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.
For instance, differentiable stacks have applications in foliation theory,[2] Poisson geometry[3] and twisted K-theory.
[4] Recall that a category fibred in groupoids (also called a groupoid fibration) consists of a category
to the category of differentiable manifolds such that These properties ensure that, for every object
A stack is a groupoid fibration satisfied further glueing properties, expressed in terms of descent.
together with a special kind of representable submersion
is called atlas, presentation or cover of the stack
[5][6] Recall that a prestack (of groupoids) on a category
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.
(often called atlas, presentation or cover of the stack
, the category of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor
with the category of affine schemes, one recovers the standard notion of algebraic stack.
Recall that a Lie groupoid consists of two differentiable manifolds
are Morita equivalent if there is a principal bi-bundle
Morita equivalence is an equivalence relation between Lie groupoids, weaker than isomorphism but strong enough to preserve many geometric properties.
, is the Morita equivalence class of some Lie groupoid
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.
{\displaystyle BG:\mathrm {Mfd} ^{\text{opp}}\to \mathrm {Grp} }
Any other Lie groupoid in the Morita class of
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.
, its quotient (differentiable) stack is the differential counterpart of the quotient (algebraic) stack in algebraic geometry.
corresponds to the Morita equivalence class of the action groupoid
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.
may be equipped with Grothendieck topology in a certain way (see the reference).
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
corresponds one-to-one to the set of gerbes over