A differentiable stack is the analogue in differential geometry of an algebraic stack in algebraic geometry. It can be described either as a stack over differentiable manifolds which admits an atlas, or as a Lie groupoid up to Morita equivalence.[1]

Differentiable stacks are particularly useful to handle spaces with singularities (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,[2] Poisson geometry[3] and twisted K-theory.[4]

Definition

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Definition 1 (via groupoid fibrations)

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Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category   together with a functor   to the category of differentiable manifolds such that

  1.   is a fibred category, i.e. for any object   of   and any arrow   of   there is an arrow   lying over  ;
  2. for every commutative triangle   in   and every arrows   over   and   over  , there exists a unique arrow   over   making the triangle   commute.

These properties ensure that, for every object   in  , one can define its fibre, denoted by   or  , as the subcategory of   made up by all objects of   lying over   and all morphisms of   lying over  . By construction,   is a groupoid, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of descent.

Any manifold   defines its slice category  , whose objects are pairs   of a manifold   and a smooth map  ; then   is a groupoid fibration which is actually also a stack. A morphism   of groupoid fibrations is called a representable submersion if

  • for every manifold   and any morphism  , the fibred product   is representable, i.e. it is isomorphic to   (for some manifold  ) as groupoid fibrations;
  • the induce smooth map   is a submersion.

A differentiable stack is a stack   together with a special kind of representable submersion   (every submersion   described above is asked to be surjective), for some manifold  . The map   is called atlas, presentation or cover of the stack  .[5][6]

Definition 2 (via 2-functors)

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Recall that a prestack (of groupoids) on a category  , also known as a 2-presheaf, is a 2-functor  , where   is the 2-category of (set-theoretical) groupoids, their morphisms, and the natural transformations between them. A stack 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, i.e. a category equipped with a Grothendieck topology.

Any object   defines a stack  , which associated to another object   the groupoid   of morphisms from   to  . A stack   is called geometric if there is an object   and a morphism of stacks   (often called atlas, presentation or cover of the stack  ) such that

  • the morphism   is representable, i.e. for every object   in   and any morphism   the fibred product   is isomorphic to   (for some object  ) as stacks;
  • the induces morphism   satisfies a further property depending on the category   (e.g., for manifold it is asked to be a submersion).

A differentiable stack is a stack on  , the category of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor  , which is also geometric, i.e. admits an atlas   as described above.[7][8]

Note that, replacing   with the category of affine schemes, one recovers the standard notion of algebraic stack. Similarly, replacing   with the category of topological spaces, one obtains the definition of topological stack.

Definition 3 (via Morita equivalences)

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Recall that a Lie groupoid consists of two differentiable manifolds   and  , together with two surjective submersions  , as well as a partial multiplication map  , a unit map  , and an inverse map  , satisfying group-like compatibilities.

Two Lie groupoids   and   are Morita equivalent if there is a principal bi-bundle   between them, i.e. a principal right  -bundle  , a principal left  -bundle  , such that the two actions on   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  , is the Morita equivalence class of some Lie groupoid  .[5][9]

Equivalence between the definitions 1 and 2

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Any fibred category   defines the 2-sheaf  . Conversely, any prestack   gives rise to a category  , whose objects are pairs   of a manifold   and an object  , and whose morphisms are maps   such that  . Such   becomes a fibred category with the functor  .

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.[5]

Equivalence between the definitions 2 and 3

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Every Lie groupoid   gives rise to the differentiable stack  , which sends any manifold   to the category of  -torsors on   (i.e.  -principal bundles). Any other Lie groupoid in the Morita class of   induces an isomorphic stack.

Conversely, any differentiable stack   is of the form  , i.e. it can be represented by a Lie groupoid. More precisely, if   is an atlas of the stack  , then one defines the Lie groupoid   and checks that   is isomorphic to  .

A theorem by Dorette Pronk states an equivalence of bicategories between differentiable stacks according to the first definition and Lie groupoids up to Morita equivalence.[10]

Examples

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  • Any manifold   defines a differentiable stack  , which is trivially presented by the identity morphism  . The stack   corresponds to the Morita equivalence class of the unit groupoid  .
  • Any Lie group   defines a differentiable stack  , which sends any manifold   to the category of  -principal bundle on  . It is presented by the trivial stack morphism  , sending a point to the universal  -bundle over the classifying space of  . The stack   corresponds to the Morita equivalence class of   seen as a Lie groupoid over a point (i.e., the Morita equivalence class of any transitive Lie groupoids with isotropy  ).
  • Any foliation   on a manifold   defines a differentiable stack via its leaf spaces. It corresponds to the Morita equivalence class of the holonomy groupoid  .
  • Any orbifold is a differentiable stack, since it is the Morita equivalence class of a proper Lie groupoid with discrete isotropies (hence finite, since isotropies of proper Lie groupoids are compact).

Quotient differentiable stack

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Given a Lie group action   on  , its quotient (differentiable) stack is the differential counterpart of the quotient (algebraic) stack in algebraic geometry. It is defined as the stack   associating to any manifold   the category of principal  -bundles   and  -equivariant maps  . It is a differentiable stack presented by the stack morphism   defined for any manifold   as

 

where   is the  -equivariant map  .[7]

The stack   corresponds to the Morita equivalence class of the action groupoid  . Accordingly, one recovers the following particular cases:

  • if   is a point, the differentiable stack   coincides with  
  • if the action is free and proper (and therefore the quotient   is a manifold), the differentiable stack   coincides with  
  • if the action is proper (and therefore the quotient   is an orbifold), the differentiable stack   coincides with the stack defined by the orbifold

Differential space

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A differentiable space is a differentiable stack with trivial stabilizers. For example, if a Lie group acts 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

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A differentiable stack   may be equipped with Grothendieck topology in a certain way (see the reference). This gives the notion of a sheaf over  . For example, the sheaf   of differential  -forms over   is given by, for any   in   over a manifold  , letting   be the space of  -forms on  . The sheaf   is called the structure sheaf on   and is denoted by  .   comes with exterior derivative and thus is a complex of sheaves of vector spaces over  : one thus has the notion of de Rham cohomology of  .

Gerbes

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An epimorphism between differentiable stacks   is called a gerbe over   if   is also an epimorphism. For example, if   is a stack,   is a gerbe. A theorem of Giraud says that   corresponds one-to-one to the set of gerbes over   that are locally isomorphic to   and that come with trivializations of their bands.[11]

References

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  1. ^ Blohmann, Christian (2008-01-01). "Stacky Lie Groups". International Mathematics Research Notices. 2008. arXiv:math/0702399. doi:10.1093/imrn/rnn082. ISSN 1687-0247.
  2. ^ Moerdijk, Ieke (1993). "Foliations, groupoids and Grothendieck étendues". Rev. Acad. Cienc. Zaragoza. 48 (2): 5–33. MR 1268130.
  3. ^ 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.
  4. ^ 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].
  5. ^ a b c 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.
  6. ^ Grégory Ginot, Introduction to Differentiable Stacks (and gerbes, moduli spaces …), 2013
  7. ^ a b Jochen Heinloth: Some notes on differentiable stacks, Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32.
  8. ^ Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008
  9. ^ Ping Xu, Differentiable Stacks, Gerbes, and Twisted K-Theory, 2017
  10. ^ Pronk, Dorette A. (1996). "Etendues and stacks as bicategories of fractions". Compositio Mathematica. 102 (3): 243–303 – via Numérisation de documents anciens mathématiques. [fr].
  11. ^ 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.
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