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<title>Differentiable stack</title>

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Differentiable stack

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A differentiable stack is the analogue in <a href="Differential_geometry" title="Differential geometry">differential geometry</a> of an <a href="Algebraic_stack" title="Algebraic stack">algebraic stack</a> in <a href="Algebraic_geometry" title="Algebraic geometry">algebraic geometry</a>. It can be described either as a <a href="Stack_(mathematics)" title="Stack (mathematics)">stack</a> over <a href="Differentiable_manifold" title="Differentiable manifold">differentiable manifolds</a> which admits an atlas, or as a <a href="Lie_groupoid" title="Lie groupoid">Lie groupoid</a> up to <a href="Morita_equivalence" title="Morita equivalence">Morita equivalence</a>.<a href="#cite_note-1">[1]</a>

Differentiable stacks are particularly useful to handle spaces with <a href="Singularity_(mathematics)" title="Singularity (mathematics)">singularities</a> (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in <a href="Foliation_theory" class="mw-redirect" title="Foliation theory">foliation theory</a>,<a href="#cite_note-2">[2]</a> <a href="Poisson_geometry" class="mw-redirect" title="Poisson geometry">Poisson geometry</a><a href="#cite_note-3">[3]</a> and <a href="Twisted_K-theory" title="Twisted K-theory">twisted K-theory</a>.<a href="#cite_note-4">[4]</a>

<div class="mw-heading mw-heading2"><h2 id="Definition">Definition</h2></div>

<div class="mw-heading mw-heading3"><h3 id="Definition_1_(via_groupoid_fibrations)">Definition 1 (via groupoid fibrations)</h3></div>

Recall that a <a href="Category_fibered_in_groupoids" class="mw-redirect" title="Category fibered in groupoids">category fibred in groupoids</a> (also called a groupoid fibration) consists of a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> together with a functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }">

</mrow>

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }</annotation>

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</math><img src="./1df8cef656d609ee78638ccd268b39f75f6aaaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.411ex; height:2.176ex;" alt="{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }" loading="lazy"> to the <a href="Category_of_manifolds" title="Category of manifolds">category of differentiable manifolds</a> such that

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> is a <a href="Fibred_category" title="Fibred category">fibred category</a>, i.e. for any object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle u}</annotation>

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</math><img src="./c3e6bb763d22c20916ed4f0bb6bd49d7470cffd8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle u}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

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</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> and any arrow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to U}</annotation>

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</math><img src="./9f9f8b9779ad0f99b17fe0bd33bedef7a5a21a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle V\to U}" loading="lazy"> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mfd} }">

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mfd} }</annotation>

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</math><img src="./95f06d99ad853c46b862405016b356664b9d1592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.289ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mfd} }" loading="lazy"> there is an arrow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\to u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\to u}</annotation>

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</math><img src="./740b8dd3faea7f652895e7f7c542b3aa00cae14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.071ex; height:1.843ex;" alt="{\displaystyle v\to u}" loading="lazy"> lying over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to U}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to U}</annotation>

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</math><img src="./9f9f8b9779ad0f99b17fe0bd33bedef7a5a21a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle V\to U}" loading="lazy">;</li>

<li>for every <a href="Commutative_diagram" title="Commutative diagram">commutative triangle</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\to V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W\to V\to U}</annotation>

</semantics>

</math><img src="./ebe8754e636d5b3f0cda25a5ac316666f4e3a34a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:13.233ex; height:2.176ex;" alt="{\displaystyle W\to V\to U}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mdf} }">

</mrow>

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mdf} }</annotation>

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</math><img src="./78e6fb40a66ab0c17051ff3019bbceb8781b3154.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.289ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mdf} }" loading="lazy"> and every arrows <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\to u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\to u}</annotation>

</semantics>

</math><img src="./629ed16988d18ef0cf0392c563b13cb5ed150f83.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.608ex; height:1.843ex;" alt="{\displaystyle w\to u}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle W\to U}</annotation>

</semantics>

</math><img src="./aa9052de53d890a619d0f2ea0ca21b14ae018c80.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.832ex; height:2.176ex;" alt="{\displaystyle W\to U}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle v\to u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle v\to u}</annotation>

</semantics>

</math><img src="./740b8dd3faea7f652895e7f7c542b3aa00cae14d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.071ex; height:1.843ex;" alt="{\displaystyle v\to u}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to U}</annotation>

</semantics>

</math><img src="./9f9f8b9779ad0f99b17fe0bd33bedef7a5a21a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle V\to U}" loading="lazy">, there exists a unique arrow <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\to v}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\to v}</annotation>

</semantics>

</math><img src="./6b276861d86632830254b1dadf05641d9b5ebd25.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.406ex; height:1.843ex;" alt="{\displaystyle w\to v}" loading="lazy"> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle W\to V}">

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</mrow>

<annotation encoding="application/x-tex">{\displaystyle W\to V}</annotation>

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</math><img src="./596b9b862eaf230df9abd76c98b775ae3f38c8f5.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.837ex; height:2.176ex;" alt="{\displaystyle W\to V}" loading="lazy"> making the triangle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle w\to v\to u}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle w\to v\to u}</annotation>

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</math><img src="./d889a6cc36a9a64bb4b528a8cd679df8f039652a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.35ex; height:1.843ex;" alt="{\displaystyle w\to v\to u}" loading="lazy"> commute.</li></ol>

These properties ensure that, for every object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

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<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mfd} }">

</mrow>

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<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mfd} }</annotation>

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</math><img src="./95f06d99ad853c46b862405016b356664b9d1592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.289ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mfd} }" loading="lazy">, one can define its fibre, denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(U)}</annotation>

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</math><img src="./52414f17b55ee456b0b6cabc0da7f5c141981c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(U)}" loading="lazy"> or <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}_{U}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}_{U}}</annotation>

</semantics>

</math><img src="./c6710aeffb0733f0aba2c4ab1afaf1c7b3a6b932.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.718ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}_{U}}" loading="lazy">, as the <a href="Subcategory" title="Subcategory">subcategory</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> made up by all objects of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> lying over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and all morphisms of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> lying over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle id_{U}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle id_{U}}</annotation>

</semantics>

</math><img src="./31aba1ec03fba82173b8dd0b88db9ba2bacc87f2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.504ex; height:2.509ex;" alt="{\displaystyle id_{U}}" loading="lazy">. By construction, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi ^{-1}(U)}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi ^{-1}(U)}</annotation>

</semantics>

</math><img src="./52414f17b55ee456b0b6cabc0da7f5c141981c09.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:7.259ex; height:3.176ex;" alt="{\displaystyle \pi ^{-1}(U)}" loading="lazy"> is a <a href="Groupoid" title="Groupoid">groupoid</a>, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of <a href="Descent_(mathematics)" title="Descent (mathematics)">descent</a>.

Any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> defines its <a href="Slice_category" class="mw-redirect" title="Slice category">slice category</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}=\mathrm {Hom} _{\mathrm {Mdf} }(-,X)}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}=\mathrm {Hom} _{\mathrm {Mdf} }(-,X)}</annotation>

</semantics>

</math><img src="./de1efe587723753d855a23c8382858c75acc0914.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:20.963ex; height:2.843ex;" alt="{\displaystyle F_{X}=\mathrm {Hom} _{\mathrm {Mdf} }(-,X)}" loading="lazy">, whose objects are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,f)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,f)}</annotation>

</semantics>

</math><img src="./ec4dbdf43357db955dc953fe357a7fc4a8566d96.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.904ex; height:2.843ex;" alt="{\displaystyle (U,f)}" loading="lazy"> of a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and a smooth map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle f:U\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle f:U\to X}</annotation>

</semantics>

</math><img src="./10758466782339104419e62ee06709090b175888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:10.592ex; height:2.509ex;" alt="{\displaystyle f:U\to X}" loading="lazy">; then <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}\to \mathrm {Mdf} ,(U,f)\mapsto U}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}\to \mathrm {Mdf} ,(U,f)\mapsto U}</annotation>

</semantics>

</math><img src="./2af6a09c8f1c81302499b00a66d2fe00c0fd2599.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:23.365ex; height:2.843ex;" alt="{\displaystyle F_{X}\to \mathrm {Mdf} ,(U,f)\mapsto U}" loading="lazy"> is a groupoid fibration which is actually also a stack. A morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\to {\mathcal {D}}}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\to {\mathcal {D}}}</annotation>

</semantics>

</math><img src="./a4afc624b5e4d7030088eca335ca0c37c84a66d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:6.645ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\to {\mathcal {D}}}" loading="lazy"> of groupoid fibrations is called a representable submersion if

<ul><li>for every manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{U}\to {\mathcal {D}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{U}\to {\mathcal {D}}}</annotation>

</semantics>

</math><img src="./cfc847e55c3bae15496a9adad4fd87f29c8a328b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.393ex; height:2.509ex;" alt="{\displaystyle F_{U}\to {\mathcal {D}}}" loading="lazy">, the <a href="Fibred_product" class="mw-redirect" title="Fibred product">fibred product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\times _{\mathcal {D}}F_{U}}">

</mrow>

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\times _{\mathcal {D}}F_{U}}</annotation>

</semantics>

</math><img src="./b420eec74222d15f5f5e0fe838e1bf95514b6680.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:8.566ex; height:2.509ex;" alt="{\displaystyle {\mathcal {C}}\times _{\mathcal {D}}F_{U}}" loading="lazy"> is representable, i.e. it is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{V}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{V}}</annotation>

</semantics>

</math><img src="./4e25a78c2450c4e9c3fbd721c1f682e9679d2f3d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:2.991ex; height:2.509ex;" alt="{\displaystyle F_{V}}" loading="lazy"> (for some manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V}</annotation>

</semantics>

</math><img src="./af0f6064540e84211d0ffe4dac72098adfa52845.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.787ex; height:2.176ex;" alt="{\displaystyle V}" loading="lazy">) as groupoid fibrations;</li>

<li>the induced smooth map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to U}</annotation>

</semantics>

</math><img src="./9f9f8b9779ad0f99b17fe0bd33bedef7a5a21a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle V\to U}" loading="lazy"> is a <a href="Submersion_(mathematics)" title="Submersion (mathematics)">submersion</a>.</li></ul>

A differentiable stack is a stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }</annotation>

</semantics>

</math><img src="./1df8cef656d609ee78638ccd268b39f75f6aaaac.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:12.411ex; height:2.176ex;" alt="{\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }" loading="lazy"> together with a special kind of representable submersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}\to {\mathcal {C}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}\to {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./898e9907bcf02cd70faf6615904ca96d6b4f03d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.98ex; height:2.509ex;" alt="{\displaystyle F_{X}\to {\mathcal {C}}}" loading="lazy"> (every submersion <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle V\to U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle V\to U}</annotation>

</semantics>

</math><img src="./9f9f8b9779ad0f99b17fe0bd33bedef7a5a21a55.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.184ex; height:2.176ex;" alt="{\displaystyle V\to U}" loading="lazy"> described above is asked to be <a href="Surjective_function" title="Surjective function">surjective</a>), for some manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. The map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle F_{X}\to {\mathcal {C}}}">

<msub>

</mrow>

</msub>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle F_{X}\to {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./898e9907bcf02cd70faf6615904ca96d6b4f03d6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:7.98ex; height:2.509ex;" alt="{\displaystyle F_{X}\to {\mathcal {C}}}" loading="lazy"> is called atlas, presentation or cover of the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.<a href="#cite_note-:0-5">[5]</a><a href="#cite_note-6">[6]</a>

<div class="mw-heading mw-heading3"><h3 id="Definition_2_(via_2-functors)">Definition 2 (via 2-functors)</h3></div>

Recall that a <a href="Prestack" title="Prestack">prestack</a> (of groupoids) on a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy">, also known as a 2-<a href="Presheaf_(category_theory)" title="Presheaf (category theory)">presheaf</a>, is a <a href="2-functor" class="mw-redirect" title="2-functor">2-functor</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./6a0247662550be93cfd7578fcd8e7a37a49c7007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.685ex; height:2.676ex;" alt="{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }" loading="lazy">, where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Grp} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./e16467c94bbc67cebf95571f7418c4512561f8c6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:4.029ex; height:2.509ex;" alt="{\displaystyle \mathrm {Grp} }" loading="lazy"> is the <a href="2_category" class="mw-redirect" title="2 category">2-category</a> of (set-theoretical) <a href="Groupoid" title="Groupoid">groupoids</a>, their morphisms, and the natural transformations between them. A <a href="Stack_(mathematics)" title="Stack (mathematics)">stack</a> is a prestack satisfying further glueing properties (analogously to the glueing properties satisfied by a sheaf). In order to state such properties precisely, one needs to define (pre)stacks on a <a href="Site_(mathematics)" class="mw-redirect" title="Site (mathematics)">site</a>, i.e. a category equipped with a <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a>.

Any object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}</annotation>

</semantics>

</math><img src="./067db81c8a840c70772d4f6aacae3eb5869fdf68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.144ex; height:2.843ex;" alt="{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}" loading="lazy"> defines a stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)}">

</munder>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)}</annotation>

</semantics>

</math><img src="./d8ef343b5aa35d4419b326b53b299ed7c16e5496.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.122ex; margin-bottom: -0.755ex; width:19.354ex; height:3.343ex;" alt="{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)}" loading="lazy">, which associated to another object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N\in \mathrm {Obj} ({\mathcal {C}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N\in \mathrm {Obj} ({\mathcal {C}})}</annotation>

</semantics>

</math><img src="./c7686e705ba776c97b482b345960ac4fcdf6303f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.765ex; height:2.843ex;" alt="{\displaystyle N\in \mathrm {Obj} ({\mathcal {C}})}" loading="lazy"> the groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hom} _{\mathcal {C}}(N,M)}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Hom} _{\mathcal {C}}(N,M)}</annotation>

</semantics>

</math><img src="./931af62251b8a5544b471c2655149152484198b9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.299ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hom} _{\mathcal {C}}(N,M)}" loading="lazy"> of <a href="Morphism" title="Morphism">morphisms</a> from <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">. A stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./6a0247662550be93cfd7578fcd8e7a37a49c7007.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:15.685ex; height:2.676ex;" alt="{\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }" loading="lazy"> is called geometric if there is an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}</annotation>

</semantics>

</math><img src="./067db81c8a840c70772d4f6aacae3eb5869fdf68.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.144ex; height:2.843ex;" alt="{\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}" loading="lazy"> and a morphism of stacks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to X}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to X}</annotation>

</semantics>

</math><img src="./a5235ae11063ca96468ce50cc27f2a57f3f1bab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.16ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\to X}" loading="lazy"> (often called atlas, presentation or cover of the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">) such that

<ul><li>the morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to X}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to X}</annotation>

</semantics>

</math><img src="./a5235ae11063ca96468ce50cc27f2a57f3f1bab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.16ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\to X}" loading="lazy"> is representable, i.e. for every object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y}</annotation>

</semantics>

</math><img src="./961d67d6b454b4df2301ac571808a3538b3a6d3f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.171ex; width:1.773ex; height:2.009ex;" alt="{\displaystyle Y}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> and any morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Y\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Y\to X}</annotation>

</semantics>

</math><img src="./772ca21e4bcba5ccedd53ff7cb37add671628747.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.367ex; height:2.176ex;" alt="{\displaystyle Y\to X}" loading="lazy"> the <a href="Fibred_product" class="mw-redirect" title="Fibred product">fibred product</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\times _{X}{\underline {Y}}}">

</munder>

</mrow>

<msub>

</mrow>

</msub>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\times _{X}{\underline {Y}}}</annotation>

</semantics>

</math><img src="./c01fb23e2fb4925af3c7664fc3c1bba0687b7ed2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.813ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\times _{X}{\underline {Y}}}" loading="lazy"> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {Z}}}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {Z}}}</annotation>

</semantics>

</math><img src="./b179ffebb62a9357ca9191457d52ada9b1a906de.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.06ex; margin-bottom: -0.827ex; width:1.742ex; height:3.176ex;" alt="{\displaystyle {\underline {Z}}}" loading="lazy"> (for some object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z}</annotation>

</semantics>

</math><img src="./1cc6b75e09a8aa3f04d8584b11db534f88fb56bd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.68ex; height:2.176ex;" alt="{\displaystyle Z}" loading="lazy">) as stacks;</li>

<li>the induces morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle Z\to Y}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle Z\to Y}</annotation>

</semantics>

</math><img src="./a2f836de5648fddbae09be6abce0cc80eb9ba3f0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.068ex; height:2.176ex;" alt="{\displaystyle Z\to Y}" loading="lazy"> satisfies a further property depending on the category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> (e.g., for manifold it is asked to be a <a href="Submersion_(mathematics)" title="Submersion (mathematics)">submersion</a>).</li></ul>

A differentiable stack is a stack on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}=\mathrm {Mfd} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}=\mathrm {Mfd} }</annotation>

</semantics>

</math><img src="./a604842eb2ff1dfa26f2f4f82700d98754f64a24.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:8.626ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}=\mathrm {Mfd} }" loading="lazy">, the <a href="Category_of_manifolds" title="Category of manifolds">category of differentiable manifolds</a> (viewed as a site with the usual open covering topology), i.e. a 2-functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./4fe805586c99075d3bbf4f0832c417053d2f104c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.731ex; height:2.676ex;" alt="{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }" loading="lazy">, which is also geometric, i.e. admits an atlas <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to X}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to X}</annotation>

</semantics>

</math><img src="./a5235ae11063ca96468ce50cc27f2a57f3f1bab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.16ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\to X}" loading="lazy"> as described above.<a href="#cite_note-:1-7">[7]</a><a href="#cite_note-8">[8]</a>

Note that, replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mfd} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mfd} }</annotation>

</semantics>

</math><img src="./95f06d99ad853c46b862405016b356664b9d1592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.289ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mfd} }" loading="lazy"> with the category of <a href="Affine_scheme" class="mw-redirect" title="Affine scheme">affine schemes</a>, one recovers the standard notion of <a href="Algebraic_stack" title="Algebraic stack">algebraic stack</a>. Similarly, replacing <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Mfd} }">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Mfd} }</annotation>

</semantics>

</math><img src="./95f06d99ad853c46b862405016b356664b9d1592.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:4.289ex; height:2.176ex;" alt="{\displaystyle \mathrm {Mfd} }" loading="lazy"> with the category of <a href="Topological_space" title="Topological space">topological spaces</a>, one obtains the definition of topological stack.

<div class="mw-heading mw-heading3"><h3 id="Definition_3_(via_Morita_equivalences)">Definition 3 (via Morita equivalences)</h3></div>

Recall that a <a href="Lie_groupoid" title="Lie groupoid">Lie groupoid</a> consists of two differentiable manifolds <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, together with two <a href="Surjective_function" title="Surjective function">surjective</a> <a href="Submersion_(mathematics)" title="Submersion (mathematics)">submersions</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle s,t:G\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle s,t:G\to M}</annotation>

</semantics>

</math><img src="./c6a7cdec3656c75b20f97db6f94e60f0ac9b5ba2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:12.784ex; height:2.509ex;" alt="{\displaystyle s,t:G\to M}" loading="lazy">, as well as a partial multiplication map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle m:G\times _{M}G\to G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle m:G\times _{M}G\to G}</annotation>

</semantics>

</math><img src="./4d937e6745e8ccb49947ab000cbf6c7f33d9af07.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:17.871ex; height:2.509ex;" alt="{\displaystyle m:G\times _{M}G\to G}" loading="lazy">, a unit map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u:M\to G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u:M\to G}</annotation>

</semantics>

</math><img src="./da5bf8afd128b1d6b51d38d94be8566936cb2806.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:11.15ex; height:2.176ex;" alt="{\displaystyle u:M\to G}" loading="lazy">, and an inverse map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle i:G\to G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle i:G\to G}</annotation>

</semantics>

</math><img src="./413e0936c524dbc4d440b73ac947d32ebd0215b1.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:10.007ex; height:2.176ex;" alt="{\displaystyle i:G\to G}" loading="lazy">, satisfying group-like compatibilities.

Two Lie groupoids <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\rightrightarrows M}</annotation>

</semantics>

</math><img src="./c39368f8c0b42847455ed449388037303b977d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.883ex; height:2.343ex;" alt="{\displaystyle G\rightrightarrows M}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H\rightrightarrows N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H\rightrightarrows N}</annotation>

</semantics>

</math><img src="./3e2bcd9ae234ec0e3d3089e6a7dad53ea730a13c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.741ex; height:2.343ex;" alt="{\displaystyle H\rightrightarrows N}" loading="lazy"> are Morita equivalent if there is a principal bi-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> between them, i.e. a principal right <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H}</annotation>

</semantics>

</math><img src="./75a9edddcca2f782014371f75dca39d7e13a9c1b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle H}" loading="lazy">-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to M}</annotation>

</semantics>

</math><img src="./890d3709f835443a68c49b48a9142616e48f4da7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.802ex; height:2.176ex;" alt="{\displaystyle P\to M}" loading="lazy">, a principal left <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-bundle <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to N}</annotation>

</semantics>

</math><img src="./9c1a12814d506b703f7c5654b25af5caeef38af9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.423ex; height:2.176ex;" alt="{\displaystyle P\to N}" loading="lazy">, such that the two actions on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P}</annotation>

</semantics>

</math><img src="./b4dc73bf40314945ff376bd363916a738548d40a.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.745ex; height:2.176ex;" alt="{\displaystyle P}" loading="lazy"> commutes. Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.

A differentiable stack, denoted as <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy">, is the Morita equivalence class of some Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\rightrightarrows M}</annotation>

</semantics>

</math><img src="./c39368f8c0b42847455ed449388037303b977d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.883ex; height:2.343ex;" alt="{\displaystyle G\rightrightarrows M}" loading="lazy">.<a href="#cite_note-:0-5">[5]</a><a href="#cite_note-9">[9]</a>

<div class="mw-heading mw-heading3"><h3 id="Equivalence_between_the_definitions_1_and_2">Equivalence between the definitions 1 and 2</h3></div>

Any fibred category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} }">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} }</annotation>

</semantics>

</math><img src="./5d5c597138f66fac80b8140273d32306004d7337.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:9.142ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} }" loading="lazy"> defines the 2-sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\mathrm {Mdf} ^{opp}\to \mathrm {Grp} ,U\mapsto \pi ^{-1}(U)}">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:\mathrm {Mdf} ^{opp}\to \mathrm {Grp} ,U\mapsto \pi ^{-1}(U)}</annotation>

</semantics>

</math><img src="./ae0079dd06600845539d44b4720a9d2bcc5ed69d.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:32.267ex; height:3.176ex;" alt="{\displaystyle X:\mathrm {Mdf} ^{opp}\to \mathrm {Grp} ,U\mapsto \pi ^{-1}(U)}" loading="lazy">. Conversely, any prestack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\mathrm {Mdf} ^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:\mathrm {Mdf} ^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./fc6d7b6d801bdae65330fd85563e1fabd9409717.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.777ex; height:2.676ex;" alt="{\displaystyle X:\mathrm {Mdf} ^{\text{opp}}\to \mathrm {Grp} }" loading="lazy"> gives rise to a category <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy">, whose objects are pairs <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle (U,x)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle (U,x)}</annotation>

</semantics>

</math><img src="./86bece9c0272e14e23c6c43e0164f292d329e19e.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.956ex; height:2.843ex;" alt="{\displaystyle (U,x)}" loading="lazy"> of a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy"> and an object <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x\in X(U)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x\in X(U)}</annotation>

</semantics>

</math><img src="./a08d4dacf6f4b8267326f0fa25698d2244594ae6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:9.742ex; height:2.843ex;" alt="{\displaystyle x\in X(U)}" loading="lazy">, and whose morphisms are maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :(U,x)\to (V,y)}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi :(U,x)\to (V,y)}</annotation>

</semantics>

</math><img src="./2a31d9f5b8d261a7878864c7a2ecd801ec4f6233.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:18.678ex; height:2.843ex;" alt="{\displaystyle \phi :(U,x)\to (V,y)}" loading="lazy"> such that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X(\phi )(y)=x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X(\phi )(y)=x}</annotation>

</semantics>

</math><img src="./eaa53ca38bae94b3324130600ed405c22db19a78.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:12.568ex; height:2.843ex;" alt="{\displaystyle X(\phi )(y)=x}" loading="lazy">. Such <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}}</annotation>

</semantics>

</math><img src="./e7b3edab7022ca9e2976651bc59c489513ee9019.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.239ex; height:2.176ex;" alt="{\displaystyle {\mathcal {C}}}" loading="lazy"> becomes a fibred category with the functor <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} ,(U,x)\mapsto U}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} ,(U,x)\mapsto U}</annotation>

</semantics>

</math><img src="./2a505f9847b49e7a2bc34711cf9941d6d6d99862.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:21.528ex; height:2.843ex;" alt="{\displaystyle {\mathcal {C}}\to \mathrm {Mdf} ,(U,x)\mapsto U}" loading="lazy">.

The glueing properties defining a stack in the first and in the second definition are equivalent; similarly, an atlas in the sense of Definition 1 induces an atlas in the sense of Definition 2 and vice versa.<a href="#cite_note-:0-5">[5]</a>

<div class="mw-heading mw-heading3"><h3 id="Equivalence_between_the_definitions_2_and_3">Equivalence between the definitions 2 and 3</h3></div>

Every Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\rightrightarrows M}</annotation>

</semantics>

</math><img src="./c39368f8c0b42847455ed449388037303b977d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.883ex; height:2.343ex;" alt="{\displaystyle G\rightrightarrows M}" loading="lazy"> gives rise to the differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./e438e38da31324cb34ac3a5cd8367eab59af1391.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:20.341ex; height:2.676ex;" alt="{\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }" loading="lazy">, which sends any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-<a href="Torsor" class="mw-redirect" title="Torsor">torsors</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> (i.e. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-<a href="Principal_bundle" title="Principal bundle">principal bundles</a>). Any other Lie groupoid in the Morita class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\rightrightarrows M}</annotation>

</semantics>

</math><img src="./c39368f8c0b42847455ed449388037303b977d42.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:7.883ex; height:2.343ex;" alt="{\displaystyle G\rightrightarrows M}" loading="lazy"> induces an isomorphic stack.

Conversely, any differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }">

<msup>

</mrow>

</mrow>

</msup>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }</annotation>

</semantics>

</math><img src="./4fe805586c99075d3bbf4f0832c417053d2f104c.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:18.731ex; height:2.676ex;" alt="{\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }" loading="lazy"> is of the form <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>

</semantics>

</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy">, i.e. it can be represented by a Lie groupoid. More precisely, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to X}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to X}</annotation>

</semantics>

</math><img src="./a5235ae11063ca96468ce50cc27f2a57f3f1bab6.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.16ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\to X}" loading="lazy"> is an atlas of the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">, then one defines the Lie groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G_{X}:=M\times _{X}M\rightrightarrows M}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G_{X}:=M\times _{X}M\rightrightarrows M}</annotation>

</semantics>

</math><img src="./c41da5a15bfb94beddf99132435f88f6b18fe404.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:22.618ex; height:2.509ex;" alt="{\displaystyle G_{X}:=M\times _{X}M\rightrightarrows M}" loading="lazy"> and checks that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG_{X}}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG_{X}}</annotation>

</semantics>

</math><img src="./0701215a27a2fcccb7ef5d149879634458f911a9.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:5.223ex; height:2.509ex;" alt="{\displaystyle BG_{X}}" loading="lazy"> is isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

A theorem by <a href="Dorette_Pronk" title="Dorette Pronk">Dorette Pronk</a> states an equivalence of <a href="Bicategories" class="mw-redirect" title="Bicategories">bicategories</a> between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.<a href="#cite_note-10">[10]</a>

<div class="mw-heading mw-heading2"><h2 id="Examples">Examples</h2></div>

<ul><li>Any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> defines a differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathrm {Hom} }(-,M)}">

</munder>

</mrow>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathrm {Hom} }(-,M)}</annotation>

</semantics>

</math><img src="./4c1dee1339ee80b8240a886296c07fdb9db5873f.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.122ex; margin-bottom: -0.755ex; width:21.902ex; height:3.343ex;" alt="{\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathrm {Hom} }(-,M)}" loading="lazy">, which is trivially presented by the identity morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to {\underline {M}}}">

</munder>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to {\underline {M}}}</annotation>

</semantics>

</math><img src="./88347918dff59105f7df5501e7cda2d81c2363cb.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:8.623ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}\to {\underline {M}}}" loading="lazy">. The stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}}">

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}}</annotation>

</semantics>

</math><img src="./6260298b54a5757766f3c1375969d2b65879dd16.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.511ex; margin-left: -0.122ex; margin-bottom: -0.827ex; width:2.566ex; height:3.176ex;" alt="{\displaystyle {\underline {M}}}" loading="lazy"> corresponds to the Morita equivalence class of the <a href="Lie_groupoid#Trivial_and_extreme_cases" title="Lie groupoid">unit groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle u(M)\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle u(M)\rightrightarrows M}</annotation>

</semantics>

</math><img src="./12e2a9fec91fe597de410cc9d2f7ccd19911dd84.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:11.637ex; height:2.843ex;" alt="{\displaystyle u(M)\rightrightarrows M}" loading="lazy">.</li>

<li>Any <a href="Lie_group" title="Lie group">Lie group</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy"> defines a differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>

</semantics>

</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy">, which sends any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy"> to the category of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-principal bundle on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle N}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle N}</annotation>

</semantics>

</math><img src="./f5e3890c981ae85503089652feb48b191b57aae3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.064ex; height:2.176ex;" alt="{\displaystyle N}" loading="lazy">. It is presented by the trivial stack morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {pt}}\to BG}">

<mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {pt}}\to BG}</annotation>

</semantics>

</math><img src="./9df6fea42c483c868c68fc440d5d745b431b7645.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.962ex; margin-left: -0.088ex; margin-bottom: -0.876ex; width:9.304ex; height:3.676ex;" alt="{\displaystyle {\underline {pt}}\to BG}" loading="lazy">, sending a point to the <a href="Universal_bundle" title="Universal bundle">universal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-bundle</a> over the <a href="Classifying_space" title="Classifying space">classifying space</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">. The stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>

</semantics>

</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy"> corresponds to the Morita equivalence class of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\rightrightarrows \{*\}}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\rightrightarrows \{*\}}</annotation>

</semantics>

</math><img src="./69b12528dda7328399b43d17017378800e074218.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:8.928ex; height:2.843ex;" alt="{\displaystyle G\rightrightarrows \{*\}}" loading="lazy"> seen as a Lie groupoid over a point (i.e., the Morita equivalence class of any transitive Lie groupoids with isotropy <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">).</li>

<li>Any <a href="Foliation" title="Foliation">foliation</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {F}}}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {F}}}</annotation>

</semantics>

</math><img src="./205d4b91000d9dcf1a5bbabdfa6a8395fa60b676.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.927ex; height:2.176ex;" alt="{\displaystyle {\mathcal {F}}}" loading="lazy"> on a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> defines a differentiable stack via its leaf spaces. It corresponds to the Morita equivalence class of the <a href="Lie_groupoid#Examples_from_differential_geometry" title="Lie groupoid">holonomy groupoid</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows M}">

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows M}</annotation>

</semantics>

</math><img src="./68af35d96c1d448e6c13bbc76e7a9621cb05cd00.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:13.345ex; height:2.843ex;" alt="{\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows M}" loading="lazy">.</li>

<li>Any <a href="Orbifold" title="Orbifold">orbifold</a> is a differentiable stack, since it is the Morita equivalence class of a proper Lie groupoid with <a href="Discrete_space" title="Discrete space">discrete</a> isotropies (hence <a href="Finite_space" class="mw-redirect" title="Finite space">finite</a>, since isotropies of proper Lie groupoids are <a href="Compact_space" title="Compact space">compact</a>).</li></ul>

<div class="mw-heading mw-heading3"><h3 id="Quotient_differentiable_stack">Quotient differentiable stack</h3></div>

Given a <a href="Lie_group_action" title="Lie group action">Lie group action</a> <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle a:M\times G\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle a:M\times G\to M}</annotation>

</semantics>

</math><img src="./39eb1380a1575feb1ac1a0c47030366f6355f7e3.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:16.333ex; height:2.176ex;" alt="{\displaystyle a:M\times G\to M}" loading="lazy"> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy">, its quotient (differentiable) stack is the differential counterpart of the <a href="Quotient_stack" title="Quotient stack">quotient (algebraic) stack</a> in algebraic geometry. It is defined as the stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> associating to any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> the category of principal <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-bundles <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle P\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle P\to X}</annotation>

</semantics>

</math><img src="./706af59e3c62c6fa253d5f87c0810d24703b99ff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.34ex; height:2.176ex;" alt="{\displaystyle P\to X}" loading="lazy"> and <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-equivariant maps <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi :P\to M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi :P\to M}</annotation>

</semantics>

</math><img src="./421dae600ac385af438f14ff7df89d2c645e7b52.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:11.124ex; height:2.509ex;" alt="{\displaystyle \phi :P\to M}" loading="lazy">. It is a differentiable stack presented by the stack morphism <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M}}\to [M/G]}">

</munder>

</mrow>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}\to [M/G]}</annotation>

</semantics>

</math><img src="./bc2b1cd0a821949e4c8dfbe72069ad6541299557.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.583ex; margin-left: -0.122ex; margin-bottom: -0.755ex; width:12.905ex; height:3.343ex;" alt="{\displaystyle {\underline {M}}\to [M/G]}" loading="lazy"> defined for any manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> as

</munder>

</mrow>

</mrow>

</mrow>

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M}}(X)=\mathrm {Hom} (X,M)\to [M/G](X),\quad f\mapsto (X\times G\to X,\phi _{f})}</annotation>

</semantics>

</math>

where <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{f}:X\times G\to M}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{f}:X\times G\to M}</annotation>

</semantics>

</math><img src="./805b300e53ae84286f136edf0081976345a3abff.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:17.162ex; height:2.843ex;" alt="{\displaystyle \phi _{f}:X\times G\to M}" loading="lazy"> is the <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G}</annotation>

</semantics>

</math><img src="./f5f3c8921a3b352de45446a6789b104458c9f90b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.827ex; height:2.176ex;" alt="{\displaystyle G}" loading="lazy">-equivariant map <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \phi _{f}=a\circ (f\circ \mathrm {pr} _{1},\mathrm {pr} _{2}):(x,g)\mapsto f(x)\cdot g}">

<msub>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \phi _{f}=a\circ (f\circ \mathrm {pr} _{1},\mathrm {pr} _{2}):(x,g)\mapsto f(x)\cdot g}</annotation>

</semantics>

</math><img src="./2be6361eb6d2be1cddc9e3648fa9c60e3a770ad8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:39.931ex; height:3.009ex;" alt="{\displaystyle \phi _{f}=a\circ (f\circ \mathrm {pr} _{1},\mathrm {pr} _{2}):(x,g)\mapsto f(x)\cdot g}" loading="lazy">.<a href="#cite_note-:1-7">[7]</a>

The stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> corresponds to the Morita equivalence class of the action groupoid <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M\times G\rightrightarrows M}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M\times G\rightrightarrows M}</annotation>

</semantics>

</math><img src="./3a7dc85cd38d0e7f71fa473d01048be654d9b096.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.505ex; width:13.166ex; height:2.343ex;" alt="{\displaystyle M\times G\rightrightarrows M}" loading="lazy">. Accordingly, one recovers the following particular cases:

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M}</annotation>

</semantics>

</math><img src="./f82cade9898ced02fdd08712e5f0c0151758a0dd.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:2.442ex; height:2.176ex;" alt="{\displaystyle M}" loading="lazy"> is a point, the differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BG}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BG}</annotation>

</semantics>

</math><img src="./773ca20b2080cb3766062a5451a01d2220e9b067.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:3.591ex; height:2.176ex;" alt="{\displaystyle BG}" loading="lazy"></li>

<li>if the action is <a href="Free_action" class="mw-redirect" title="Free action">free</a> and <a href="Proper_action" class="mw-redirect" title="Proper action">proper</a> (and therefore the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/G}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M/G}</annotation>

</semantics>

</math><img src="./8af27fe581fa061d68e3a6fa10664cada5e3b550.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.431ex; height:2.843ex;" alt="{\displaystyle M/G}" loading="lazy"> is a manifold), the differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> coincides with <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\underline {M/G}}}">

<mrow>

</mrow>

</mrow>

</munder>

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\underline {M/G}}}</annotation>

</semantics>

</math><img src="./2028bb4cd00da9bb01685cbc1bf5d2a33fe26e75.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.092ex; margin-bottom: -0.746ex; width:5.434ex; height:3.843ex;" alt="{\displaystyle {\underline {M/G}}}" loading="lazy"></li>

<li>if the action is proper (and therefore the quotient <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle M/G}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle M/G}</annotation>

</semantics>

</math><img src="./8af27fe581fa061d68e3a6fa10664cada5e3b550.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:5.431ex; height:2.843ex;" alt="{\displaystyle M/G}" loading="lazy"> is an orbifold), the differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle [M/G]}">

</mrow>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle [M/G]}</annotation>

</semantics>

</math><img src="./375110835b60a898de6e63062ea75344f6de53ef.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:6.725ex; height:2.843ex;" alt="{\displaystyle [M/G]}" loading="lazy"> coincides with the stack defined by the orbifold</li></ul>

<div class="mw-heading mw-heading2"><h2 id="Differential_space">Differential space</h2></div>

A differentiable space is a differentiable stack with trivial stabilizers. For example, if a <a href="Lie_group" title="Lie group">Lie group</a> <a href="Lie_group_action" title="Lie group action">acts</a> freely but not necessarily properly on a manifold, then the quotient by it is in general not a manifold but a differentiable space.

<div class="mw-heading mw-heading2"><h2 id="With_Grothendieck_topology">With Grothendieck topology</h2></div>

A differentiable stack <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> may be equipped with <a href="Grothendieck_topology" title="Grothendieck topology">Grothendieck topology</a> in a certain way (see the reference). This gives the notion of a <a href="Sheaf_(mathematics)" title="Sheaf (mathematics)">sheaf</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">. For example, the sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}^{p}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{X}^{p}}</annotation>

</semantics>

</math><img src="./c1970a566cdb1a20d936ea1524dce92e0f6071c2.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.31ex; height:3.176ex;" alt="{\displaystyle \Omega _{X}^{p}}" loading="lazy"> of differential <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-forms over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is given by, for any <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle x}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle x}</annotation>

</semantics>

</math><img src="./87f9e315fd7e2ba406057a97300593c4802b53e4.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.33ex; height:1.676ex;" alt="{\displaystyle x}" loading="lazy"> in <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> over a manifold <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">, letting <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}^{p}(x)}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{X}^{p}(x)}</annotation>

</semantics>

</math><img src="./b6d0b21c6d68f8337199ddd8708349e8a2b5f84b.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:6.449ex; height:3.176ex;" alt="{\displaystyle \Omega _{X}^{p}(x)}" loading="lazy"> be the space of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle p}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle p}</annotation>

</semantics>

</math><img src="./81eac1e205430d1f40810df36a0edffdc367af36.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; margin-left: -0.089ex; width:1.259ex; height:2.009ex;" alt="{\displaystyle p}" loading="lazy">-forms on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle U}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle U}</annotation>

</semantics>

</math><img src="./458a728f53b9a0274f059cd695e067c430956025.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.783ex; height:2.176ex;" alt="{\displaystyle U}" loading="lazy">. The sheaf <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}^{0}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{X}^{0}}</annotation>

</semantics>

</math><img src="./c498fd92e2f346b86a4c2079c98bb336dd8c1dc8.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.31ex; height:3.176ex;" alt="{\displaystyle \Omega _{X}^{0}}" loading="lazy"> is called the <a href="Structure_sheaf" class="mw-redirect" title="Structure sheaf">structure sheaf</a> on <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> and is denoted by <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle {\mathcal {O}}_{X}}">

<msub>

</mrow>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle {\mathcal {O}}_{X}}</annotation>

</semantics>

</math><img src="./9fed6a46b79218af44f23e5d6f487fb7e0d6cd01.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:3.482ex; height:2.509ex;" alt="{\displaystyle {\mathcal {O}}_{X}}" loading="lazy">. <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle \Omega _{X}^{*}}">

</mrow>

</mrow>

</msubsup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle \Omega _{X}^{*}}</annotation>

</semantics>

</math><img src="./b19d76eeccd4170004491b8ed1c05982d660eba7.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -1.005ex; width:3.31ex; height:2.843ex;" alt="{\displaystyle \Omega _{X}^{*}}" loading="lazy"> comes with <a href="Exterior_derivative" title="Exterior derivative">exterior derivative</a> and thus is a <a href="Complex_of_sheaves" class="mw-redirect" title="Complex of sheaves">complex of sheaves</a> of <a href="Vector_space" title="Vector space">vector spaces</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">: one thus has the notion of <a href="De_Rham_cohomology" title="De Rham cohomology">de Rham cohomology</a> of <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy">.

<div class="mw-heading mw-heading2"><h2 id="Gerbes">Gerbes</h2></div>

An epimorphism between differentiable stacks <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\to X}</annotation>

</semantics>

</math><img src="./92ac8b29f3aa93549c3871f007d9186523c967a0.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:7.421ex; height:2.176ex;" alt="{\displaystyle G\to X}" loading="lazy"> is called a <a href="Gerbe" title="Gerbe">gerbe</a> over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle G\to G\times _{X}G}">

<msub>

</mrow>

</msub>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle G\to G\times _{X}G}</annotation>

</semantics>

</math><img src="./aaf0d7be7b2a993772e0f002529e571705f28888.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.671ex; width:13.567ex; height:2.509ex;" alt="{\displaystyle G\to G\times _{X}G}" loading="lazy"> is also an epimorphism. For example, if <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> is a stack, <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BS^{1}\times X\to X}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BS^{1}\times X\to X}</annotation>

</semantics>

</math><img src="./0c4208e3fa179461b78c1263978528c451379e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.754ex; height:2.676ex;" alt="{\displaystyle BS^{1}\times X\to X}" loading="lazy"> is a gerbe. A theorem of <a href="Jean_Giraud_(mathematician)" title="Jean Giraud (mathematician)">Giraud</a> says that <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle H^{2}(X,S^{1})}">

<msup>

</mrow>

</msup>

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle H^{2}(X,S^{1})}</annotation>

</semantics>

</math><img src="./b88a98a1eecd49f8a026a4fffadfd174dfae3913.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.838ex; width:10.557ex; height:3.176ex;" alt="{\displaystyle H^{2}(X,S^{1})}" loading="lazy"> corresponds one-to-one to the set of gerbes over <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle X}">

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle X}</annotation>

</semantics>

</math><img src="./68baa052181f707c662844a465bfeeb135e82bab.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:1.98ex; height:2.176ex;" alt="{\displaystyle X}" loading="lazy"> that are locally isomorphic to <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="{\displaystyle BS^{1}\times X\to X}">

<msup>

</mrow>

</msup>

</mstyle>

</mrow>

<annotation encoding="application/x-tex">{\displaystyle BS^{1}\times X\to X}</annotation>

</semantics>

</math><img src="./0c4208e3fa179461b78c1263978528c451379e61.svg" class="mwe-math-fallback-image-inline mw-invert skin-invert" aria-hidden="true" style="vertical-align: -0.338ex; width:14.754ex; height:2.676ex;" alt="{\displaystyle BS^{1}\times X\to X}" loading="lazy"> and that come with trivializations of their bands.<a href="#cite_note-11">[11]</a>

<div class="mw-heading mw-heading2"><h2 id="References">References</h2></div>

/* start https://en.wikipedia.org/ */

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/* end https://en.wikipedia.org/ */

</style><cite id="CITEREFBlohmann2008" class="citation journal cs1">Blohmann, Christian (2008-01-01). <a rel="nofollow" class="external text" href="https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnn082/705350">"Stacky Lie Groups"</a>. International Mathematics Research Notices. 2008. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0702399">math/0702399</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1093%2Fimrn%2Frnn082">10.1093/imrn/rnn082</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1687-0247">1687-0247</a>.</cite>

</li>

<li id="cite_note-2"><a href="#cite_ref-2">^</a> <cite id="CITEREFMoerdijk1993" class="citation journal cs1">Moerdijk, Ieke (1993). "Foliations, groupoids and Grothendieck étendues". Rev. Acad. Cienc. Zaragoza. 48 (2): 5–33. <a href="MR_(identifier)" class="mw-redirect" title="MR (identifier)">MR</a> <a rel="nofollow" class="external text" href="https://mathscinet.ams.org/mathscinet-getitem?mr=1268130">1268130</a>.</cite>

</li>

<li id="cite_note-3"><a href="#cite_ref-3">^</a> <cite id="CITEREFBlohmannWeinstein2008" class="citation book cs1">Blohmann, Christian; <a href="Alan_Weinstein" title="Alan Weinstein">Weinstein, Alan</a> (2008). "Group-like objects in Poisson geometry and algebra". <a rel="nofollow" class="external text" href="http://www.ams.org/conm/450/">Poisson Geometry in Mathematics and Physics</a>. Contemporary Mathematics. Vol. 450. American Mathematical Society. pp. 25–39. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0701499">math/0701499</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.1090%2Fconm%2F450">10.1090/conm/450</a>. <a href="ISBN_(identifier)" class="mw-redirect" title="ISBN (identifier)">ISBN</a> <bdi>978-0-8218-4423-6</bdi>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:16778766">16778766</a>.</cite>

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<li id="cite_note-:0-5">^ <a href="#cite_ref-:0_5-0">a</a> <a href="#cite_ref-:0_5-1">b</a> <a href="#cite_ref-:0_5-2">c</a> <cite id="CITEREFBehrendXu2011" class="citation journal cs1"><a href="Kai_Behrend" title="Kai Behrend">Behrend, Kai</a>; Xu, Ping (2011). <a rel="nofollow" class="external text" href="https://www.intlpress.com/site/pub/pages/journals/items/jsg/content/vols/0009/0003/a002/abstract.php">"Differentiable stacks and gerbes"</a>. Journal of Symplectic Geometry. 9 (3): 285–341. <a href="ArXiv_(identifier)" class="mw-redirect" title="ArXiv (identifier)">arXiv</a>:<a rel="nofollow" class="external text" href="https://arxiv.org/abs/math/0605694">math/0605694</a>. <a href="Doi_(identifier)" class="mw-redirect" title="Doi (identifier)">doi</a>:<a rel="nofollow" class="external text" href="https://doi.org/10.4310%2FJSG.2011.v9.n3.a2">10.4310/JSG.2011.v9.n3.a2</a>. <a href="ISSN_(identifier)" class="mw-redirect" title="ISSN (identifier)">ISSN</a> <a rel="nofollow" class="external text" href="https://search.worldcat.org/issn/1540-2347">1540-2347</a>. <a href="S2CID_(identifier)" class="mw-redirect" title="S2CID (identifier)">S2CID</a> <a rel="nofollow" class="external text" href="https://api.semanticscholar.org/CorpusID:17281854">17281854</a>.</cite>

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</ol></div></div>

<div class="mw-heading mw-heading2"><h2 id="External_links">External links</h2></div>

<ul><li><a rel="nofollow" class="external free" href="http://ncatlab.org/nlab/show/differentiable+stack">http://ncatlab.org/nlab/show/differentiable+stack</a></li></ul></div><div><div class="zim-footer">

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