></li> </ul> <li><a href='#MorphismsOfLieGroupoids'>Morphisms of Lie groupoids</a></li> <li><a href='#morphisms_of_lie_algebroids'>Morphisms of Lie algebroids</a></li> <ul> <li><a href='#example'>Example</a></li> <li><a href='#example_2'>Example</a></li> <li><a href='#open_problem'>Open problem</a></li> </ul> <li><a href='#higher_lie_groupoids'>Higher Lie groupoids</a></li> <li><a href='#examples'>Examples</a></li> <li><a href='#related_concepts'>Related concepts</a></li> <li><a href='#references'>References</a></li> </ul> </div> <h2 id="idea">Idea</h2> <p>A <em>Lie groupoid</em> is a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal</a> to <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>. This is a joint generalization of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a> and <a class="existingWikiWord" href="/nlab/show/Lie+groups">Lie groups</a> to <a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a>.</p> <p>Regarded in the more general context of <a class="existingWikiWord" href="/nlab/show/smooth+groupoids">smooth groupoids</a>/<a class="existingWikiWord" href="/nlab/show/smooth+stacks">smooth stacks</a>, Lie groupoids present certain well-behaved such objects, often called <em><a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a></em>.</p> <h2 id="definition">Definition</h2> <div class="num_defn" id="ExplicitDefinition"> <h6 id="definition_2">Definition</h6> <p>A <em>Lie groupoid</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo lspace="mediummathspace" rspace="mediummathspace">∶−</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>⇉</mo><msub><mi>X</mi> <mn>0</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X \coloneq (X_1 \rightrightarrows X_0)</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/groupoid">groupoid</a> such that both the space of arrows <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1</annotation></semantics></math> and the space of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_0</annotation></semantics></math> are smooth manifolds, all structure maps are smooth, and source and target maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>:</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>⇉</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">s, t: X_1\rightrightarrows X_0</annotation></semantics></math> are surjective submersions.</p> </div> <p>A <em>Lie groupoid</em> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi></mrow><annotation encoding="application/x-tex">X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> in the <a class="existingWikiWord" href="/nlab/show/category">category</a> <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> of <a class="existingWikiWord" href="/nlab/show/smooth+manifolds">smooth manifolds</a>.</p> <p>Since <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> does not have all <a class="existingWikiWord" href="/nlab/show/pullbacks">pullbacks</a>, to ensure that this definition makes sense, one needs to ensure that the space <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>s</mi><mo>,</mo><mi>t</mi></mrow></msub><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \times_{s,t} X_1</annotation></semantics></math> of composable <a class="existingWikiWord" href="/nlab/show/morphism">morphism</a>s is an object of <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>. This is achieved either by adopting the definition of <a class="existingWikiWord" href="/nlab/show/internal+groupoid">internal groupoid</a> in the sense of Ehresmann, which includes as data the <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a> of <a class="existingWikiWord" href="/nlab/show/composable+pairs">composable pairs</a>, or by taking the conventional route and demanding that the source and target maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>,</mo><mi>t</mi><mo>:</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">s,t : X_0 \to X_1</annotation></semantics></math> are <a class="existingWikiWord" href="/nlab/show/submersion">submersion</a>s. This ensures the <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> exists to define said manifold of composable pairs. Therefore a definition used in most modern differential geometry literature is as we see above.</p> <p>But for most practical purposes, the apparently evident <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd(Diff)</annotation></semantics></math> of such internal groupoids, <a class="existingWikiWord" href="/nlab/show/internal+functor">internal functor</a>s and internal <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s is <em>not</em> the correct 2-category to consider. One way to see this is that the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> fails in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a>, which means that an internal functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \to Y</annotation></semantics></math> of internal groupoids which is <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective</a> and <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful</a> may nevertheless not be an equivalence, in that it may not have a weak inverse in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Grpd</mi><mo stretchy="false">(</mo><mi>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Grpd(Diff)</annotation></semantics></math>.</p> <p>See the section <a href="#2CatOfGrpds">2-Category of Lie groupoids</a> below.</p> <p>A bit more general than a Lie groupoid is a <a class="existingWikiWord" href="/nlab/show/diffeological+groupoid">diffeological groupoid</a>.</p> <h3 id="examples_for_lie_groupoids">Examples for Lie groupoids</h3> <ul> <li> <p>A Lie group <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 Lie groupoid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X_1=G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>pt</mi></mrow><annotation encoding="application/x-tex">X_0=pt</annotation></semantics></math> a point. Composition 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> is provided by the multiplication 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> </li> <li> <p>A manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is a Lie groupoid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_1=M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math>. Source and target maps are identities and we only have identity arrows in this example.</p> </li> <li> <p>Given a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> and an open cover <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i\}</annotation></semantics></math>, we can form a Lie groupoid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mo>⊔</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>×</mo> <mi>M</mi></msub><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">X_1=\sqcup U_i\times_M U_j</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mo>⊔</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">X_0=\sqcup U_i</annotation></semantics></math>. Then for an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>ij</mi></msub><mo>:</mo><mo>=</mo><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>,</mo><msub><mi>x</mi> <mi>j</mi></msub><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub><msub><mo>×</mo> <mi>M</mi></msub><msub><mi>U</mi> <mi>j</mi></msub><mo>⊂</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">x_{ij}:=(x_i, x_j)\in U_i\times_M U_j \subset X_1</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>ij</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi> <mi>i</mi></msub><mo>∈</mo><msub><mi>U</mi> <mi>i</mi></msub></mrow><annotation encoding="application/x-tex">t(x_{ij})=x_i \in U_i</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><msub><mi>x</mi> <mi>ij</mi></msub><mo stretchy="false">)</mo><mo>=</mo><msub><mi>x</mi> <mi>j</mi></msub><mo>∈</mo><msub><mi>U</mi> <mi>j</mi></msub></mrow><annotation encoding="application/x-tex">s(x_{ij})=x_j \in U_j</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>x</mi> <mi>ij</mi></msub><mo>⋅</mo><msub><mi>x</mi> <mi>jk</mi></msub><mo>=</mo><msub><mi>x</mi> <mi>ik</mi></msub></mrow><annotation encoding="application/x-tex">x_{ij} \cdot x_{jk}= x_{ik}</annotation></semantics></math>. This is sometimes called the <a class="existingWikiWord" href="/nlab/show/%C4%8Cech+groupoid">Čech groupoid</a> or <strong>covering groupoid</strong>.</p> </li> <li> <p>Given a Lie group <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> (right) action on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, then we may form an associated <a class="existingWikiWord" href="/nlab/show/action+groupoid">action groupoid</a> (or sometimes called <strong>transformation groupoid</strong>) as follows: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>M</mi><mo>×</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">X_1 = M \times G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math>. For an element <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>∈</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">(x, g) \in X_1</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi></mrow><annotation encoding="application/x-tex">t(x, g) = x</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>=</mo><mi>x</mi><mo>⋅</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">s(x, g)=x\cdot g^{-1}</annotation></semantics></math>, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>h</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>g</mi><mo>⋅</mo><mi>h</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, g)\cdot (y, h) = (x, g\cdot h)</annotation></semantics></math> (we must have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>y</mi><mo>=</mo><mi>x</mi><mo>⋅</mo><msup><mi>g</mi> <mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow></msup></mrow><annotation encoding="application/x-tex">y=x\cdot g^{-1}</annotation></semantics></math> for the multiplication to happen). Action groupoid presents the quotient stack <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>M</mi><mo stretchy="false">/</mo><mi>G</mi><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[M/G]</annotation></semantics></math>. Roughly speaking, it is a good replacement for quotient space even if the action is not as nice as you want.</p> </li> <li> <p>Given a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, we may also form so-called <a class="existingWikiWord" href="/nlab/show/pair+groupoid">pair groupoid</a>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>=</mo><mi>M</mi><mo>×</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_1= M\times M</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math>. Source and target are projections, and multiplication is given by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo>⋅</mo><mo stretchy="false">(</mo><mi>y</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo><mo>=</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>z</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(x, y) \cdot (y , z)= (x, z)</annotation></semantics></math>. Pair groupoid may be interpreted as the global object of tangent bundle (think why? see the section below on Lie algebroid).</p> </li> <li> <p>Given a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, we have also an associated <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> or <strong>homotopy groupoid</strong> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Pi(M)</annotation></semantics></math>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Π</mi><mo stretchy="false">(</mo><mi>M</mi><msub><mo stretchy="false">)</mo> <mn>1</mn></msub><mo>=</mo><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">\Pi(M)_1=\{</annotation></semantics></math>paths in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi><mo stretchy="false">}</mo><mo stretchy="false">/</mo></mrow><annotation encoding="application/x-tex">M\}/</annotation></semantics></math> homotopies, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Π</mi> <mn>0</mn></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\Pi_0(M)=M</annotation></semantics></math>. Source and target are end points of a path. Multiplication is concatenation of paths (think why associative?).</p> </li> <li> <p>Given a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> with a <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>, we may form various groupoids associated with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>.</p> </li> </ul> <ol> <li> <p><strong><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-pair groupoid</strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo><mo stretchy="false">(</mo><mi>x</mi><mo>,</mo><mi>y</mi><mo stretchy="false">)</mo><mo stretchy="false">|</mo><mi>x</mi><mo>,</mo><mi>y</mi><mspace width="thickmathspace"></mspace><mtext>are in the same leaf in</mtext><mspace width="thickmathspace"></mspace><mi>F</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">X_1:=\{(x, y)| x, y \;\text{are in the same leaf in}\; F \}</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math>. Source and target are obvious projections and multiplication is like in the case of pair groupoid. The problem for this groupoid is that it might not be a Lie groupoid. (why not? for counter example, we refer to <a href="https://math.berkeley.edu/~alanw/Models.pdf">Section 13.5 of Geometric Models for Noncommutative Algebras</a> ).</p> </li> <li> <p><strong>monodromy groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Mon</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Mon_F(M)</annotation></semantics></math></strong> (it is a foliation version of fundamental groupoid, thus it is also sometimes called <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>F</mi></mrow><annotation encoding="application/x-tex">F</annotation></semantics></math>-fundamental groupoid): <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">X_1:=\{</annotation></semantics></math> leaf-wise paths<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo><mo stretchy="false">/</mo></mrow><annotation encoding="application/x-tex">\}/</annotation></semantics></math> leaf-wise homotopy, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math> and the rest is like in the case of fundamental groupoid.</p> </li> <li> <p><strong><a class="existingWikiWord" href="/nlab/show/holonomy+groupoid">holonomy groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>Hol</mi> <mi>F</mi></msub><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Hol_F(M)</annotation></semantics></math></strong>: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>:</mo><mo>=</mo><mo stretchy="false">{</mo></mrow><annotation encoding="application/x-tex">X_1:=\{</annotation></semantics></math> leaf-wise paths<math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">}</mo><mo stretchy="false">/</mo></mrow><annotation encoding="application/x-tex">\}/</annotation></semantics></math> holonomy, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>0</mn></msub><mo>=</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">X_0=M</annotation></semantics></math> and the rest is like in the case of fundamental groupoid. Here, the holonomy of a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> is defined to the germ of diffeomorphisms induced by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi></mrow><annotation encoding="application/x-tex">\gamma</annotation></semantics></math> between the transversals at the end points.</p> </li> </ol> <p>Among all possible Lie groupoids associated to a foliation, monodromy groupoid is the biggest and holonomy groupoid is the smallest.</p> <h3 id="terminology">Terminology</h3> <p>Originally Lie groupoids were called (by Ehresmann) <em>differentiable groupoids</em> (and also one considered differentiable <em>categories</em>). Sometime in the 1980s there was a change of terminology to <em>Lie groupoid</em> and <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>s. (reference?)</p> <h3 id="specialisations">Specialisations</h3> <p>One definition which Ehresmann introduced in his paper <em>Catégories topologiques et catégories différentiables</em> (see below) is that of <a class="existingWikiWord" href="/nlab/show/locally+trivial+category">locally trivial groupoid</a>. It is defined more generally for topological categories, and extends in an obvious way to topological groupoids, and Lie categories and groupoids. For a topological (resp. Lie) category <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>, let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>X</mi> <mn>1</mn> <mi>iso</mi></msubsup></mrow><annotation encoding="application/x-tex">X_1^{iso}</annotation></semantics></math> denote the subspace (resp. submanifold) of invertible arrows . (This always exists, by general abstract nonsense - I should look up the reference, it’s in Bunge-Pare I think - <a class="existingWikiWord" href="/nlab/show/David+Roberts">DR</a>)</p> <div class="num_defn"> <h6 id="definition_3">Definition</h6> <p>A topological groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mn>1</mn></msub><mo>⇉</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">X_1 \rightrightarrows X_0</annotation></semantics></math> is <strong>locally trivial</strong> if for every point <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi><mo>∈</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">p\in X_0</annotation></semantics></math> there is a neighbourhood <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> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>p</mi></mrow><annotation encoding="application/x-tex">p</annotation></semantics></math> and a lift of the inclusion <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><mi>p</mi><mo stretchy="false">}</mo><mo>×</mo><mi>U</mi><mo>↪</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\{p\} \times U \hookrightarrow X_0 \times X_0</annotation></semantics></math> through <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo><mo>:</mo><msubsup><mi>X</mi> <mn>1</mn> <mi>iso</mi></msubsup><mo>→</mo><msub><mi>X</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">(s,t):X_1^{iso}\to X_0 \times X_0</annotation></semantics></math>.</p> </div> <p>Clearly for a Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msubsup><mi>X</mi> <mn>1</mn> <mi>iso</mi></msubsup><mo>=</mo><msub><mi>X</mi> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">X_1^{iso} = X_1</annotation></semantics></math>. It is simple to show from the definition that for a transitive Lie groupoid, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>,</mo><mi>t</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(s,t)</annotation></semantics></math> has local sections. Ehresmann goes on to show a link between smooth <a class="existingWikiWord" href="/nlab/show/principal+bundles">principal bundles</a> and transitive, locally trivial Lie groupoids. See <a class="existingWikiWord" href="/nlab/show/locally+trivial+category">locally trivial category</a> for details.</p> <h3 id="2CatOfGrpds">The (2,1)-category of Lie groupoids</h3> <p>As usual for internal categories, the naive 2-category of internal groupoids, <a class="existingWikiWord" href="/nlab/show/internal+functor">internal functor</a>s and internal <a class="existingWikiWord" href="/nlab/show/natural+transformation">natural transformation</a>s is not quite “correct”. One sign of this is that the <a class="existingWikiWord" href="/nlab/show/axiom+of+choice">axiom of choice</a> fails in <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> so that an internal functor which is an <a class="existingWikiWord" href="/nlab/show/essentially+surjective+functor">essentially surjective functor</a> and a <a class="existingWikiWord" href="/nlab/show/full+and+faithful+functor">full and faithful functor</a> may still not have an internal weak inverse.</p> <p>One way to deal with this is to equip the 2-category with some structure of a <a class="existingWikiWord" href="/nlab/show/homotopical+category">homotopical category</a> and allow morphisms of Lie groupoids to be <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>s, i.e. <a class="existingWikiWord" href="/nlab/show/span">span</a>s of internal functors <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math>.</p> <p>Such generalized morphisms – called <em><a class="existingWikiWord" href="/nlab/show/Morita+morphisms">Morita morphisms</a></em> or <em>generalized morphisms</em> in the literature – are sometimes modeled as <a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a>s and then called <a class="existingWikiWord" href="/nlab/show/Hilsum-Skandalis+morphism">Hilsum-Skandalis morphism</a>s.</p> <p>Another equivalent approach is to embed Lie groupoids into the context of <a class="existingWikiWord" href="/nlab/show/2-topos">2-topos</a> theory:</p> <p>The <a class="existingWikiWord" href="/nlab/show/2-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>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(Diff)</annotation></semantics></math> of <a class="existingWikiWord" href="/nlab/show/stack">stack</a>s/<a class="existingWikiWord" href="/nlab/show/2-sheaves">2-sheaves</a> on <a class="existingWikiWord" href="/nlab/show/Diff">Diff</a> may be understood as a nice <a class="existingWikiWord" href="/nlab/show/2-category">2-category</a> of general groupoids <em>modeled on</em> <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth manifold</a>s. The degreewise <a class="existingWikiWord" href="/nlab/show/Yoneda+embedding">Yoneda embedding</a> allows to emebed groupoids internal to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math> into stacks on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Diff</mi></mrow><annotation encoding="application/x-tex">Diff</annotation></semantics></math>. this wider context contains for instance also <a class="existingWikiWord" href="/nlab/show/diffeological+groupoid">diffeological groupoid</a>s.</p> <p>Regarded inside this wider context, Lie groupoids are identified with <a class="existingWikiWord" href="/nlab/show/differentiable+stack">differentiable stack</a>s. The <a class="existingWikiWord" href="/nlab/show/%28n%2Cr%29-category">(2,1)-category</a> of Lie groupoids is then the full sub-<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>-category of <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>Diff</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">Sh_{(2,1)}(Diff)</annotation></semantics></math> on differentiable stacks.</p> <p>For more comments on this, see also the beginning of <a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a>.</p> <h2 id="lie_algebroid">Lie algebroid</h2> <p>As the <a class="existingWikiWord" href="/nlab/show/infinitesimal+space">infinitesimal</a> approximation to a <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> is a <a class="existingWikiWord" href="/nlab/show/Lie+algebra">Lie algebra</a>, so the infinitesimal approximation to a Lie groupoid is a <a class="existingWikiWord" href="/nlab/show/Lie+algebroid">Lie algebroid</a>.</p> <div class="num_defn"> <h6 id="definition_4">Definition</h6> <p>A Lie algebroid is a vector bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">A\to M</annotation></semantics></math> together with a vector bundle morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>:</mo><mi>A</mi><mo>→</mo><mi>TM</mi></mrow><annotation encoding="application/x-tex">\rho: A\to TM</annotation></semantics></math> (called anchor map), and a Lie bracket <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> on the space of sections of <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>, satisfying the Leibniz rule</p> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>fY</mi><mo stretchy="false">]</mo><mo>=</mo><mi>f</mi><mo stretchy="false">[</mo><mi>X</mi><mo>,</mo><mi>Y</mi><mo stretchy="false">]</mo><mo>+</mo><mi>ρ</mi><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mi>Y</mi><mo>.</mo></mrow><annotation encoding="application/x-tex">[X, fY]=f[X,Y]+\rho(X)(f) Y. </annotation></semantics></math></p> </div> <div class="num_defn"> <h6 id="remark">Remark</h6> <p>You would expect <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> to preserve <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math>, wouldn’t you? It is actually automatic! (see Y. Kosmann-Shwarzbah and F. Magri. Poisson-Nijenhuis strutures. Ann. Inst. H. Poinaré Phys. Théor., 53(1):3581, 1990.)</p> <p>Recent progress: it turns out that one may link Lie algebroid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>L</mi> <mn>∞</mn></msub></mrow><annotation encoding="application/x-tex">L_\infty</annotation></semantics></math>-spaces (ask Owen Gwilliam for it)</p> </div> <h3 id="examples_of_lie_algebroids">Examples of Lie algebroids</h3> <ul> <li> <p>A Lie algebra is a Lie algebroid with base space being a point.</p> </li> <li> <p><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>-bundle over a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> is certainly a Lie algebroid in a trivial way.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/action+Lie+algebroid">action Lie algebroid</a></p> </li> <li> <p>Tangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>TM</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">TM\to M</annotation></semantics></math> is a Lie algebroid with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo>=</mo><mi>id</mi></mrow><annotation encoding="application/x-tex">\rho=id</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">[-,-]</annotation></semantics></math> the usual Lie bracket for vector fields. See <a class="existingWikiWord" href="/nlab/show/tangent+Lie+algebroid">tangent Lie algebroid</a>.</p> </li> <li> <p>Given a <a class="existingWikiWord" href="/nlab/show/Poisson+manifold">Poisson manifold</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math> with Poisson bivector field <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>π</mi></mrow><annotation encoding="application/x-tex">\pi</annotation></semantics></math>, the cotangent bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>T</mi> <mo>*</mo></msup><mi>P</mi></mrow><annotation encoding="application/x-tex">T^*P</annotation></semantics></math> is equipped with a Lie algebroid structure: <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ρ</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo>=</mo><mi>π</mi><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\rho(\xi)= \pi(\xi)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>ξ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ξ</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>=</mo><mi>d</mi><mi>π</mi><mo stretchy="false">(</mo><msub><mi>ξ</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>ξ</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">[\xi_1, \xi_2]=d\pi(\xi_1, \xi_2)</annotation></semantics></math> (or you may have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mi>df</mi><mo>,</mo><mi>dg</mi><mo stretchy="false">]</mo><mo>=</mo><mi>d</mi><mo stretchy="false">{</mo><mi>f</mi><mo>,</mo><mi>g</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">[df, dg]=d\{ f, g\}</annotation></semantics></math> if you prefer to think in Poisson bracket). See <a class="existingWikiWord" href="/nlab/show/Poisson+Lie+algebroid">Poisson Lie algebroid</a>.</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+algebroid">Atiyah Lie algebroid</a></p> </li> </ul> <h2 id="MorphismsOfLieGroupoids">Morphisms of Lie groupoids</h2> <p>There are several versions of Lie groupoid morphisms, some of them are equivalent in a correct sense, some of them are not.</p> <ul> <li> <p>strict morphism: a strict morphism from Lie groupoid <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a functor from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> as categories and preserving the smooth structures.</p> </li> <li> <p>generalised morphism: a generalised morphism from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/span">span</a> of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \stackrel{\simeq}{\rightarrow} X</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> of Lie groupoids, defined as below (see also at <em><a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a></em>).</p> </li> </ul> <p>A Lie groupoid functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">f : G\to H</annotation></semantics></math> is a <strong>weak equivalence</strong> if it is</p> <ol> <li> <p>essentially surjective; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>t</mi><mo>∘</mo><msub><mi>pr</mi> <mn>2</mn></msub><mo>:</mo><msub><mi>G</mi> <mn>0</mn></msub><msub><mo>×</mo> <mrow><msub><mi>H</mi> <mn>0</mn></msub><mo>,</mo><mi>s</mi></mrow></msub><msub><mi>H</mi> <mn>1</mn></msub><mo>→</mo><msub><mi>H</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">t \circ pr_2 : G_0 \times_{H_0,s} H_1 \to H_0</annotation></semantics></math> is a surjective submersion;</p> </li> <li> <p>fully faithful; that is, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>G</mi> <mn>1</mn></msub><mo>≅</mo><msub><mi>H</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><mi>t</mi><mo>×</mo><mi>s</mi><mo>,</mo><msub><mi>H</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>H</mi> <mn>0</mn></msub></mrow></msub><msub><mi>G</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">G_1 \cong H_1\times_{t\times s, H_0\times H_0} G_0 \times G_0</annotation></semantics></math>.</p> </li> </ol> <p>Composition of generalised morphism is given by weak <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of Lie groupoids (see also <a class="existingWikiWord" href="/nlab/show/weak+limit">weak limit</a>). Given (strict) morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X\to Y</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X' \to Y</annotation></semantics></math>, the <strong>weak <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></strong> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X\to Y</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X' \to Y</annotation></semantics></math> is a groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><msubsup><mo>×</mo> <mi>Y</mi> <mi>w</mi></msubsup><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mrow><annotation encoding="application/x-tex">\hat X \times_{Y}^w \hat X'</annotation></semantics></math> with space of objects <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover> <mn>0</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover><msub><mo>′</mo> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\hat X_0 \times_{ Y_0} Y_1 \times_{Y_0} \hat X'_0</annotation></semantics></math> and space of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><msub><mi>Y</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>Y</mi> <mn>0</mn></msub></mrow></msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover><msub><mo>′</mo> <mn>1</mn></msub></mrow><annotation encoding="application/x-tex">\hat X_1 \times_{Y_0} Y_1 \times_{Y_0} \hat X'_1</annotation></semantics></math>. When <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">\hat X' \to Y</annotation></semantics></math> is a weak equivalence, the weak pullback is a Lie groupoid thank to the property of essentially surjective. (Is this composition associative?)</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>: an anafunctor from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Y</mi></mrow><annotation encoding="application/x-tex">Y</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/span">span</a> of morphisms <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mover><mo>→</mo><mo>≃</mo></mover><mi>X</mi></mrow><annotation encoding="application/x-tex">\hat X \stackrel{\simeq}{\rightarrow} X</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/acyclic+fibration">acyclic fibration</a> of Lie groupoids. That is, this map is a <a class="existingWikiWord" href="/nlab/show/weak+equivalence">weak equivalence</a> of Lie groupoids and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mover><mi>X</mi><mo stretchy="false">^</mo></mover> <mn>0</mn></msub><mo>→</mo><msub><mi>X</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">\hat X_0 \to X_0</annotation></semantics></math> is a surjective submersion.</li> </ul> <p>Composition of anafunctors is given through strong <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a> of Lie groupoids, that is level-wise pullback.</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a> functor (or H.S. bibundle, or Hilsum-Skandalis bibundle): a bibundle functor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G\to H</annotation></semantics></math> is a <a class="existingWikiWord" href="/nlab/show/groupoid+principal+bundle">groupoid principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> (with right action) such that <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> acts on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> from left and <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> action commutes with <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> action. If both <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>H</mi></mrow><annotation encoding="application/x-tex">H</annotation></semantics></math> actions are principal, then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi></mrow><annotation encoding="application/x-tex">E</annotation></semantics></math> gives arise to <a class="existingWikiWord" href="/nlab/show/Morita+equivalence">Morita equivalence</a> between them.</li> </ul> <p>The last three morphisms are more or less equivalent, that is they give arise to equivalent 2-categories (in fact (2,1)-categories) of Lie groupoids. To make it explicit, we need to talk about <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a>s between them.</p> <p>A <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> between bibundle functors is simply a bibundle isomorphism (of course preserving all the structures of bibundles).</p> <p>A strict <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> from generalised morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X' \to Y</annotation></semantics></math> is given by a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mrow><annotation encoding="application/x-tex">\hat X \to \hat X'</annotation></semantics></math> such that the following diagram commutes</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="false" rowspacing="0.5ex"><mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{matrix} & & \hat X \\ & \swarrow & & \searrow \\ X & & \downarrow & & Y \\ & \searrow & & \swarrow \\ & & \hat X' \end{matrix} </annotation></semantics></math></div> <p>This forces the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mrow><annotation encoding="application/x-tex">\hat X \to \hat X'</annotation></semantics></math> to be a weak equivalence by <a class="existingWikiWord" href="/nlab/show/2-out-of-3">2-out-of-3</a> for weak equivalences. A <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X' \to Y</annotation></semantics></math> is provided by a <a class="existingWikiWord" href="/nlab/show/span">span</a> of strict 2-morphisms:</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable displaystyle="false" rowspacing="0.5ex"><mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↙</mo></mtd> <mtd><mo stretchy="false">↑</mo></mtd> <mtd><mo>↘</mo></mtd></mtr> <mtr><mtd><mi>X</mi></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>″</mo></mtd> <mtd></mtd> <mtd><mi>Y</mi></mtd></mtr> <mtr><mtd></mtd> <mtd><mo>↘</mo></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd><mo>↙</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \begin{matrix} & & \hat X \\ & \swarrow & \uparrow & \searrow \\ X & & \hat X'' & & Y \\ & \searrow & \downarrow & \swarrow \\ & & \hat X' \end{matrix} </annotation></semantics></math></div> <p>A <a class="existingWikiWord" href="/nlab/show/2-morphism">2-morphism</a> between <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a>s are defined like above, however the left legs are required to be <a class="existingWikiWord" href="/nlab/show/acyclic+fibration">acyclic fibration</a>s between Lie groupoids. (think this time what may you say about the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>′</mo></mrow><annotation encoding="application/x-tex">\hat X \to \hat X'</annotation></semantics></math>?)</p> <p>Then these three (2,1)-categories, which we denote by <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GEN</mi></mrow><annotation encoding="application/x-tex">GEN</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ANA</mi></mrow><annotation encoding="application/x-tex">ANA</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BUN</mi></mrow><annotation encoding="application/x-tex">BUN</annotation></semantics></math>, are all <strong><a class="existingWikiWord" href="/nlab/show/equivalent">equivalent</a></strong> to each other. For a nice survey on this statement, we refer to Section 1.5 of <a href="http://ediss.uni-goettingen.de/handle/11858/00-1735-0000-0022-5F4F-A">Du Li’s thesis</a>.</p> <p>The idea is that <a href="http://ncatlab.org/nlab/show/bibundle#bundlisation">Bundlisation</a> may extend to an equivalence of <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>-categories between <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GEN</mi></mrow><annotation encoding="application/x-tex">GEN</annotation></semantics></math>, the <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>-category made by generalised morphisms, and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>BUN</mi></mrow><annotation encoding="application/x-tex">BUN</annotation></semantics></math>. The inverse is given by the following construction: given a bibundle functor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>:</mo><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">E: G\to H</annotation></semantics></math>, we pull back <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> along the map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>E</mi><mo>→</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow><annotation encoding="application/x-tex">E\to G_0</annotation></semantics></math> and obtain a Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo stretchy="false">|</mo> <mi>E</mi></msub><mo>:</mo><mo>=</mo><msub><mi>G</mi> <mn>1</mn></msub><msub><mo>×</mo> <mrow><msub><mi>G</mi> <mn>0</mn></msub><mo>×</mo><msub><mi>G</mi> <mn>0</mn></msub></mrow></msub><mi>E</mi><mo>×</mo><mi>E</mi><mo>⇒</mo><mi>E</mi></mrow><annotation encoding="application/x-tex">G|_E:=G_1\times_{G_0\times G_0} E \times E \Rightarrow E</annotation></semantics></math>. Then the natural projection <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><msub><mo stretchy="false">|</mo> <mi>E</mi></msub><mo>→</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">G|_E \to G</annotation></semantics></math> is an <a class="existingWikiWord" href="/nlab/show/acyclic+fibration">acyclic fibration</a>. Thus we obtain a generalised morphism which is also an anafunctor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi><mo>→</mo><mi>H</mi></mrow><annotation encoding="application/x-tex">G \to H</annotation></semantics></math>.</p> <p>Even though <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GEN</mi></mrow><annotation encoding="application/x-tex">GEN</annotation></semantics></math> contains more morphisms than <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ANA</mi></mrow><annotation encoding="application/x-tex">ANA</annotation></semantics></math>, a generalised morphism maybe equivalently replaced by an anafunctor. In fact a generalised morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} \hat X \to Y</annotation></semantics></math> gives arise to an anafunctor <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>X</mi><msubsup><mo>×</mo> <mi>X</mi> <mi>w</mi></msubsup><mover><mi>X</mi><mo stretchy="false">^</mo></mover><mo>→</mo><mi>Y</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} X \times_{X}^w \hat X \to Y</annotation></semantics></math>.</p> <p>As a consequence of the universal property of the <a class="existingWikiWord" href="/nlab/show/calculus+of+fractions">calculus of fractions</a>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>GEN</mi></mrow><annotation encoding="application/x-tex">GEN</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ANA</mi></mrow><annotation encoding="application/x-tex">ANA</annotation></semantics></math> are equivalent.</p> <h2 id="morphisms_of_lie_algebroids">Morphisms of Lie algebroids</h2> <p>Morphisms of Lie algebroids are counter-intuitive: they are not morphisms of vector bundles which preserve the algebroid structure. To define a Lie algebroid morphism, we first need to introduce the <em>Chevalley-Eilenberg algebra</em> associated to a Lie algebroid <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>We consider <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> to be a trivially graded vector bundle, i.e. concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>. Then <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A[1]</annotation></semantics></math> is concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">-1</annotation></semantics></math>. The functions on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo></mrow><annotation encoding="application/x-tex">A[1]</annotation></semantics></math> are given as</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>=</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>⊕</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mo>∧</mo> <mn>2</mn></msup><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>⊕</mo><mi>…</mi><mo>,</mo></mrow><annotation encoding="application/x-tex">C(A[1])=C^\infty(M)\oplus \Gamma(A^*)\oplus \Gamma(\wedge^2 A^*)\oplus \ldots ,</annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(M)</annotation></semantics></math> is considered to be of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(A^*)</annotation></semantics></math> to be of degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, and so forth.</p> <p>Now we can define a degree-one derivation on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(A[1])</annotation></semantics></math> as follows: For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>ξ</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mo>∧</mo> <mi>n</mi></msup><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\xi \in \Gamma(\wedge^n A^*)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>X</mi> <mi>i</mi></msub><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">X_i\in \Gamma(A)</annotation></semantics></math>, let</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>…</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>X</mi> <mi>n</mi></msub><mo stretchy="false">)</mo><mo>:</mo><mo>=</mo><munder><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mn>0</mn><mo>≤</mo><mi>i</mi><mo><</mo><mi>j</mi><mo>≤</mo><mi>n</mi></mrow></munder><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mrow><mi>i</mi><mo>+</mo><mi>j</mi></mrow></msup><mi>ξ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mi>i</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>X</mi> <mi>j</mi></msub><msub><mo stretchy="false">]</mo> <mi>A</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><mi>…</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mover><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><mo>^</mo></mover><mo>,</mo><mspace width="thinmathspace"></mspace><mi>…</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mover><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><mo>^</mo></mover><mo>,</mo><mspace width="thinmathspace"></mspace><mi>…</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>+</mo><munderover><mo lspace="thinmathspace" rspace="thinmathspace">∑</mo> <mrow><mi>i</mi><mo>=</mo><mn>0</mn></mrow> <mi>n</mi></munderover><mo stretchy="false">(</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mn>1</mn><msup><mo stretchy="false">)</mo> <mi>i</mi></msup><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mi>i</mi></msub><mo stretchy="false">)</mo><mi>ξ</mi><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>…</mi><mspace width="thinmathspace"></mspace><mo>,</mo><mspace width="thinmathspace"></mspace><mover><mrow><msub><mi>X</mi> <mi>i</mi></msub></mrow><mo>^</mo></mover><mo>,</mo><mspace width="thinmathspace"></mspace><mi>…</mi><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> d_A(\xi)(X_1,\,\ldots\,,\,X_n) := \sum_{0\leq i \lt j\leq n} (-1)^{i+j} \xi\bigl([X_i,\,X_j]_A,\, \ldots\,,\, \widehat{X_i},\,\ldots\,,\,\widehat{X_i},\,\ldots\bigr) + \sum_{i=0}^n (-1)^i \rho_A(X_i) \xi\bigl(\ldots\,,\,\widehat{X_i},\,\ldots\bigr). </annotation></semantics></math></div> <p>The condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">]</mo><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">[d_A,\,d_A] = 0</annotation></semantics></math> is not automatically fulfilled: since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>deg</mi><msub><mi>d</mi> <mi>A</mi></msub><mo>=</mo><mn>1</mn></mrow><annotation encoding="application/x-tex">\deg d_A = 1</annotation></semantics></math>, we have <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>,</mo><mspace width="thinmathspace"></mspace><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">]</mo><mo>=</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>+</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>=</mo><mn>2</mn><msub><mi>d</mi> <mi>A</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">[d_A,\,d_A] = d_A \circ d_A + d_A \circ d_A = 2 d_A \circ d_A</annotation></semantics></math>. The condition <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mo>∘</mo><msub><mi>d</mi> <mi>A</mi></msub><mo>=</mo><mn>0</mn></mrow><annotation encoding="application/x-tex">d_A \circ d_A = 0</annotation></semantics></math> is actually equivalent to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>A</mi><mo>,</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>A</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\bigl(A, \rho_A, [ - , - ]_A\bigr)</annotation></semantics></math> being a Lie algebroid; that is, it is fulfilled if and only if</p> <ul> <li><math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">[- , - ]_A</annotation></semantics></math> satisfies the Jacobi identity</li> <li>and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">[ - , - ]_A</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>A</mi></msub></mrow><annotation encoding="application/x-tex">\rho_A</annotation></semantics></math> together satisfy the Leibniz identity.</li> </ul> <p>(Proof: calculation gives the restriction <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msup><mi>ρ</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>d</mi><mi>f</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A(f) = \rho^\ast(d f)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo maxsize="1.2em" minsize="1.2em">⟨</mo><mi>ξ</mi><mo>,</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo maxsize="1.2em" minsize="1.2em">⟩</mo><mo>+</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ξ</mi><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo stretchy="false">(</mo><mi>ξ</mi><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A(\xi)(X_1, X_2) = - \bigl\langle \xi, [X_1, X_2]\bigr\rangle + \rho_A(X_1)(\xi X_2) - \rho_A(X_2)(\xi X_1)</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>ξ</mi><mo stretchy="false">)</mo><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>=</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mi>ξ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><mi>ξ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>−</mo><mi>ξ</mi><mo stretchy="false">(</mo><mo stretchy="false">[</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">]</mo><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo stretchy="false">)</mo><mi>ξ</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>−</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo><mi>ξ</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mo>+</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>3</mn></msub><mo stretchy="false">)</mo><mi>ξ</mi><mo stretchy="false">(</mo><msub><mi>X</mi> <mn>1</mn></msub><mo>,</mo><msub><mi>X</mi> <mn>2</mn></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">d_A(\xi)(X_1, X_2, X_3) = - \xi([X_1, X_2], X_3) + \xi([X_1, X_3], X_2) - \xi([X_2, X_3], X_2) + \rho_A(X_1)\xi(X_2, X_3) - \rho_A(X_2)\xi(X_1, X_3) + \rho_A(X_3)\xi(X_1, X_2)</annotation></semantics></math>.)</p> <p>This point of view also applies to <a class="existingWikiWord" href="/nlab/show/higher+Courant+Lie+algebroids">higher Courant Lie algebroids</a> and <a class="existingWikiWord" href="/nlab/show/L-infinity-algebra">L-infinity-algebra</a>s.</p> <p>Example: For the tangent Lie algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mo>=</mo><mi>T</mi><mi>M</mi></mrow><annotation encoding="application/x-tex">A = T M</annotation></semantics></math>, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo maxsize="1.2em" minsize="1.2em">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>T</mi><mi>M</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>A</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo><mo>=</mo><mo maxsize="1.2em" minsize="1.2em">(</mo><msup><mi>Ω</mi> <mo>*</mo></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo maxsize="1.2em" minsize="1.2em">)</mo></mrow><annotation encoding="application/x-tex">\bigl(C(T M[1]), d_A\bigr) = \bigl(\Omega^\ast(M), d_{dR}\bigr)</annotation></semantics></math>.</p> <p>Then a morphism from a Lie algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \rho_A, [-,-]_A)</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>B</mi><mo>,</mo><msub><mi>ρ</mi> <mi>B</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>B</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(B, \rho_B, [-,-]_B)</annotation></semantics></math> is a morphism of the associated differential graded commutative algebras</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>A</mi></msub><mo stretchy="false">)</mo><mo>←</mo><mo stretchy="false">(</mo><mi>C</mi><mo stretchy="false">(</mo><mi>B</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>B</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (C(A[1]), d_A) \leftarrow (C(B[1]), d_B). </annotation></semantics></math></div> <p>Such a morphism of c.d.g.a.‘s is determined by maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>N</mi><mo stretchy="false">)</mo><mo>→</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^\infty(N) \to C^\infty (M)</annotation></semantics></math> on degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and a map <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>B</mi> <mo>*</mo></msup><mo stretchy="false">)</mo><mo>→</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Gamma(B^*)\to \Gamma(A^*)</annotation></semantics></math> on degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>. Thus a morphism of vector bundles <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>A</mi><mover><mo>→</mo><mi>f</mi></mover><mi>B</mi></mrow><annotation encoding="application/x-tex">A\xrightarrow{f} B</annotation></semantics></math> give rise to a morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> of c.g.a. For <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to be a Lie algebroid morphism, we further need <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math> to satisfy additional conditions so that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>f</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">f^*</annotation></semantics></math> preserves the differential.</p> <p>This way to explain morphisms of Lie algebroids is described in <a class="existingWikiWord" href="/nlab/show/Kirill+Mackenzie">Kirill Mackenzie</a>, chapter 4.3. If the Lie algebroids are over the same manifold <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>, then a morphism from <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> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> can be described as a morphism of vector bundles that respects the anchor maps and the Lie bracket. If, however, <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> is over a different manifold <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>, this direct approach does not work. In this situation we have to pull back the Lie algebroid to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> (Note that this is not simply the vector bundle pullback of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>B</mi></mrow><annotation encoding="application/x-tex">B</annotation></semantics></math> along <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi></mrow><annotation encoding="application/x-tex">f</annotation></semantics></math>, but a more involved construction, see <a class="existingWikiWord" href="/nlab/show/Kirill+Mackenzie">Kirill Mackenzie</a>). Using the defintion of a morphism on a common base manifold one arrives at two conditions on the bundle morphism to be a morphism of Lie algebroids. For details see the linked book.</p> <h3 id="example">Example</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>I</mi></mrow><annotation encoding="application/x-tex">I</annotation></semantics></math> be an interval with the tangent bundle Lie algebroid <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>TI</mi><mo>,</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>TI</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(TI, \id_{TI}, [-,-])</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>A</mi><mo>,</mo><msub><mi>ρ</mi> <mi>A</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><msub><mo stretchy="false">]</mo> <mi>A</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(A, \rho_A, [-,-]_A)</annotation></semantics></math> an arbitrary Lie algebroid on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. Then a path <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>A</mi></mrow><annotation encoding="application/x-tex">a\colon I \to A</annotation></semantics></math> defines a map from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi><mo lspace="verythinmathspace">:</mo><mi>C</mi><mo stretchy="false">(</mo><mi>A</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>→</mo><mi>C</mi><mo stretchy="false">(</mo><mi>TI</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo><mo>≅</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\varphi\colon C(A[1]) \to C(TI[1]) \cong \Omega(I)</annotation></semantics></math> which respects the commutative graded algebra structure. A function <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">f\in C^{\infty}(M)</annotation></semantics></math> gets mapped to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>f</mi><mo>∘</mo><mi>γ</mi></mrow><annotation encoding="application/x-tex">f\circ \gamma</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>γ</mi><mo lspace="verythinmathspace">:</mo><mi>I</mi><mo>→</mo><mi>M</mi></mrow><annotation encoding="application/x-tex">\gamma\colon I \to M</annotation></semantics></math> is the projection of <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>. A section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>A</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s\in\Gamma(A^*)</annotation></semantics></math> gets mapped to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>s</mi><mo>∘</mo><mi>γ</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>dt</mi></mrow><annotation encoding="application/x-tex">(s\circ \gamma (a) dt</annotation></semantics></math> under <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>φ</mi></mrow><annotation encoding="application/x-tex">\varphi</annotation></semantics></math>, where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>dt</mi></mrow><annotation encoding="application/x-tex">dt</annotation></semantics></math> is the canonical section of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>Ω</mi> <mn>1</mn></msup><mo stretchy="false">(</mo><mi>I</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">\Omega^1(I)</annotation></semantics></math>. Since <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>TI</mi><mo stretchy="false">[</mo><mn>1</mn><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(TI[1])</annotation></semantics></math> is concentrated in degree <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math> and <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>1</mn></mrow><annotation encoding="application/x-tex">1</annotation></semantics></math>, the other degrees get mapped to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mn>0</mn></mrow><annotation encoding="application/x-tex">0</annotation></semantics></math>.</p> <p>For this map to be a morphism of Lie algebroids, it has to respect the differentials. As explained above this only needs to be checked for a smooth function in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C^{\infty}(M)</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>φ</mi><mo stretchy="false">(</mo><msub><mi>d</mi> <mi>A</mi></msub><mi>f</mi><mo stretchy="false">)</mo><mo>=</mo><msub><mi>d</mi> <mi>dR</mi></msub><mi>φ</mi><mo stretchy="false">(</mo><mi>f</mi><mo stretchy="false">)</mo><mspace width="2em"></mspace><mi>f</mi><mo>∈</mo><msup><mi>C</mi> <mn>∞</mn></msup><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \varphi(d_A f) = d_{dR} \varphi(f) \qquad f\in C^{\infty}(M). </annotation></semantics></math></div> <p>A quick calculation shows that this is true if and only if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><msub><mi>ρ</mi> <mi>A</mi></msub><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo stretchy="false">)</mo><mo>=</mo><mfrac><mi>d</mi><mi>dt</mi></mfrac><mi>γ</mi><mo stretchy="false">(</mo><mi>t</mi><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> \rho_A(a(t)) = \frac{d}{dt}\gamma(t). </annotation></semantics></math></div> <h3 id="example_2">Example</h3> <p>Let <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math> the unit square in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>ℝ</mi> <mn>2</mn></msup></mrow><annotation encoding="application/x-tex">\mathbb{R}^2</annotation></semantics></math>. Then a Lie algebroid morphism from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>TM</mi><mo>,</mo><msub><mo lspace="0em" rspace="thinmathspace">id</mo> <mi>TM</mi></msub><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>,</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo stretchy="false">]</mo><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(TM, \id_{TM}, [-,-])</annotation></semantics></math> to <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>𝔤</mi><mo>,</mo><mi>T</mi><mo>⋅</mo><mo>,</mo><mo stretchy="false">[</mo><mo lspace="verythinmathspace" rspace="0em">−</mo><mo>.</mo><mo>−</mo><msub><mo stretchy="false">]</mo> <mi>𝔤</mi></msub><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(\mathfrak{g}, T\cdot, [-.-]_{\mathfrak{g}})</annotation></semantics></math>, where <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">\mathfrak{g}</annotation></semantics></math> is a Lie algebra, is given by a morphism of c.d.g.a.</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>CE</mi><mo stretchy="false">(</mo><mi>𝔤</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>𝔤</mi></msub><mo stretchy="false">)</mo><mo>→</mo><mo stretchy="false">(</mo><mi>Ω</mi><mo stretchy="false">(</mo><mi>M</mi><mo stretchy="false">)</mo><mo>,</mo><msub><mi>d</mi> <mi>dR</mi></msub><mo stretchy="false">)</mo><mo>.</mo></mrow><annotation encoding="application/x-tex"> (CE(\mathfrak{g}), d_{\mathfrak{g}}) \to (\Omega(M), d_{dR}). </annotation></semantics></math></div> <p>Two smooth maps <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>,</mo><mi>b</mi><mo lspace="verythinmathspace">:</mo><mi>M</mi><mo>→</mo><mi>𝔤</mi></mrow><annotation encoding="application/x-tex">a,b\colon M\to \mathfrak{g}</annotation></semantics></math> give a map between c.g.a. spaces above. Here a section <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo>∈</mo><mi>Γ</mi><mo stretchy="false">(</mo><msup><mi>𝔤</mi> <mo>*</mo></msup><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">s\in\Gamma(\mathfrak{g}^*)</annotation></semantics></math>, i.e. an element of <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msup><mi>𝔤</mi> <mo>*</mo></msup></mrow><annotation encoding="application/x-tex">\mathfrak{g}^*</annotation></semantics></math> gets mapped to the one form</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mi>s</mi><mo stretchy="false">(</mo><mi>a</mi><mo stretchy="false">)</mo><mi>dt</mi><mo>+</mo><mi>s</mi><mo stretchy="false">(</mo><mi>b</mi><mo stretchy="false">)</mo><mi>ds</mi><mo>,</mo></mrow><annotation encoding="application/x-tex"> s(a) dt + s(b) ds, </annotation></semantics></math></div> <p>where <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">(</mo><mi>t</mi><mo>,</mo><mi>s</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">(t,s)</annotation></semantics></math> are the coordinates on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. A quick calculation shows that this map respects the differential if and only if</p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mfrac><mi>da</mi><mi>ds</mi></mfrac><mo>−</mo><mfrac><mi>db</mi><mi>ds</mi></mfrac><mo>=</mo><mo stretchy="false">[</mo><mi>a</mi><mo>,</mo><mi>b</mi><msub><mo stretchy="false">]</mo> <mi>𝔤</mi></msub><mo>.</mo></mrow><annotation encoding="application/x-tex"> \frac{da}{ds} - \frac{db}{ds} = [a,b]_{\mathfrak{g}}. </annotation></semantics></math></div> <p>This formula also shows that <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>a</mi><mo>⋅</mo><mi>dt</mi><mo>+</mo><mi>b</mi><mo>⋅</mo><mi>ds</mi></mrow><annotation encoding="application/x-tex">a\cdot dt + b\cdot ds</annotation></semantics></math> is a flat connection on the trivial principal <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>-bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>M</mi></mrow><annotation encoding="application/x-tex">M</annotation></semantics></math>. Therefore Lie algebroid morphisms open a way to talk about higher connections and flat conditions.</p> <h3 id="open_problem">Open problem</h3> <p>It is unclear how to use the idea of <a class="existingWikiWord" href="/nlab/show/bibundle">bibundle</a> or <a class="existingWikiWord" href="/nlab/show/span">span</a> to define a more general version of Lie algebroid morphisms so that they really correspond to the case of Lie groupoid morphisms.</p> <h2 id="higher_lie_groupoids">Higher Lie groupoids</h2> <p>See</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/internal+%E2%88%9E-groupoid">internal ∞-groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/%E2%88%9E-Lie+groupoid">∞-Lie groupoid</a></p> </li> </ul> <h2 id="examples">Examples</h2> <ul> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/smooth+manifold">smooth 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> is a 0-<a class="existingWikiWord" href="/nlab/show/0-groupoid">truncated</a> Lie groupoid.</p> </li> <li> <p>For every <a class="existingWikiWord" href="/nlab/show/Lie+group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math> the one-object <a class="existingWikiWord" href="/nlab/show/delooping">delooping</a> groupoid <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 Lie groupoid.</p> </li> <li> <p>The Lie group <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> itself is a 0-<a class="existingWikiWord" href="/nlab/show/truncated">truncated</a> <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the 2-category or Lie groupoids.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/Lie+2-group">Lie 2-group</a> is in particular a Lie groupoid: a <a class="existingWikiWord" href="/nlab/show/group+object">group object</a> in the category of Lie groupoids.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/inner+automorphism+2-group">inner automorphism 2-group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>=</mo><mi>INN</mi><mo stretchy="false">(</mo><mi>G</mi><mo stretchy="false">)</mo><mo>=</mo><mi>G</mi><mo stretchy="false">/</mo><mo stretchy="false">/</mo><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G = INN(G) = G//G</annotation></semantics></math> is a Lie groupoid. There is an obvious morphism <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">\mathbf{E}G \to \mathbf{B}G</annotation></semantics></math>.</p> </li> <li> <p>For every <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>-<a class="existingWikiWord" href="/nlab/show/principal+bundle">principal bundle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi><mo>→</mo><mi>X</mi></mrow><annotation encoding="application/x-tex">P \to X</annotation></semantics></math> there is its <a class="existingWikiWord" href="/nlab/show/Atiyah+Lie+groupoid">Atiyah Lie groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>At</mi><mo stretchy="false">(</mo><mi>P</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">At(P)</annotation></semantics></math>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/fundamental+groupoid">fundamental groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><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">\Pi_1(X)</annotation></semantics></math> of a smooth manifold is naturally a Lie groupoid.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/path+groupoid">path groupoid</a> of a smooth manifold is naturally a <a class="existingWikiWord" href="/nlab/show/diffeological+groupoid">diffeological groupoid</a>.</p> </li> <li> <p>The <a class="existingWikiWord" href="/nlab/show/Cech+groupoid">Cech groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">C(U)</annotation></semantics></math> of a <a class="existingWikiWord" href="/nlab/show/cover">cover</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mo stretchy="false">{</mo><msub><mi>U</mi> <mi>i</mi></msub><mo>→</mo><mi>X</mi><mo stretchy="false">}</mo></mrow><annotation encoding="application/x-tex">\{U_i \to X\}</annotation></semantics></math> of a smooth manifold is a Lie groupoid.</p> </li> <li> <p>Every <a class="existingWikiWord" href="/nlab/show/foliation">foliation</a> gives rise to its <a class="existingWikiWord" href="/nlab/show/holonomy+groupoid">holonomy groupoid</a>.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a> is a Lie groupoid.</p> </li> <li> <p>An <a class="existingWikiWord" href="/nlab/show/anafunctor">anafunctor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>X</mi><mover><mo>←</mo><mo>≃</mo></mover><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo><mo>→</mo><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mrow><annotation encoding="application/x-tex">X \stackrel{\simeq}{\leftarrow} C(U) \to \mathbf{B}G</annotation></semantics></math> from a smooth manifold <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> to <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/Cech+cohomology">Cech cocycle</a> in degree 1 with values in <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>G</mi></mrow><annotation encoding="application/x-tex">G</annotation></semantics></math>, classifying <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> <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>.</p> </li> <li> <p>The (1-categorical) <a class="existingWikiWord" href="/nlab/show/pullback">pullback</a></p> <div class="maruku-equation"><math xmlns="http://www.w3.org/1998/Math/MathML" display="block" class="maruku-mathml"><semantics><mrow><mrow><mtable><mtr><mtd><mi>P</mi></mtd> <mtd><mover><mo>←</mo><mo>≃</mo></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>P</mi></mstyle></mtd> <mtd><mo>→</mo></mtd> <mtd><mstyle mathvariant="bold"><mi>E</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd> <mtd></mtd> <mtd><mo stretchy="false">↓</mo></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>C</mi><mo stretchy="false">(</mo><mi>U</mi><mo stretchy="false">)</mo></mtd> <mtd><mover><mo>→</mo><mrow></mrow></mover></mtd> <mtd><mstyle mathvariant="bold"><mi>B</mi></mstyle><mi>G</mi></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><msup><mo stretchy="false">↓</mo> <mpadded width="0"><mo>≃</mo></mpadded></msup></mtd></mtr> <mtr><mtd></mtd> <mtd></mtd> <mtd><mi>X</mi></mtd></mtr></mtable></mrow></mrow><annotation encoding="application/x-tex"> \array{ P &\stackrel{\simeq}{\leftarrow}& \mathbf{P} &\to& \mathbf{E}G \\ && \downarrow && \downarrow \\ && C(U) &\stackrel{}{\to}& \mathbf{B}G \\ && \downarrow^{\mathrlap{\simeq}} \\ && X } </annotation></semantics></math></div> <p>is a Lie groupoid equivalent to this principal bundle <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><mi>P</mi></mrow><annotation encoding="application/x-tex">P</annotation></semantics></math>.</p> <p>(For more on the general phenomenon of which this is a special case see <a class="existingWikiWord" href="/nlab/show/principal+%E2%88%9E-bundle">principal ∞-bundle</a> and <a class="existingWikiWord" href="/nlab/show/universal+principal+%E2%88%9E-bundle">universal principal ∞-bundle</a>.)</p> </li> <li> <p>Similarly an anafunctor from <math xmlns="http://www.w3.org/1998/Math/MathML" display="inline" class="maruku-mathml"><semantics><mrow><msub><mi>P</mi> <mn>1</mn></msub><mo stretchy="false">(</mo><mi>X</mi><mo stretchy="false">)</mo></mrow><annotation encoding="application/x-tex">P_1(X)</annotation></semantics></math> to <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/connection+on+a+bundle">connection on a bundle</a> (see there for details).</p> </li> </ul> <h2 id="related_concepts">Related concepts</h2> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/smooth+groupoid">smooth groupoid</a>, <a class="existingWikiWord" href="/nlab/show/smooth+functor">smooth functor</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/LieGrpd">LieGrpd</a>, <a class="existingWikiWord" href="/nlab/show/SmoothGrpd">SmoothGrpd</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/bisection+of+a+Lie+groupoid">bisection of a Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/orbifold">orbifold</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/effective+Lie+groupoid">effective Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/proper+Lie+groupoid">proper Lie groupoid</a>, <a class="existingWikiWord" href="/nlab/show/etale+groupoid">etale groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/homotopy+groups+of+a+Lie+groupoid">homotopy groups of a Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Morita+morphism">Morita morphism</a>, <a class="existingWikiWord" href="/nlab/show/Hilsum-Skandalis+morphism">Hilsum-Skandalis morphism</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Tannaka+duality+for+Lie+groupoids">Tannaka duality for Lie groupoids</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/double+Lie+groupoid">double Lie groupoid</a></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/higher+differential+geometry">higher differential geometry</a></p> </li> </ul> <h2 id="references">References</h2> <p>The notion of Lie categories, hence of Lie groupoids, goes back to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Catégories topologiques et categories différentiables</em>, Colloque de Géométrie différentielle globale, Bruxelles, C.B.R.M., (1959) pp. 137-150 (<a class="existingWikiWord" href="/nlab/files/EhresmannCategoriesTopologiques.pdf" title="pdf">pdf</a>, <a href="https://zbmath.org/?q=an:0205.28202">zbMath:0205.28202</a>)</li> </ul> <p>Their understanding as <a class="existingWikiWord" href="/nlab/show/internal+categories">internal categories</a>/<a class="existingWikiWord" href="/nlab/show/internal+groupoids">internal groupoids</a> <a class="existingWikiWord" href="/nlab/show/internalization">in</a> <a class="existingWikiWord" href="/nlab/show/SmoothManifolds">SmoothManifolds</a> is often attributed to</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Charles+Ehresmann">Charles Ehresmann</a>, <em>Catégories structurées</em>, Annales scientifiques de l’École Normale Supérieure, Série 3, Tome 80 (1963) no. 4, pp. 349-426 (<a href="http://www.numdam.org/item/ASENS_1963_3_80_4_349_0">numdam:ASENS_1963_3_80_4_349_0</a>)</li> </ul> <p>but the simple notion of <a class="existingWikiWord" href="/nlab/show/internalization">internalization</a> and <a class="existingWikiWord" href="/nlab/show/internal+groupoids">internal groupoids</a> (<a href="internalization#Grothendieck60">Grothendieck 1960</a>, <a href="internalization#Grothendieck61">61</a>) is hardly recognizable in this account.</p> <p>Textbook accounts:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/Kirill+Mackenzie">Kirill Mackenzie</a>, <em>Lie groupoids and Lie algebroids in differential geometry</em>, London Mathematical Society Lecture Note Series, 124. Cambridge University Press, Cambridge, 1987. xvi+327 pp (<a href="https://doi.org/10.1017/CBO9780511661839">doi:10.1017/CBO9780511661839</a>, <a href="http://www.ams.org/mathscinet-getitem?mr=896907">MR:896907</a>)</p> </li> <li id="MoerdijkMrcun03"> <p><a class="existingWikiWord" href="/nlab/show/Ieke+Moerdijk">Ieke Moerdijk</a>, <a class="existingWikiWord" href="/nlab/show/Janez+Mr%C4%8Dun">Janez Mrčun</a> <em>Introduction to Foliations and Lie Groupoids</em>, Cambridge Studies in Advanced Mathematics 91, Cambridge University Press, 2003 (<a href="https://doi.org/10.1017/CBO9780511615450">doi:10.1017/CBO9780511615450</a>)</p> <blockquote> <p>(in the context of <a class="existingWikiWord" href="/nlab/show/foliation+theory">foliation theory</a>: <a class="existingWikiWord" href="/nlab/show/foliation+groupoids">foliation groupoids</a>)</p> </blockquote> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Kirill+Mackenzie">Kirill Mackenzie</a>, <em>General Theory of Lie Groupoids and Lie Algebroids,</em> Cambridge University Press, 2005 (<a href="https://doi.org/10.1017/CBO9781107325883">doi:10.1017/CBO9781107325883</a>)</p> </li> </ul> <p>Historical review:</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Jean+Pradines">Jean Pradines</a>, <em>In <a class="existingWikiWord" href="/nlab/show/Ehresmann">Ehresmann</a>‘s footsteps: from Group Geometries to Groupoid Geometries</em>, Banach Center Publications, vol. 76, Warsawa 2007, 87-157 (<a href="http://arxiv.org/abs/0711.1608">arXiv:0711.1608</a>, <a href="https://www.impan.pl/en/publishing-house/banach-center-publications/all/76/0/86184/in-ehresmann-s-footsteps-from-group-geometries-to-groupoid-geometries">doi:10.4064/bc76-0-5</a>)</li> </ul> <p>The relation to <a class="existingWikiWord" href="/nlab/show/differentiable+stacks">differentiable stacks</a> is discussed/reviewed in section 2 of</p> <ul> <li id="Blohmann"><a class="existingWikiWord" href="/nlab/show/Christian+Blohmann">Christian Blohmann</a>, <em>Stacky Lie groups</em>, Int. Mat. Res. Not. (2008) Vol. 2008: article ID rnn082 (<a href="http://arxiv.org/abs/math/0702399">arXiv:math/0702399</a>)</li> </ul> <p>Lie groupoids as a source for <a class="existingWikiWord" href="/nlab/show/groupoid+convolution+algebras">groupoid convolution</a> <a class="existingWikiWord" href="/nlab/show/C%2A-algebras">C*-algebras</a> are discussed in</p> <ul> <li><a class="existingWikiWord" href="/nlab/show/Alain+Connes">Alain Connes</a>, <em><a class="existingWikiWord" href="/nlab/show/Noncommutative+Geometry">Noncommutative Geometry</a></em></li> </ul> <p>Review:</p> <ul> <li> <p><a class="existingWikiWord" href="/nlab/show/John+Baez">John Baez</a><em><a href="http://math.ucr.edu/home/baez/TWF.html">TWF</a> <a href="http://math.ucr.edu/home/baez/week256.html">256</a></em></p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Alan+Weinstein">Alan Weinstein</a>, <em>Groupoids: Unifying Internal and External Symmetry – A Tour through some Examples</em>, Notices of the AMS <strong>43</strong> 7 (1996) [<a href="http://www.ams.org/notices/199607/weinstein.pdf">pdf</a>, <a class="existingWikiWord" href="/nlab/files/Weinstein_Groupoids.pdf" title="pdf">pdf</a>]</p> </li> <li> <p><a class="existingWikiWord" href="/nlab/show/Henrique+Bursztyn">Henrique Bursztyn</a>, Matias del Hoyo, <em>Lie Groupoids</em>, in <em><a class="existingWikiWord" href="/nlab/show/Encyclopedia+of+Mathematical+Physics+2nd+ed">Encyclopedia of Mathematical Physics 2nd ed</a></em>, Elsevier (2024) [<a href="https://arxiv.org/abs/2309.14105">arXiv:2309.14105</a>]</p> </li> </ul> <p>Groupoids and their various morphisms between them in different categories, including in Diff, is also in</p> <ul> <li>Ralf Meyer and <a class="existingWikiWord" href="/nlab/show/Chenchang+Zhu">Chenchang Zhu</a>, <em>Groupoids in categories with pretopology</em>, <a href="http://arxiv.org/pdf/1408.5220.pdf">arXiv:math/1408.5220</a></li> </ul> </body></html> </div> <div class="revisedby"> <p> Last revised on March 26, 2024 at 19:56:40. 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