Differentiable stack
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
Definition 1 (via groupoid fibrations)
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
- is a fibred category, i.e. for any object of and any arrow of there is an arrow lying over ;
- 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)
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)
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
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
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
- 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
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
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
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
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
- ^ Blohmann, Christian (2008-01-01). "Stacky Lie Groups". International Mathematics Research Notices. 2008. arXiv:math/0702399. doi:10.1093/imrn/rnn082. ISSN 1687-0247.
- ^ Moerdijk, Ieke (1993). "Foliations, groupoids and Grothendieck étendues". Rev. Acad. Cienc. Zaragoza. 48 (2): 5–33. MR 1268130.
- ^ 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.
- ^ 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. .
- ^ 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.
- ^ Grégory Ginot, Introduction to Differentiable Stacks (and gerbes, moduli spaces …), 2013
- ^ a b Jochen Heinloth: Some notes on differentiable stacks, Mathematisches Institut Seminars, Universität Göttingen, 2004-05, p. 1-32.
- ^ Eugene Lerman, Anton Malkin, Differential characters as stacks and prequantization, 2008
- ^ Ping Xu, Differentiable Stacks, Gerbes, and Twisted K-Theory, 2017
- ^ Pronk, Dorette A. (1996). "Etendues and stacks as bicategories of fractions". Compositio Mathematica. 102 (3): 243–303 – via Numérisation de documents anciens mathématiques. .
- ^ 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.