Differentiable stack

<h2 id="definition">Definition</h2>
<h3>Definition 1 (via groupoid fibrations)</h3>
<p>Recall that a <a href="/facts/Category_fibered_in_groupoids/MkASN1rA">category fibred in groupoids</a> (also called a groupoid fibration) consists of a category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 together with a functor 
  
    
      
        π
        :
        
          
            C
          
        
        →
        
          M
          f
          d
        
      
    
    {\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }
  
 to the <a href="/facts/Category_of_manifolds/3fWHf2vB">category of differentiable manifolds</a> such that
</p>
<ol><li>
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 is a <a href="/facts/Fibred_category/LBN8vcbG">fibred category</a>, i.e. for any object 
  
    
      
        u
      
    
    {\displaystyle u}
  
 of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 and any arrow 
  
    
      
        V
        →
        U
      
    
    {\displaystyle V\to U}
  
 of 
  
    
      
        
          M
          f
          d
        
      
    
    {\displaystyle \mathrm {Mfd} }
  
 there is an arrow 
  
    
      
        v
        →
        u
      
    
    {\displaystyle v\to u}
  
 lying over 
  
    
      
        V
        →
        U
      
    
    {\displaystyle V\to U}
  
;</li>
<li>for every <a href="/facts/Commutative_diagram/bdG8xuer">commutative triangle</a> 
  
    
      
        W
        →
        V
        →
        U
      
    
    {\displaystyle W\to V\to U}
  
 in 
  
    
      
        
          M
          d
          f
        
      
    
    {\displaystyle \mathrm {Mdf} }
  
 and every arrows 
  
    
      
        w
        →
        u
      
    
    {\displaystyle w\to u}
  
 over 
  
    
      
        W
        →
        U
      
    
    {\displaystyle W\to U}
  
 and 
  
    
      
        v
        →
        u
      
    
    {\displaystyle v\to u}
  
 over 
  
    
      
        V
        →
        U
      
    
    {\displaystyle V\to U}
  
, there exists a unique arrow 
  
    
      
        w
        →
        v
      
    
    {\displaystyle w\to v}
  
 over 
  
    
      
        W
        →
        V
      
    
    {\displaystyle W\to V}
  
 making the triangle 
  
    
      
        w
        →
        v
        →
        u
      
    
    {\displaystyle w\to v\to u}
  
 commute.</li></ol>
<p>These properties ensure that, for every object 
  
    
      
        U
      
    
    {\displaystyle U}
  
 in 
  
    
      
        
          M
          f
          d
        
      
    
    {\displaystyle \mathrm {Mfd} }
  
, one can define its fibre, denoted by 
  
    
      
        
          π
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle \pi ^{-1}(U)}
  
 or 
  
    
      
        
          
            
              C
            
          
          
            U
          
        
      
    
    {\displaystyle {\mathcal {C}}_{U}}
  
, as the <a href="/facts/Subcategory/k4lEeDDb">subcategory</a> of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 made up by all objects of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 lying over 
  
    
      
        U
      
    
    {\displaystyle U}
  
 and all morphisms of 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 lying over 
  
    
      
        i
        
          d
          
            U
          
        
      
    
    {\displaystyle id_{U}}
  
. By construction, 
  
    
      
        
          π
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle \pi ^{-1}(U)}
  
 is a <a href="/facts/Groupoid/hvj8wvWe">groupoid</a>, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of <a href="/facts/Descent_(mathematics)/YTMNz2vH">descent</a>.
</p><p>Any manifold 
  
    
      
        X
      
    
    {\displaystyle X}
  
 defines its <a href="/facts/Slice_category/wTa3tURp">slice category</a> 
  
    
      
        
          F
          
            X
          
        
        =
        
          
            H
            o
            m
          
          
            
              M
              d
              f
            
          
        
        (
        −
        ,
        X
        )
      
    
    {\displaystyle F_{X}=\mathrm {Hom} _{\mathrm {Mdf} }(-,X)}
  
, whose objects are pairs 
  
    
      
        (
        U
        ,
        f
        )
      
    
    {\displaystyle (U,f)}
  
 of a manifold 
  
    
      
        U
      
    
    {\displaystyle U}
  
 and a smooth map 
  
    
      
        f
        :
        U
        →
        X
      
    
    {\displaystyle f:U\to X}
  
; then 
  
    
      
        
          F
          
            X
          
        
        →
        
          M
          d
          f
        
        ,
        (
        U
        ,
        f
        )
        ↦
        U
      
    
    {\displaystyle F_{X}\to \mathrm {Mdf} ,(U,f)\mapsto U}
  
 is a groupoid fibration which is actually also a stack. A morphism 
  
    
      
        
          
            C
          
        
        →
        
          
            D
          
        
      
    
    {\displaystyle {\mathcal {C}}\to {\mathcal {D}}}
  
 of groupoid fibrations is called a representable submersion if
</p>
<ul><li>for every manifold 
  
    
      
        U
      
    
    {\displaystyle U}
  
 and any morphism 
  
    
      
        
          F
          
            U
          
        
        →
        
          
            D
          
        
      
    
    {\displaystyle F_{U}\to {\mathcal {D}}}
  
, the <a href="/facts/Fibred_product/bC0QoX5U">fibred product</a> 
  
    
      
        
          
            C
          
        
        
          ×
          
            
              D
            
          
        
        
          F
          
            U
          
        
      
    
    {\displaystyle {\mathcal {C}}\times _{\mathcal {D}}F_{U}}
  
 is representable, i.e. it is isomorphic to 
  
    
      
        
          F
          
            V
          
        
      
    
    {\displaystyle F_{V}}
  
 (for some manifold 
  
    
      
        V
      
    
    {\displaystyle V}
  
) as groupoid fibrations;</li>
<li>the induced smooth map 
  
    
      
        V
        →
        U
      
    
    {\displaystyle V\to U}
  
 is a <a href="/facts/Submersion_(mathematics)/ARWB4sfr">submersion</a>.</li></ul>
<p>A differentiable stack is a stack 
  
    
      
        π
        :
        
          
            C
          
        
        →
        
          M
          f
          d
        
      
    
    {\displaystyle \pi :{\mathcal {C}}\to \mathrm {Mfd} }
  
 together with a special kind of representable submersion 
  
    
      
        
          F
          
            X
          
        
        →
        
          
            C
          
        
      
    
    {\displaystyle F_{X}\to {\mathcal {C}}}
  
 (every submersion 
  
    
      
        V
        →
        U
      
    
    {\displaystyle V\to U}
  
 described above is asked to be <a href="/facts/Surjective_function/KezhLpCb">surjective</a>), for some manifold 
  
    
      
        X
      
    
    {\displaystyle X}
  
. The map 
  
    
      
        
          F
          
            X
          
        
        →
        
          
            C
          
        
      
    
    {\displaystyle F_{X}\to {\mathcal {C}}}
  
 is called atlas, presentation or cover of the stack 
  
    
      
        X
      
    
    {\displaystyle X}
  
.<a class="footnote-ref" id="fnref:5" href="#fn:5"><sup>5</sup></a><a class="footnote-ref" id="fnref:6" href="#fn:6"><sup>6</sup></a>
</p>
<h3>Definition 2 (via 2-functors)</h3>
<p>Recall that a <a href="/facts/Prestack/2byjNpEA">prestack</a> (of groupoids) on a category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
, also known as a 2-<a href="/facts/Presheaf_(category_theory)/gWTNAFo3">presheaf</a>, is a <a href="/facts/2-functor/pMtYoHqx">2-functor</a> 
  
    
      
        X
        :
        
          
            
              C
            
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }
  
, where 
  
    
      
        
          G
          r
          p
        
      
    
    {\displaystyle \mathrm {Grp} }
  
 is the <a href="/facts/2_category/YeMVfoAc">2-category</a> of (set-theoretical) <a href="/facts/Groupoid/hvj8wvWe">groupoids</a>, their morphisms, and the natural transformations between them. A <a href="/facts/Stack_(mathematics)/MkASN1rA">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="/facts/Site_(mathematics)/Przxx5sH">site</a>, i.e. a category equipped with a <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a>.
</p><p>Any object 
  
    
      
        M
        ∈
        
          O
          b
          j
        
        (
        
          
            C
          
        
        )
      
    
    {\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}
  
 defines a stack 
  
    
      
        
          
            M
            _
          
        
        :=
        
          
            H
            o
            m
          
          
            
              C
            
          
        
        (
        −
        ,
        M
        )
      
    
    {\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathcal {C}}(-,M)}
  
, which associated to another object 
  
    
      
        N
        ∈
        
          O
          b
          j
        
        (
        
          
            C
          
        
        )
      
    
    {\displaystyle N\in \mathrm {Obj} ({\mathcal {C}})}
  
 the groupoid 
  
    
      
        
          
            H
            o
            m
          
          
            
              C
            
          
        
        (
        N
        ,
        M
        )
      
    
    {\displaystyle \mathrm {Hom} _{\mathcal {C}}(N,M)}
  
 of <a href="/facts/Morphism/Kdpuyz3S">morphisms</a> from 
  
    
      
        N
      
    
    {\displaystyle N}
  
 to 
  
    
      
        M
      
    
    {\displaystyle M}
  
. A stack 
  
    
      
        X
        :
        
          
            
              C
            
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle X:{\mathcal {C}}^{\text{opp}}\to \mathrm {Grp} }
  
 is called geometric if there is an object 
  
    
      
        M
        ∈
        
          O
          b
          j
        
        (
        
          
            C
          
        
        )
      
    
    {\displaystyle M\in \mathrm {Obj} ({\mathcal {C}})}
  
 and a morphism of stacks 
  
    
      
        
          
            M
            _
          
        
        →
        X
      
    
    {\displaystyle {\underline {M}}\to X}
  
 (often called atlas, presentation or cover of the stack 
  
    
      
        X
      
    
    {\displaystyle X}
  
) such that
</p>
<ul><li>the morphism 
  
    
      
        
          
            M
            _
          
        
        →
        X
      
    
    {\displaystyle {\underline {M}}\to X}
  
 is representable, i.e. for every object 
  
    
      
        Y
      
    
    {\displaystyle Y}
  
 in 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 and any morphism 
  
    
      
        Y
        →
        X
      
    
    {\displaystyle Y\to X}
  
 the <a href="/facts/Fibred_product/bC0QoX5U">fibred product</a> 
  
    
      
        
          
            M
            _
          
        
        
          ×
          
            X
          
        
        
          
            Y
            _
          
        
      
    
    {\displaystyle {\underline {M}}\times _{X}{\underline {Y}}}
  
 is isomorphic to 
  
    
      
        
          
            Z
            _
          
        
      
    
    {\displaystyle {\underline {Z}}}
  
 (for some object 
  
    
      
        Z
      
    
    {\displaystyle Z}
  
) as stacks;</li>
<li>the induces morphism 
  
    
      
        Z
        →
        Y
      
    
    {\displaystyle Z\to Y}
  
 satisfies a further property depending on the category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 (e.g., for manifold it is asked to be a <a href="/facts/Submersion_(mathematics)/ARWB4sfr">submersion</a>).</li></ul>
<p>A differentiable stack is a stack on 
  
    
      
        
          
            C
          
        
        =
        
          M
          f
          d
        
      
    
    {\displaystyle {\mathcal {C}}=\mathrm {Mfd} }
  
, the <a href="/facts/Category_of_manifolds/3fWHf2vB">category of differentiable manifolds</a> (viewed as a site with the usual open covering topology), i.e. a 2-functor 
  
    
      
        X
        :
        
          
            M
            f
            d
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }
  
, which is also geometric, i.e. admits an atlas 
  
    
      
        
          
            M
            _
          
        
        →
        X
      
    
    {\displaystyle {\underline {M}}\to X}
  
 as described above.<a class="footnote-ref" id="fnref:7" href="#fn:7"><sup>7</sup></a><a class="footnote-ref" id="fnref:8" href="#fn:8"><sup>8</sup></a>
</p><p>Note that, replacing 
  
    
      
        
          M
          f
          d
        
      
    
    {\displaystyle \mathrm {Mfd} }
  
 with the category of <a href="/facts/Affine_scheme/2k2nDE5w">affine schemes</a>, one recovers the standard notion of <a href="/facts/Algebraic_stack/flrtwvwU">algebraic stack</a>. Similarly, replacing 
  
    
      
        
          M
          f
          d
        
      
    
    {\displaystyle \mathrm {Mfd} }
  
 with the category of <a href="/facts/Topological_space/v2ut7CbK">topological spaces</a>, one obtains the definition of topological stack.
</p>
<h3>Definition 3 (via Morita equivalences)</h3>
<p>Recall that a <a href="/facts/Lie_groupoid/11ARLE9M">Lie groupoid</a> consists of two differentiable manifolds 
  
    
      
        G
      
    
    {\displaystyle G}
  
 and 
  
    
      
        M
      
    
    {\displaystyle M}
  
, together with two <a href="/facts/Surjective_function/KezhLpCb">surjective</a> <a href="/facts/Submersion_(mathematics)/ARWB4sfr">submersions</a> 
  
    
      
        s
        ,
        t
        :
        G
        →
        M
      
    
    {\displaystyle s,t:G\to M}
  
, as well as a partial multiplication map 
  
    
      
        m
        :
        G
        
          ×
          
            M
          
        
        G
        →
        G
      
    
    {\displaystyle m:G\times _{M}G\to G}
  
, a unit map 
  
    
      
        u
        :
        M
        →
        G
      
    
    {\displaystyle u:M\to G}
  
, and an inverse map 
  
    
      
        i
        :
        G
        →
        G
      
    
    {\displaystyle i:G\to G}
  
, satisfying group-like compatibilities.
</p><p>Two Lie groupoids 
  
    
      
        G
        ⇉
        M
      
    
    {\displaystyle G\rightrightarrows M}
  
 and 
  
    
      
        H
        ⇉
        N
      
    
    {\displaystyle H\rightrightarrows N}
  
 are Morita equivalent if there is a principal bi-bundle 
  
    
      
        P
      
    
    {\displaystyle P}
  
 between them, i.e. a principal right 
  
    
      
        H
      
    
    {\displaystyle H}
  
-bundle 
  
    
      
        P
        →
        M
      
    
    {\displaystyle P\to M}
  
, a principal left 
  
    
      
        G
      
    
    {\displaystyle G}
  
-bundle 
  
    
      
        P
        →
        N
      
    
    {\displaystyle P\to N}
  
, such that the two actions on 
  
    
      
        P
      
    
    {\displaystyle P}
  
 commutes. Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.
</p><p>A differentiable stack, denoted as 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
, is the Morita equivalence class of some Lie groupoid 
  
    
      
        G
        ⇉
        M
      
    
    {\displaystyle G\rightrightarrows M}
  
.<a class="footnote-ref" id="fnref:9" href="#fn:9"><sup>9</sup></a><a class="footnote-ref" id="fnref:10" href="#fn:10"><sup>10</sup></a>
</p>
<h3>Equivalence between the definitions 1 and 2</h3>
<p>Any fibred category 
  
    
      
        
          
            C
          
        
        →
        
          M
          d
          f
        
      
    
    {\displaystyle {\mathcal {C}}\to \mathrm {Mdf} }
  
 defines the 2-sheaf 
  
    
      
        X
        :
        
          
            M
            d
            f
          
          
            o
            p
            p
          
        
        →
        
          G
          r
          p
        
        ,
        U
        ↦
        
          π
          
            −
            1
          
        
        (
        U
        )
      
    
    {\displaystyle X:\mathrm {Mdf} ^{opp}\to \mathrm {Grp} ,U\mapsto \pi ^{-1}(U)}
  
. Conversely, any prestack 
  
    
      
        X
        :
        
          
            M
            d
            f
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle X:\mathrm {Mdf} ^{\text{opp}}\to \mathrm {Grp} }
  
 gives rise to a category 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
, whose objects are pairs 
  
    
      
        (
        U
        ,
        x
        )
      
    
    {\displaystyle (U,x)}
  
 of a manifold 
  
    
      
        U
      
    
    {\displaystyle U}
  
 and an object 
  
    
      
        x
        ∈
        X
        (
        U
        )
      
    
    {\displaystyle x\in X(U)}
  
, and whose morphisms are maps 
  
    
      
        ϕ
        :
        (
        U
        ,
        x
        )
        →
        (
        V
        ,
        y
        )
      
    
    {\displaystyle \phi :(U,x)\to (V,y)}
  
 such that 
  
    
      
        X
        (
        ϕ
        )
        (
        y
        )
        =
        x
      
    
    {\displaystyle X(\phi )(y)=x}
  
. Such 
  
    
      
        
          
            C
          
        
      
    
    {\displaystyle {\mathcal {C}}}
  
 becomes a fibred category with the functor 
  
    
      
        
          
            C
          
        
        →
        
          M
          d
          f
        
        ,
        (
        U
        ,
        x
        )
        ↦
        U
      
    
    {\displaystyle {\mathcal {C}}\to \mathrm {Mdf} ,(U,x)\mapsto U}
  
.
</p><p>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 class="footnote-ref" id="fnref:11" href="#fn:11"><sup>11</sup></a>
</p>
<h3>Equivalence between the definitions 2 and 3</h3>
<p>Every Lie groupoid 
  
    
      
        G
        ⇉
        M
      
    
    {\displaystyle G\rightrightarrows M}
  
 gives rise to the differentiable stack 
  
    
      
        B
        G
        :
        
          
            M
            f
            d
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }
  
, which sends any manifold 
  
    
      
        N
      
    
    {\displaystyle N}
  
 to the category of 
  
    
      
        G
      
    
    {\displaystyle G}
  
-<a href="/facts/Torsor/N27maCdL">torsors</a> on 
  
    
      
        N
      
    
    {\displaystyle N}
  
 (i.e. 
  
    
      
        G
      
    
    {\displaystyle G}
  
-<a href="/facts/Principal_bundle/bbB2D94B">principal bundles</a>). Any other Lie groupoid in the Morita class of 
  
    
      
        G
        ⇉
        M
      
    
    {\displaystyle G\rightrightarrows M}
  
 induces an isomorphic stack.
</p><p>Conversely, any differentiable stack 
  
    
      
        X
        :
        
          
            M
            f
            d
          
          
            opp
          
        
        →
        
          G
          r
          p
        
      
    
    {\displaystyle X:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }
  
 is of the form 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
, i.e. it can be represented by a Lie groupoid. More precisely, if 
  
    
      
        
          
            M
            _
          
        
        →
        X
      
    
    {\displaystyle {\underline {M}}\to X}
  
 is an atlas of the stack 
  
    
      
        X
      
    
    {\displaystyle X}
  
, then one defines the Lie groupoid 
  
    
      
        
          G
          
            X
          
        
        :=
        M
        
          ×
          
            X
          
        
        M
        ⇉
        M
      
    
    {\displaystyle G_{X}:=M\times _{X}M\rightrightarrows M}
  
 and checks that 
  
    
      
        B
        
          G
          
            X
          
        
      
    
    {\displaystyle BG_{X}}
  
 is isomorphic to 
  
    
      
        X
      
    
    {\displaystyle X}
  
.
</p><p>A theorem by <a href="/facts/Dorette_Pronk/Xjs06jKM">Dorette Pronk</a> states an equivalence of <a href="/facts/Bicategories/vaYsrIbg">bicategories</a> between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.<a class="footnote-ref" id="fnref:12" href="#fn:12"><sup>12</sup></a>
</p>
<h2 id="examples">Examples</h2>
<ul><li>Any manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
 defines a differentiable stack 
  
    
      
        
          
            M
            _
          
        
        :=
        
          
            H
            o
            m
          
          
            
              H
              o
              m
            
          
        
        (
        −
        ,
        M
        )
      
    
    {\displaystyle {\underline {M}}:=\mathrm {Hom} _{\mathrm {Hom} }(-,M)}
  
, which is trivially presented by the identity morphism 
  
    
      
        
          
            M
            _
          
        
        →
        
          
            M
            _
          
        
      
    
    {\displaystyle {\underline {M}}\to {\underline {M}}}
  
. The stack 
  
    
      
        
          
            M
            _
          
        
      
    
    {\displaystyle {\underline {M}}}
  
 corresponds to the Morita equivalence class of the <a href="/facts/Lie_groupoid/11ARLE9M">unit groupoid</a> 
  
    
      
        u
        (
        M
        )
        ⇉
        M
      
    
    {\displaystyle u(M)\rightrightarrows M}
  
.</li>
<li>Any <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> 
  
    
      
        G
      
    
    {\displaystyle G}
  
 defines a differentiable stack 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
, which sends any manifold 
  
    
      
        N
      
    
    {\displaystyle N}
  
 to the category of 
  
    
      
        G
      
    
    {\displaystyle G}
  
-principal bundle on 
  
    
      
        N
      
    
    {\displaystyle N}
  
. It is presented by the trivial stack morphism 
  
    
      
        
          
            
              p
              t
            
            _
          
        
        →
        B
        G
      
    
    {\displaystyle {\underline {pt}}\to BG}
  
, sending a point to the <a href="/facts/Universal_bundle/N6zTy9fn">universal 
  
    
      
        G
      
    
    {\displaystyle G}
  
-bundle</a> over the <a href="/facts/Classifying_space/gGIAmhtL">classifying space</a> of 
  
    
      
        G
      
    
    {\displaystyle G}
  
. The stack 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
 corresponds to the Morita equivalence class of 
  
    
      
        G
        ⇉
        {
        ∗
        }
      
    
    {\displaystyle G\rightrightarrows \{*\}}
  
 seen as a Lie groupoid over a point (i.e., the Morita equivalence class of any transitive Lie groupoids with isotropy 
  
    
      
        G
      
    
    {\displaystyle G}
  
).</li>
<li>Any <a href="/facts/Foliation/cxml9vyJ">foliation</a> 
  
    
      
        
          
            F
          
        
      
    
    {\displaystyle {\mathcal {F}}}
  
 on a manifold 
  
    
      
        M
      
    
    {\displaystyle M}
  
 defines a differentiable stack via its leaf spaces. It corresponds to the Morita equivalence class of the <a href="/facts/Lie_groupoid/11ARLE9M">holonomy groupoid</a> 
  
    
      
        
          H
          o
          l
        
        (
        
          
            F
          
        
        )
        ⇉
        M
      
    
    {\displaystyle \mathrm {Hol} ({\mathcal {F}})\rightrightarrows M}
  
.</li>
<li>Any <a href="/facts/Orbifold/KR6dQNzH">orbifold</a> is a differentiable stack, since it is the Morita equivalence class of a proper Lie groupoid with <a href="/facts/Discrete_space/hpvXA23u">discrete</a> isotropies (hence <a href="/facts/Finite_space/HKD3e0qp">finite</a>, since isotropies of proper Lie groupoids are <a href="/facts/Compact_space/d0cgXJH7">compact</a>).</li></ul>
<h3>Quotient differentiable stack</h3>
<p>Given a <a href="/facts/Lie_group_action/IIlu4h7M">Lie group action</a> 
  
    
      
        a
        :
        M
        ×
        G
        →
        M
      
    
    {\displaystyle a:M\times G\to M}
  
 on 
  
    
      
        M
      
    
    {\displaystyle M}
  
, its quotient (differentiable) stack is the differential counterpart of the <a href="/facts/Quotient_stack/lwme8cqw">quotient (algebraic) stack</a> in algebraic geometry. It is defined as the stack 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
 associating to any manifold 
  
    
      
        X
      
    
    {\displaystyle X}
  
 the category of principal 
  
    
      
        G
      
    
    {\displaystyle G}
  
-bundles 
  
    
      
        P
        →
        X
      
    
    {\displaystyle P\to X}
  
 and 
  
    
      
        G
      
    
    {\displaystyle G}
  
-equivariant maps 
  
    
      
        ϕ
        :
        P
        →
        M
      
    
    {\displaystyle \phi :P\to M}
  
. It is a differentiable stack presented by the stack morphism 
  
    
      
        
          
            M
            _
          
        
        →
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle {\underline {M}}\to [M/G]}
  
 defined for any manifold 
  
    
      
        X
      
    
    {\displaystyle X}
  
 as
</p><p>
  
    
      
        
          
            M
            _
          
        
        (
        X
        )
        =
        
          H
          o
          m
        
        (
        X
        ,
        M
        )
        →
        [
        M
        
          /
        
        G
        ]
        (
        X
        )
        ,
        
        f
        ↦
        (
        X
        ×
        G
        →
        X
        ,
        
          ϕ
          
            f
          
        
        )
      
    
    {\displaystyle {\underline {M}}(X)=\mathrm {Hom} (X,M)\to [M/G](X),\quad f\mapsto (X\times G\to X,\phi _{f})}

</p><p>where 
  
    
      
        
          ϕ
          
            f
          
        
        :
        X
        ×
        G
        →
        M
      
    
    {\displaystyle \phi _{f}:X\times G\to M}
  
 is the 
  
    
      
        G
      
    
    {\displaystyle G}
  
-equivariant map 
  
    
      
        
          ϕ
          
            f
          
        
        =
        a
        ∘
        (
        f
        ∘
        
          
            p
            r
          
          
            1
          
        
        ,
        
          
            p
            r
          
          
            2
          
        
        )
        :
        (
        x
        ,
        g
        )
        ↦
        f
        (
        x
        )
        ⋅
        g
      
    
    {\displaystyle \phi _{f}=a\circ (f\circ \mathrm {pr} _{1},\mathrm {pr} _{2}):(x,g)\mapsto f(x)\cdot g}
  
.<a class="footnote-ref" id="fnref:13" href="#fn:13"><sup>13</sup></a>
</p><p>The stack 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
 corresponds to the Morita equivalence class of the action groupoid 
  
    
      
        M
        ×
        G
        ⇉
        M
      
    
    {\displaystyle M\times G\rightrightarrows M}
  
. Accordingly, one recovers the following particular cases:
</p>
<ul><li>if 
  
    
      
        M
      
    
    {\displaystyle M}
  
 is a point, the differentiable stack 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
 coincides with 
  
    
      
        B
        G
      
    
    {\displaystyle BG}
  
</li>
<li>if the action is <a href="/facts/Free_action/Vh9VtUUw">free</a> and <a href="/facts/Proper_action/Vh9VtUUw">proper</a> (and therefore the quotient 
  
    
      
        M
        
          /
        
        G
      
    
    {\displaystyle M/G}
  
 is a manifold), the differentiable stack 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
 coincides with 
  
    
      
        
          
            
              M
              
                /
              
              G
            
            _
          
        
      
    
    {\displaystyle {\underline {M/G}}}
  
</li>
<li>if the action is proper (and therefore the quotient 
  
    
      
        M
        
          /
        
        G
      
    
    {\displaystyle M/G}
  
 is an orbifold), the differentiable stack 
  
    
      
        [
        M
        
          /
        
        G
        ]
      
    
    {\displaystyle [M/G]}
  
 coincides with the stack defined by the orbifold</li></ul>
<h2 id="differential-space">Differential space</h2>
<p>A differentiable space is a differentiable stack with trivial stabilizers. For example, if a <a href="/facts/Lie_group/a9jCQyjF">Lie group</a> <a href="/facts/Lie_group_action/IIlu4h7M">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.
</p>
<h2 id="with-grothendieck-topology">With Grothendieck topology</h2>
<p>A differentiable stack <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> may be equipped with <a href="/facts/Grothendieck_topology/Przxx5sH">Grothendieck topology</a> in a certain way (see the reference). This gives the notion of a <a href="/facts/Sheaf_(mathematics)/MM3hcRw4">sheaf</a> over <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i>. For example, the sheaf 
  
    
      
        
          Ω
          
            X
          
          
            p
          
        
      
    
    {\displaystyle \Omega _{X}^{p}}
  
 of differential 
  
    
      
        p
      
    
    {\displaystyle p}
  
-forms over <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> is given by, for any <i>
  
    
      
        x
      
    
    {\displaystyle x}
  
</i> in <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> over a manifold <i>
  
    
      
        U
      
    
    {\displaystyle U}
  
</i>, letting 
  
    
      
        
          Ω
          
            X
          
          
            p
          
        
        (
        x
        )
      
    
    {\displaystyle \Omega _{X}^{p}(x)}
  
 be the space of <i>
  
    
      
        p
      
    
    {\displaystyle p}
  
</i>-forms on <i>
  
    
      
        U
      
    
    {\displaystyle U}
  
</i>. The sheaf 
  
    
      
        
          Ω
          
            X
          
          
            0
          
        
      
    
    {\displaystyle \Omega _{X}^{0}}
  
 is called the <a href="/facts/Structure_sheaf/JyeaTMtX">structure sheaf</a> on <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> and is denoted by 
  
    
      
        
          
            
              O
            
          
          
            X
          
        
      
    
    {\displaystyle {\mathcal {O}}_{X}}
  
. 
  
    
      
        
          Ω
          
            X
          
          
            ∗
          
        
      
    
    {\displaystyle \Omega _{X}^{*}}
  
 comes with <a href="/facts/Exterior_derivative/p7HA0wze">exterior derivative</a> and thus is a <a href="/facts/Complex_of_sheaves/MM3hcRw4">complex of sheaves</a> of <a href="/facts/Vector_space/M1pxMLx2">vector spaces</a> over <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i>: one thus has the notion of <a href="/facts/De_Rham_cohomology/4qGKeJq0">de Rham cohomology</a> of <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i>.
</p>
<h2 id="gerbes">Gerbes</h2>
<p>An epimorphism between differentiable stacks 
  
    
      
        G
        →
        X
      
    
    {\displaystyle G\to X}
  
 is called a <a href="/facts/Gerbe/nh4e2Pyc">gerbe</a> over <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> if 
  
    
      
        G
        →
        G
        
          ×
          
            X
          
        
        G
      
    
    {\displaystyle G\to G\times _{X}G}
  
 is also an epimorphism. For example, if <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> is a stack, 
  
    
      
        B
        
          S
          
            1
          
        
        ×
        X
        →
        X
      
    
    {\displaystyle BS^{1}\times X\to X}
  
 is a gerbe. A theorem of <a href="/facts/Jean_Giraud_(mathematician)/Ca1bx8CV">Giraud</a> says that 
  
    
      
        
          H
          
            2
          
        
        (
        X
        ,
        
          S
          
            1
          
        
        )
      
    
    {\displaystyle H^{2}(X,S^{1})}
  
 corresponds one-to-one to the set of gerbes over <i>
  
    
      
        X
      
    
    {\displaystyle X}
  
</i> that are locally isomorphic to 
  
    
      
        B
        
          S
          
            1
          
        
        ×
        X
        →
        X
      
    
    {\displaystyle BS^{1}\times X\to X}
  
 and that come with trivializations of their bands.<a class="footnote-ref" id="fnref:14" href="#fn:14"><sup>14</sup></a>
</p>

<h2 id="external-links">External links</h2>
<ul><li><a href="http://ncatlab.org/nlab/show/differentiable+stack">http://ncatlab.org/nlab/show/differentiable+stack</a></li></ul>

<h2 id="references">References</h2>

<ol>
<li id="fn:1"><p>Blohmann, Christian (2008-01-01). "Stacky Lie Groups". International Mathematics Research Notices. 2008. arXiv:math/0702399. doi:10.1093/imrn/rnn082. ISSN 1687-0247. <a href="https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnn082/705350" target="_blank">https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnn082/705350</a> <a href="#fnref:1" class="footnote-back-ref">↩</a></p></li>
<li id="fn:2"><p>Moerdijk, Ieke (1993). "Foliations, groupoids and Grothendieck étendues". Rev. Acad. Cienc. Zaragoza. 48 (2): 5–33. MR 1268130. <a href="/wiki/MR_(identifier)" target="_blank">/wiki/MR_(identifier)</a> <a href="#fnref:2" class="footnote-back-ref">↩</a></p></li>
<li id="fn:3"><p>Blohmann, Christian; Weinstein, Alan (2008). "Group-like objects in Poisson geometry and algebra". Poisson Geometry in Mathematics and Physics. Contemporary Mathematics. Vol. 450. American Mathematical Society. pp. 25–39. arXiv:math/0701499. doi:10.1090/conm/450. ISBN 978-0-8218-4423-6. S2CID 16778766. <a href="978-0-8218-4423-6" target="_blank">978-0-8218-4423-6</a> <a href="#fnref:3" class="footnote-back-ref">↩</a></p></li>
<li id="fn:4"><p>Tu, Jean-Louis; Xu, Ping; Laurent-Gengoux, Camille (2004-11-01). "Twisted K-theory of differentiable stacks". Annales Scientifiques de l'École Normale Supérieure. 37 (6): 841–910. arXiv:math/0306138. doi:10.1016/j.ansens.2004.10.002. ISSN 0012-9593. S2CID 119606908 – via Numérisation de documents anciens mathématiques. [fr]. <a href="http://www.numdam.org/item/ASENS_2004_4_37_6_841_0/" target="_blank">http://www.numdam.org/item/ASENS_2004_4_37_6_841_0/</a> <a href="#fnref:4" class="footnote-back-ref">↩</a></p></li>
<li id="fn:5"><p>Behrend, Kai; Xu, Ping (2011). "Differentiable stacks and gerbes". Journal of Symplectic Geometry. 9 (3): 285–341. arXiv:math/0605694. doi:10.4310/JSG.2011.v9.n3.a2. ISSN 1540-2347. S2CID 17281854. <a href="/wiki/Kai_Behrend" target="_blank">/wiki/Kai_Behrend</a> <a href="#fnref:5" class="footnote-back-ref">↩</a></p></li>
<li id="fn:6"><p>Grégory Ginot, Introduction to Differentiable Stacks (and gerbes, moduli spaces …), 2013 <a href="https://ncatlab.org/nlab/files/GinotDifferentiableStacks.pdf" target="_blank">https://ncatlab.org/nlab/files/GinotDifferentiableStacks.pdf</a> <a href="#fnref:6" class="footnote-back-ref">↩</a></p></li>
<li id="fn:7"><p>Jochen Heinloth: Some notes on differentiable stacks, Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32. <a href="https://www.uni-due.de/~mat903/preprints/heinloth.pdf" target="_blank">https://www.uni-due.de/~mat903/preprints/heinloth.pdf</a> <a href="#fnref:7" class="footnote-back-ref">↩</a></p></li>
<li id="fn:8"><p>Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008 <a href="https://arxiv.org/abs/0710.4340" target="_blank">https://arxiv.org/abs/0710.4340</a> <a href="#fnref:8" class="footnote-back-ref">↩</a></p></li>
<li id="fn:9"><p>Behrend, Kai; Xu, Ping (2011). "Differentiable stacks and gerbes". Journal of Symplectic Geometry. 9 (3): 285–341. arXiv:math/0605694. doi:10.4310/JSG.2011.v9.n3.a2. ISSN 1540-2347. S2CID 17281854. <a href="/wiki/Kai_Behrend" target="_blank">/wiki/Kai_Behrend</a> <a href="#fnref:9" class="footnote-back-ref">↩</a></p></li>
<li id="fn:10"><p>Ping Xu, Differentiable Stacks, Gerbes, and Twisted K-Theory, 2017 <a href="https://personal.psu.edu/pxx2/book.pdf" target="_blank">https://personal.psu.edu/pxx2/book.pdf</a> <a href="#fnref:10" class="footnote-back-ref">↩</a></p></li>
<li id="fn:11"><p>Behrend, Kai; Xu, Ping (2011). "Differentiable stacks and gerbes". Journal of Symplectic Geometry. 9 (3): 285–341. arXiv:math/0605694. doi:10.4310/JSG.2011.v9.n3.a2. ISSN 1540-2347. S2CID 17281854. <a href="/wiki/Kai_Behrend" target="_blank">/wiki/Kai_Behrend</a> <a href="#fnref:11" class="footnote-back-ref">↩</a></p></li>
<li id="fn:12"><p>Pronk, Dorette A. (1996). "Etendues and stacks as bicategories of fractions". Compositio Mathematica. 102 (3): 243–303 – via Numérisation de documents anciens mathématiques. [fr]. <a href="/wiki/Dorette_Pronk" target="_blank">/wiki/Dorette_Pronk</a> <a href="#fnref:12" class="footnote-back-ref">↩</a></p></li>
<li id="fn:13"><p>Jochen Heinloth: Some notes on differentiable stacks, Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32. <a href="https://www.uni-due.de/~mat903/preprints/heinloth.pdf" target="_blank">https://www.uni-due.de/~mat903/preprints/heinloth.pdf</a> <a href="#fnref:13" class="footnote-back-ref">↩</a></p></li>
<li id="fn:14"><p>Giraud, Jean (1971). "Cohomologie non abélienne". Grundlehren der Mathematischen Wissenschaften. 179. doi:10.1007/978-3-662-62103-5. ISBN 978-3-540-05307-1. ISSN 0072-7830. <a href="978-3-540-05307-1" target="_blank">978-3-540-05307-1</a> <a href="#fnref:14" class="footnote-back-ref">↩</a></p></li>
</ol>

Differentiable stack open-in-new

Differentiable stack