Differentiable stack explained

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

l{C}

together with a functor

\pi:l{C}\toMfd

to the category of differentiable manifolds such that

l{C}

is a fibred category, i.e. for any object

l{C}

and any arrow

V\toU

Mfd

there is an arrow

v\tou

lying over

V\toU

;

W\toV\toU

Mdf

and every arrows

w\tou

over

W\toU

and

v\tou

over

V\toU

, there exists a unique arrow

w\tov

over

W\toV

making the triangle

w\tov\tou

commute.

These properties ensure that, for every object

Mfd

, one can define its fibre, denoted by

\pi^-1(U)

l{C}_U

, as the subcategory of

l{C}

made up by all objects of

l{C}

lying over

and all morphisms of

l{C}

lying over

id_U

. By construction,

\pi^-1(U)

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

F_X=Hom_Mdf(-,X)

, whose objects are pairs

(U,f)

of a manifold

and a smooth map

f:U\toX

; then

F_X\toMdf,(U,f)\mapstoU

is a groupoid fibration which is actually also a stack. A morphism

l{C}\tol{D}

of groupoid fibrations is called a representable submersion if

for every manifold

and any morphism

F_U\tol{D}

, the fibred product

l{C} x _l{D

} F_U is representable, i.e. it is isomorphic to

F_V

(for some manifold

) as groupoid fibrations;

the induce smooth map

V\toU

is a submersion.

A differentiable stack is a stack

\pi:l{C}\toMfd

together with a special kind of representable submersion

F_X\tol{C}

(every submersion

V\toU

described above is asked to be surjective), for some manifold

. The map

F_X\tol{C}

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

l{C}

, also known as a 2-presheaf, is a 2-functor

X:l{C}^opp\toGrp

, where

Grp

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

M\inObj(l{C})

defines a stack

\underline{M}:=Hom_l{C

}(-,M), which associated to another object

N\inObj(l{C})

the groupoid

Hom_l{C

}(N,M) of morphisms from

. A stack

X:l{C}^opp\toGrp

is called geometric if there is an object

M\inObj(l{C})

and a morphism of stacks

\underline{M}\toX

(often called atlas, presentation or cover of the stack

) such that

the morphism

\underline{M}\toX

is representable, i.e. for every object

l{C}

and any morphism

Y\toX

the fibred product

\underline{M} x _X\underline{Y}

is isomorphic to

\underline{Z}

(for some object

) as stacks;

the induces morphism

Z\toY

satisfies a further property depending on the category

l{C}

(e.g., for manifold it is asked to be a submersion).

A differentiable stack is a stack on

l{C}=Mfd

, the category of differentiable manifolds (viewed as a site with the usual open covering topology), i.e. a 2-functor

X:Mfd^opp\toGrp

, which is also geometric, i.e. admits an atlas

\underline{M}\toX

as described above.^[7] ^[8]

Note that, replacing

Mfd

with the category of affine schemes, one recovers the standard notion of algebraic stack. Similarly, replacing

Mfd

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

s,t:G\toM

, as well as a partial multiplication map

m:G x _MG\toG

, a unit map

u:M\toG

, and an inverse map

i:G\toG

, satisfying group-like compatibilities.

Two Lie groupoids

G\rightrightarrowsM

and

H\rightrightarrowsN

are Morita equivalent if there is a principal bi-bundle

between them, i.e. a principal right

-bundle

P\toM

, a principal left

-bundle

P\toN

, 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

[M/G]

, is the Morita equivalence class of some Lie groupoid

G\rightrightarrowsM

.^[9]

Equivalence between the definitions 1 and 2

Any fibred category

l{C}\toMdf

defines the 2-sheaf

X:Mdf^opp\toGrp,U\mapsto\pi^-1(U)

. Conversely, any prestack

X:Mdf^opp\toGrp

gives rise to a category

l{C}

, whose objects are pairs

(U,x)

of a manifold

and an object

x\inX(U)

, and whose morphisms are maps

\phi:(U,x)\to(V,y)

such that

X(\phi)(y)=x

. Such

l{C}

becomes a fibred category with the functor

l{C}\toMdf,(U,x)\mapstoU

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

G\rightrightarrowsM

gives rise to the differentiable stack

BG:Mfd^opp\toGrp

, which sends any manifold

to the category of

-torsors on

(i.e.

-principal bundles). Any other Lie groupoid in the Morita class of

G\rightrightarrowsM

induces an isomorphic stack.

Conversely, any differentiable stack

X:Mfd^opp\toGrp

is of the form

, i.e. it can be represented by a Lie groupoid. More precisely, if

\underline{M}\toX

is an atlas of the stack

, then one defines the Lie groupoid

G_X:=M x _XM\rightrightarrowsM

and checks that

BG_X

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

\underline{M}:=Hom_Hom(-,M)

, which is trivially presented by the identity morphism

\underline{M}\to\underline{M}

. The stack

\underline{M}

corresponds to the Morita equivalence class of the unit groupoid

u(M)\rightrightarrowsM

defines a differentiable stack

, which sends any manifold

to the category of

-principal bundle on

. It is presented by the trivial stack morphism

\underline{pt}\toBG

, sending a point to the universal

-bundle over the classifying space of

. The stack

corresponds to the Morita equivalence class of

G\rightrightarrows\{*\}

seen as a Lie groupoid over a point (i.e., the Morita equivalence class of any transitive Lie groupoids with isotropy

l{F}

on a manifold

defines a differentiable stack via its leaf spaces. It corresponds to the Morita equivalence class of the holonomy groupoid

Hol(l{F})\rightrightarrowsM

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

a:M x G\toM

, its quotient (differentiable) stack is the differential counterpart of the quotient (algebraic) stack in algebraic geometry. It is defined as the stack

[M/G]

associating to any manifold

the category of principal

-bundles

P\toX

and

-equivariant maps

\phi:P\toM

. It is a differentiable stack presented by the stack morphism

\underline{M}\to[M/G]

defined for any manifold

$\underline(X) = \mathrm(X,M) \to [M/G](X), \quad f \mapsto (X \times G \to X, \phi_f)$

where

\phi_f:X x G\toM

is the

-equivariant map

\phi_f=a\circ(f\circpr_1,pr_2):(x,g)\mapstof(x) ⋅ g

The stack

[M/G]

corresponds to the Morita equivalence class of the action groupoid

M x G\rightrightarrowsM

. Accordingly, one recovers the following particular cases:

is a point, the differentiable stack

[M/G]

coincides with

if the action is free and proper (and therefore the quotient

M/G

is a manifold), the differentiable stack

[M/G]

coincides with

\underline{M/G}

if the action is proper (and therefore the quotient

M/G

is an orbifold), the differentiable stack

[M/G]

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
p
\Omega
X

of differential
p

-forms over

is given by, for any

over a manifold

, letting
p(x)
\Omega
X

be the space of

-forms on

. The sheaf
0
\Omega
X

is called the structure sheaf on

and is denoted by
l{O}_X

.
*
\Omega
X

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

G\toX

is called a gerbe over X

if

G\toG x _XG

is also an epimorphism. For example, if X

is a stack,

BS¹ x X\toX

is a gerbe. A theorem of Giraud says that

H^2(X,S¹⁾

corresponds one-to-one to the set of gerbes over X

that are locally isomorphic to

BS¹ x X\toX

and that come with trivializations of their bands.^[11]

External links

http://ncatlab.org/nlab/show/differentiable+stack

Notes and References

Blohmann. Christian. 2008-01-01. Stacky Lie Groups. International Mathematics Research Notices. en. 2008. math/0702399. 10.1093/imrn/rnn082. 1687-0247.
Moerdijk. Ieke. 1993. Foliations, groupoids and Grothendieck étendues. Rev. Acad. Cienc. Zaragoza. 48. 2. 5–33. 1268130.
Book: Blohmann. Christian. Weinstein. Alan. Poisson Geometry in Mathematics and Physics . Alan Weinstein. 2008. Group-like objects in Poisson geometry and algebra . Contemporary Mathematics . en. American Mathematical Society. 450. 25–39. math/0701499. 10.1090/conm/450. 978-0-8218-4423-6. 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. en. 37. 6. 841–910. math/0306138. 10.1016/j.ansens.2004.10.002. 119606908. 0012-9593. .
Behrend . Kai . Kai Behrend . Xu . Ping . 2011 . Differentiable stacks and gerbes . Journal of Symplectic Geometry . EN . 9 . 3 . 285–341 . math/0605694 . 10.4310/JSG.2011.v9.n3.a2 . 1540-2347 . 17281854.
Grégory Ginot, Introduction to Differentiable Stacks (and gerbes, moduli spaces …), 2013
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.. Dorette Pronk. 1996. Etendues and stacks as bicategories of fractions. Compositio Mathematica. 102. 3. 243–303. .
Giraud. Jean. 1971. Cohomologie non abélienne. Grundlehren der Mathematischen Wissenschaften. 179. en-gb. 10.1007/978-3-662-62103-5. 978-3-540-05307-1. 0072-7830.