# Differentiable stack

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{{Short description|Concept in differential geometry}}
A '''differentiable stack''' is the analogue in [differential geometry](/source/differential_geometry) of an [algebraic stack](/source/algebraic_stack) in [algebraic geometry](/source/algebraic_geometry). It can be described either as a [stack](/source/stack_(mathematics)) over [differentiable manifold](/source/differentiable_manifold)s which admits an atlas, or as a [Lie groupoid](/source/Lie_groupoid) up to [Morita equivalence](/source/Morita_equivalence).<ref>{{Cite journal|last=Blohmann|first=Christian|date=2008-01-01|title=Stacky Lie Groups|url=https://academic.oup.com/imrn/article/doi/10.1093/imrn/rnn082/705350|journal=International Mathematics Research Notices|language=en|volume=2008|arxiv=math/0702399|doi=10.1093/imrn/rnn082|issn=1687-0247}}</ref>

Differentiable stacks are particularly useful to handle spaces with [singularities](/source/Singularity_(mathematics)) (i.e. orbifolds, leaf spaces, quotients), which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in [foliation theory](/source/foliation_theory),<ref>{{Cite journal|last=Moerdijk|first=Ieke|author-link=Ieke Moerdijk|date=1993|title=Foliations, groupoids and Grothendieck étendues|journal=Rev. Acad. Cienc. Zaragoza|volume=48|issue=2|pages=5–33|mr=1268130}}</ref> [Poisson geometry](/source/Poisson_geometry)<ref>{{Cite book|last1=Blohmann|first1=Christian|last2=Weinstein|first2=Alan|title=Poisson Geometry in Mathematics and Physics |author-link2=Alan Weinstein|date=2008|chapter=Group-like objects in Poisson geometry and algebra |series=Contemporary Mathematics |url=http://www.ams.org/conm/450/ |language=en|publisher=American Mathematical Society|volume=450|pages=25–39|arxiv=math/0701499|doi=10.1090/conm/450|isbn=978-0-8218-4423-6|s2cid=16778766 }}</ref> and [twisted K-theory](/source/twisted_K-theory).<ref>{{Cite journal|last1=Tu|first1=Jean-Louis|last2=Xu|first2=Ping|last3=Laurent-Gengoux|first3=Camille|date=2004-11-01|title=Twisted K-theory of differentiable stacks|url=http://www.numdam.org/item/ASENS_2004_4_37_6_841_0/|journal=Annales Scientifiques de l'École Normale Supérieure|language=en|volume=37|issue=6|pages=841–910|arxiv=math/0306138|doi=10.1016/j.ansens.2004.10.002|s2cid=119606908|issn=0012-9593|via={{Interlanguage link|Numérisation de documents anciens mathématiques.|lt=Numérisation de documents anciens mathématiques.|fr}}}}</ref>

==Definition==

=== Definition 1 (via groupoid fibrations) ===
Recall that a [category fibred in groupoids](/source/Category_fibered_in_groupoids) (also called a '''groupoid fibration''') consists of a category <math>\mathcal{C}</math> together with a functor <math>\pi: \mathcal{C} \to \mathrm{Mfd}</math> to the [category of differentiable manifolds](/source/Category_of_manifolds) such that

# <math>\mathcal{C}</math> is a [fibred category](/source/fibred_category), i.e. for any object <math>u</math> of <math>\mathcal{C}</math> and any arrow <math>V \to U</math> of <math>\mathrm{Mfd}</math> there is an arrow <math>v \to u</math> lying over <math>V \to U</math>;
# for every [commutative triangle](/source/Commutative_diagram) <math>W \to V \to U</math> in <math>\mathrm{Mfd}</math> and every arrows <math>w \to u</math> over <math>W \to U</math> and <math>v \to u</math> over <math>V \to U</math>, there exists a unique arrow <math>w \to v</math> over <math>W \to V</math> making the triangle <math>w \to v \to u</math> commute.

These properties ensure that, for every object <math>U</math> in <math>\mathrm{Mfd}</math>, one can define its '''fibre''', denoted by <math>\pi^{-1}(U)</math> or <math>\mathcal{C}_U</math>, as the [subcategory](/source/subcategory) of <math>\mathcal{C}</math> made up by all objects of <math>\mathcal{C}</math> lying over <math>U</math> and all morphisms of <math>\mathcal{C}</math> lying over <math>id_U</math>. By construction, <math>\pi^{-1}(U)</math> is a [groupoid](/source/groupoid), thus explaining the name. A '''stack''' is a groupoid fibration satisfied further glueing properties, expressed in terms of [descent](/source/Descent_(mathematics)).

Any manifold <math>X</math> defines its [slice category](/source/slice_category) <math>F_X = \mathrm{Hom}_{\mathrm{Mfd}} (-, X)</math>, whose objects are pairs <math>(U,f)</math> of a manifold <math>U</math> and a smooth map <math>f: U \to X</math>; then <math>F_X \to \mathrm{Mfd}, (U,f) \mapsto U</math> is a groupoid fibration which is actually also a stack. A morphism <math>\mathcal{C} \to \mathcal{D}</math> of groupoid fibrations is called a '''representable submersion''' if

* for every manifold <math>U</math> and any morphism <math>F_U \to \mathcal{D}</math>, the [fibred product](/source/fibred_product) <math>\mathcal{C} \times_{\mathcal{D}} F_U</math> is representable, i.e. it is isomorphic to <math>F_V</math> (for some manifold <math>V</math>) as groupoid fibrations;
* the induced smooth map <math>V \to U</math> is a [submersion](/source/Submersion_(mathematics)).

A '''differentiable stack''' is a stack <math>\pi: \mathcal{C} \to \mathrm{Mfd}</math> together with a special kind of representable submersion <math>F_X \to \mathcal{C}</math> (every submersion <math>V \to U</math> described above is asked to be [surjective](/source/Surjective_function)), for some manifold <math>X</math>. The map <math>F_X \to \mathcal{C}</math> is called atlas, presentation or cover of the stack <math>X</math>.<ref name=":0">{{Cite journal |last1=Behrend |first1=Kai |author-link=Kai Behrend |last2=Xu |first2=Ping |date=2011 |title=Differentiable stacks and gerbes |url=https://www.intlpress.com/site/pub/pages/journals/items/jsg/content/vols/0009/0003/a002/abstract.php |journal=Journal of Symplectic Geometry |language=EN |volume=9 |issue=3 |pages=285–341 |arxiv=math/0605694 |doi=10.4310/JSG.2011.v9.n3.a2 |issn=1540-2347 |s2cid=17281854}}</ref><ref>Grégory Ginot, [https://ncatlab.org/nlab/files/GinotDifferentiableStacks.pdf ''Introduction to Differentiable Stacks (and gerbes, moduli spaces …)''], 2013</ref>

=== Definition 2 (via 2-functors) ===
Recall that a '''[prestack](/source/prestack)''' (of groupoids) on a category <math>\mathcal{C}</math>, also known as a 2-[presheaf](/source/Presheaf_(category_theory)), is a [2-functor](/source/2-functor) <math>X: \mathcal{C}^\text{opp} \to \mathrm{Grp}</math>, where <math>\mathrm{Grp}</math> is the [2-category](/source/2_category) of (set-theoretical) [groupoid](/source/groupoid)s, their morphisms, and the natural transformations between them. A [stack](/source/Stack_(mathematics)) 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 [site](/source/Site_(mathematics)), i.e. a category equipped with a [Grothendieck topology](/source/Grothendieck_topology).

Any object <math>M \in \mathrm{Obj}(\mathcal{C})</math> defines a stack <math>\underline{M} := \mathrm{Hom}_{\mathcal{C}}(-,M)</math>, which associated to another object <math>N \in \mathrm{Obj}(\mathcal{C})</math> the groupoid <math>\mathrm{Hom}_{\mathcal{C}}(N,M)</math> of [morphism](/source/morphism)s from <math>N</math> to <math>M</math>. A stack <math>X: \mathcal{C}^\text{opp} \to \mathrm{Grp}</math> is called '''geometric''' if there is an object <math>M \in \mathrm{Obj}(\mathcal{C})</math> and a morphism of stacks <math>\underline{M} \to X</math> (often called atlas, presentation or cover of the stack <math>X</math>) such that

* the morphism <math>\underline{M} \to X</math> is representable, i.e. for every object <math>Y</math> in <math>\mathcal{C}</math> and any morphism <math>Y \to X</math> the [fibred product](/source/fibred_product) <math>\underline{M} \times_X \underline{Y}</math> is isomorphic to <math>\underline{Z}</math> (for some object <math>Z</math>) as stacks;
* the induces morphism <math>Z \to Y</math> satisfies a further property depending on the category <math>\mathcal{C}</math> (e.g., for manifold it is asked to be a [submersion](/source/Submersion_(mathematics))).

A '''differentiable stack''' is a stack on <math>\mathcal{C} = \mathrm{Mfd}</math>, the [category of differentiable manifolds](/source/Category_of_manifolds) (viewed as a site with the usual open covering topology), i.e. a 2-functor <math>X: \mathrm{Mfd}^\text{opp} \to \mathrm{Grp}</math>, which is also geometric, i.e. admits an atlas <math>\underline{M} \to X</math> as described above.<ref name=":1">Jochen Heinloth: ''[https://www.uni-due.de/~mat903/preprints/heinloth.pdf Some notes on differentiable stacks]'', Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32.</ref><ref>Eugene Lerman, Anton Malkin, [Differential characters as stacks and prequantization](/source/arxiv%3A0710.4340), 2008</ref>

Note that, replacing <math>\mathrm{Mfd}</math> with the category of [affine scheme](/source/affine_scheme)s, one recovers the standard notion of [algebraic stack](/source/algebraic_stack). Similarly, replacing <math>\mathrm{Mfd}</math> with the category of [topological space](/source/topological_space)s, one obtains the definition of topological stack.

=== Definition 3 (via Morita equivalences) ===
Recall that a '''[Lie groupoid](/source/Lie_groupoid)''' consists of two differentiable manifolds <math>G</math> and <math>M</math>, together with two [surjective](/source/Surjective_function) [submersions](/source/Submersion_(mathematics)) <math>s,t: G \to M</math>, as well as a partial multiplication map <math>m: G \times_M G \to G</math>, a unit map <math>u: M \to G</math>, and an inverse map <math>i: G \to G</math>, satisfying group-like compatibilities.

Two Lie groupoids <math>G \rightrightarrows M</math> and <math>H \rightrightarrows N</math> are '''Morita equivalent''' if there is a principal bi-bundle <math>P</math> between them, i.e. a principal right <math>H</math>-bundle <math>P \to M</math>, a principal left <math>G</math>-bundle <math>P \to N</math>, such that the two actions on <math>P</math> commutes. Morita equivalence is an [equivalence relation](/source/equivalence_relation) between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.

A '''differentiable stack''', denoted as <math>[M/G]</math>, is the Morita equivalence class of some Lie groupoid <math>G \rightrightarrows M</math>.<ref name=":0" /><ref>Ping Xu, [https://personal.psu.edu/pxx2/book.pdf Differentiable Stacks, Gerbes, and Twisted K-Theory], 2017</ref>

=== Equivalence between the definitions 1 and 2 ===

Any fibred category <math>\mathcal{C} \to \mathrm{Mdf}</math> defines the 2-sheaf <math>X: \mathrm{Mdf}^{opp} \to \mathrm{Grp}, U \mapsto \pi^{-1}(U)</math>. Conversely, any prestack <math>X: \mathrm{Mdf}^\text{opp} \to \mathrm{Grp}</math> gives rise to a category <math>\mathcal{C}</math>, whose objects are pairs <math>(U,x)</math> of a manifold <math>U</math> and an object <math>x \in X(U)</math>, and whose morphisms are maps <math>\phi: (U,x) \to (V,y)</math> such that <math>X (\phi) (y) = x</math>. Such <math>\mathcal{C}</math> becomes a fibred category with the functor <math>\mathcal{C} \to \mathrm{Mdf}, (U,x) \mapsto U</math>.

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.<ref name=":0" />

=== Equivalence between the definitions 2 and 3 ===

Every Lie groupoid <math>G \rightrightarrows M</math> gives rise to the differentiable stack <math>BG: \mathrm{Mfd}^\text{opp} \to \mathrm{Grp}</math>, which sends any manifold <math>N</math> to the category of <math>G</math>-[torsor](/source/torsor)s on <math>N</math> (i.e. <math>G</math>-[principal bundle](/source/principal_bundle)s). Any other Lie groupoid in the Morita class of <math>G \rightrightarrows M</math> induces an isomorphic stack.

Conversely, any differentiable stack <math>X: \mathrm{Mfd}^\text{opp} \to \mathrm{Grp}</math> is of the form <math>BG</math>, i.e. it can be represented by a Lie groupoid. More precisely, if <math>\underline{M} \to X</math> is an atlas of the stack <math>X</math>, then one defines the Lie groupoid <math>G_X:= M \times_{X} M \rightrightarrows M</math> and checks that <math>BG_X</math> is isomorphic to <math>X</math>.

A theorem by [Dorette Pronk](/source/Dorette_Pronk) states an equivalence of [bicategories](/source/bicategories) between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.<ref>{{Cite journal|last=Pronk|first=Dorette A.|author-link=Dorette Pronk|date=1996|title=Etendues and stacks as bicategories of fractions|url=http://www.numdam.org/item/CM_1996__102_3_243_0/|journal=Compositio Mathematica|volume=102|issue=3|pages=243–303|via={{Interlanguage link|Numérisation de documents anciens mathématiques.|lt=Numérisation de documents anciens mathématiques.|fr}}}}</ref>

==Examples==

* Any manifold <math>M</math> defines a differentiable stack <math>\underline{M} := \mathrm{Hom}_{\mathrm{Hom}}(-,M)</math>, which is trivially presented by the identity morphism <math>\underline{M} \to \underline{M}</math>. The stack <math>\underline{M}</math> corresponds to the Morita equivalence class of the [unit groupoid](/source/Lie_groupoid) <math>u(M) \rightrightarrows M</math>.
* Any [Lie group](/source/Lie_group) <math>G</math> defines a differentiable stack <math>BG</math>, which sends any manifold <math>N</math> to the category of <math>G</math>-principal bundle on <math>N</math>. It is presented by the trivial stack morphism <math>\underline{pt} \to BG</math>, sending a point to the [universal <math>G</math>-bundle](/source/Universal_bundle) over the [classifying space](/source/classifying_space) of <math>G</math>. The stack <math>BG</math> corresponds to the Morita equivalence class of <math>G \rightrightarrows \{ *\}</math> seen as a Lie groupoid over a point (i.e., the Morita equivalence class of any transitive Lie groupoids with isotropy <math>G</math>).
* Any [foliation](/source/foliation) <math>\mathcal{F}</math> on a manifold <math>M</math> defines a differentiable stack via its leaf spaces. It corresponds to the Morita equivalence class of the [holonomy groupoid](/source/Lie_groupoid) <math>\mathrm{Hol} (\mathcal{F}) \rightrightarrows M</math>.
* Any [orbifold](/source/orbifold) is a differentiable stack, since it is the Morita equivalence class of a proper Lie groupoid with [discrete](/source/Discrete_space) isotropies (hence [finite](/source/Finite_space), since isotropies of proper Lie groupoids are [compact](/source/Compact_space)).

=== Quotient differentiable stack ===
Given a [Lie group action](/source/Lie_group_action) <math>a: M \times G \to M</math> on <math>M</math>, its '''quotient (differentiable) stack''' is the differential counterpart of the [quotient (algebraic) stack](/source/Quotient_stack) in algebraic geometry. It is defined as the stack <math>[M/G]</math> associating to any manifold <math>X</math> the category of principal <math>G</math>-bundles <math>P \to X</math> and <math>G</math>-equivariant maps <math>\phi: P \to M</math>. It is a differentiable stack presented by the stack morphism <math>\underline{M} \to [M/G]</math> defined for any manifold <math>X</math> as

<math display="block">\underline{M}(X) = \mathrm{Hom}(X,M) \to [M/G](X), \quad f \mapsto (X \times G \to X, \phi_f)</math>

where <math>\phi_f: X \times G \to M</math> is the <math>G</math>-equivariant map <math>\phi_f = a \circ (f \circ \mathrm{pr}_1, \mathrm{pr}_2): (x,g) \mapsto f(x) \cdot g</math>.<ref name=":1" />

The stack <math>[M/G]</math> corresponds to the Morita equivalence class of the action groupoid <math>M \times G \rightrightarrows M</math>. Accordingly, one recovers the following particular cases:

* if <math>M</math> is a point, the differentiable stack <math>[M/G]</math> coincides with <math>BG</math>
* if the action is [free](/source/Free_action) and [proper](/source/Proper_action) (and therefore the quotient <math>M/G</math> is a manifold), the differentiable stack <math>[M/G]</math> coincides with <math>\underline{M/G}</math> 
* if the action is proper (and therefore the quotient <math>M/G</math> is an orbifold), the differentiable stack <math>[M/G]</math> coincides with the stack defined by the orbifold

==Differential space==

A '''differentiable space''' is a differentiable stack with trivial stabilizers. For example, if a [Lie group](/source/Lie_group) [acts](/source/Lie_group_action) freely but not necessarily properly on a manifold, then the quotient by it is in general not a manifold but a differentiable space.

==With Grothendieck topology==
A differentiable stack ''<math>X</math>'' may be equipped with [Grothendieck topology](/source/Grothendieck_topology) in a certain way (see the reference). This gives the notion of a [sheaf](/source/sheaf_(mathematics)) over ''<math>X</math>''. For example, the sheaf <math>\Omega_X^p</math> of differential <math>p</math>-forms over ''<math>X</math>'' is given by, for any ''<math>x</math>'' in ''<math>X</math>'' over a manifold ''<math>U</math>'', letting <math>\Omega_X^p(x)</math> be the space of ''<math>p</math>''-forms on ''<math>U</math>''. The sheaf <math>\Omega_X^0</math> is called the [structure sheaf](/source/structure_sheaf) on ''<math>X</math>'' and is denoted by <math>\mathcal{O}_X</math>. <math>\Omega_X^*</math> comes with [exterior derivative](/source/exterior_derivative) and thus is a [complex of sheaves](/source/complex_of_sheaves) of [vector space](/source/vector_space)s over ''<math>X</math>'': one thus has the notion of [de Rham cohomology](/source/de_Rham_cohomology) of ''<math>X</math>''.

==Gerbes==
An [epimorphism](/source/epimorphism) between differentiable stacks <math>G \to X</math> is called a [gerbe](/source/gerbe) over ''<math>X</math>'' if <math>G \to G \times_X G</math> is also an epimorphism. For example, if ''<math>X</math>'' is a stack, <math>BS^1 \times X \to X</math> is a gerbe. A theorem of [Giraud](/source/Jean_Giraud_(mathematician)) says that <math>H^2(X, S^1)</math> corresponds one-to-one to the set of gerbes over ''<math>X</math>'' that are locally isomorphic to <math>BS^1 \times X \to X</math> and that come with trivializations of their [band](/source/band_(geometry))s.<ref>{{Cite journal|last=Giraud|first=Jean|date=1971|title=Cohomologie non abélienne|url=https://doi.org/10.1007/978-3-662-62103-5|journal=Grundlehren der Mathematischen Wissenschaften|volume=179|language=en-gb|doi=10.1007/978-3-662-62103-5|isbn=978-3-540-05307-1|issn=0072-7830|url-access=subscription}}</ref>

== References ==
{{reflist}}

== External links ==
*http://ncatlab.org/nlab/show/differentiable+stack

Category:Differential geometry
Category:Stacks (mathematics)

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Adapted from the Wikipedia article [Differentiable stack](https://en.wikipedia.org/wiki/Differentiable_stack) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Differentiable_stack?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.