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 (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, Poisson geometry and twisted K-theory. Definition. Definition 1 (via groupoid fibrations). Recall that a category fibred in groupoids (also called a groupoid  fibration) consists of a category formula_1 together with a functor formula_2 to the category of differentiable manifolds such that These properties ensure that, for every object formula_19 in formula_7, one can define its fibre, denoted by formula_21 or formula_22, as the subcategory of formula_1 made up by all objects of formula_1 lying over formula_19 and all morphisms of formula_1 lying over formula_27. By construction, formula_21 is a groupoid, thus explaining the name. A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of descent. Any manifold formula_29 defines its slice category formula_30, whose objects are pairs formula_31  of a manifold formula_19 and a smooth map formula_33; then formula_34 is a groupoid fibration which is actually also a stack. A morphism formula_35 of groupoid fibrations is called a representable submersion if A differentiable stack is a stack formula_2 together with a special kind of representable submersion formula_43 (every submersion formula_6 described above is asked to be surjective), for some manifold formula_29. The map formula_43 is called atlas, presentation or cover of the stack formula_29. Definition 2 (via 2-functors). Recall that a prestack (of groupoids) on a category formula_1, also known as a 2-presheaf, is a 2 -functor formula_49, where formula_50 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 Grothendiek topology. Any object formula_51 defines a stack formula_52, which associated to another object formula_53 the groupoid formula_54 of morphisms from formula_55 to formula_56. A stack formula_49 is called geometric if there is an object formula_51 and a morphism of stacks  formula_59 (often called atlas, presentation or cover of the stack formula_29) such that A differentiable stack is a stack on formula_70, the category of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor formula_71, which is also geometric, i.e. admits an atlas formula_59 as described above. Note that, replacing formula_7 with the category of affine schemes, one recovers the standard notion of algebraic stack. Similarly, replacing formula_7 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 formula_75 and formula_56, together with two surjective submersions formula_77, as well as a partial multiplication map formula_78, a unit map formula_79, and an inverse map formula_80, satisfying group-like compatibilities. Two Lie groupoids formula_81 and formula_82 are Morita equivalent if there is a principal bi-bundle formula_83 between them, i.e. a principal right formula_84-bundle formula_85, a principal left formula_75-bundle formula_87, such that the two actions on formula_83 commutes. Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties. A differentiable stack is a Morita equivalence class  of a Lie groupoid. Equivalence between the definitions 1 and 2. Any fibred category formula_89 defines the 2-sheaf formula_90. Conversely, any prestack formula_91 gives rise to a category formula_1, whose objects are pairs formula_93 of a manifold formula_19 and an object formula_95, and whose morphisms are maps formula_96 such that formula_97. Such formula_1 becomes a fibred category with the functor formula_99. 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 viceversa. Equivalence between  the definitions 2 and 3. Every Lie groupoid formula_81 gives rise to the differentiable stack formula_101, which sends any manifold formula_55 to the category of formula_75-torsors on formula_55 (i.e. formula_75-principal bundles). Any other Lie groupoid in the Morita class of formula_81 induces an isomorphic stack. Conversely, any differentiable stack formula_71 is of the form formula_108, i.e. it can be represented by a Lie groupoid. More precisely, if formula_59 is an atlas of the stack formula_29, then one defines the Lie groupoid formula_111 and checks that formula_112 is isomorphic to formula_29. 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. 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 "formula_29" may be equipped with Grothendieck topology in a certain way (see the reference). This gives the notion of a sheaf over "formula_29". For example, the sheaf formula_127 of differential formula_128-forms over "formula_29" is  given by, for any "formula_130" in "formula_29" over a manifold "formula_19", letting formula_133 be the space of "formula_128"-forms on "formula_19". The sheaf formula_136 is called the structure sheaf on "formula_29" and is denoted by formula_138. formula_139 comes with exterior derivative and thus is a complex of sheaves of vector spaces over "formula_29": one thus has the notion of de Rham cohomology of "formula_29". Gerbes. An epimorphism between differentiable stacks formula_142 is called a gerbe over "formula_29" if formula_144 is also an epimorphism. For example, if "formula_29" is a stack, formula_146 is a gerbe. A theorem of Giraud says that formula_147  corresponds one-to-one to the set of gerbes over "formula_29" that are locally isomorphic to formula_146 and that come with trivializations of their bands.