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.
Differentiable stacks are particularly useful to handle spaces with singularities, which appear naturally in differential geometry but are not differentiable manifolds. For instance, differentiable stacks have applications in foliation theory, Poisson geometry and twisted K-theory.

Definition

Definition 1 (via groupoid fibrations)

Recall that a category fibred in groupoids 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 as groupoid fibrations;
  • the induced smooth map is a submersion.
A differentiable stack is a stack together with a special kind of representable submersion , for some manifold. The map is called atlas, presentation or cover of the stack.

Definition 2 (via 2-functors)

Recall that a prestack on a category, also known as a 2-presheaf, is a 2-functor, where is the 2-category of groupoids, their morphisms, and the natural transformations between them. A stack is a prestack satisfying further glueing properties. In order to state such properties precisely, one needs to define 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 such that
  • the morphism is representable, i.e. for every object in and any morphism the fibred product is isomorphic to as stacks;
  • the induces morphism satisfies a further property depending on the category .
A differentiable stack is a stack on, the category of differentiable manifolds, i.e. a 2-functor, which is also geometric, i.e. admits an atlas as described above.
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.

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.

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 . 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.

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 Lie groupoid#Trivial and [extreme cases|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.
  • 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.

    Quotient differentiable stack

Given a Lie group action on, its quotient stack is the differential counterpart of the quotient 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.
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, the differentiable stack coincides with
  • if the action is proper, 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. 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 |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.