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Groupoid Representations for Generalized Quotients

Part 2: Morita Equivalence, Foliations, and Orbifolds Dorette Pronk

Dalhousie University

May 3, 2010

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Effective Descent and Groupoids

If we write Sh(Y) ≃ Sh(X) ×E Sh(X) and Sh(Y)

• s
• t

Sh(X)

πX

• Sh(X)

πX

E

, and we write u for δ, the truncated simplicial topos becomes Y ×s,X,t Y

π1

• m
• π2

Y

s

• t

X

,

u

• i.e., a localic groupoid.

[Butz-Moerdijk] If E has enough points, we can get a topological groupoid with the same properties..

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Description of E in terms of G

Let G be a topological groupoid with source and target maps

• pen or proper surjections.

Definition

• A G-space is a space p: X → G0 with a left action by G1,

α: G1 ×s,G0,p X → X, α(g, x) = g · x, such that

• p(g · x) = t(g);
• g1 · (g2 · x) = (g1g2) · x (where g1g2 is composition in G)

(cocycle condition);

• u(p(x)) · x = x (unit condition).
• An (equivariant) G-sheaf is a local homeomorphism

p: X → G0 with a left G1-action.

Remark

We could also have given α: G1 ×s,G0,p X

∼

→ X ×p,G0,t G1.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

The Topos Sh(G)

• A morphism ϕ: E → E′ between G-sheaves is a morphism
• f spaces over G0,

E

p

• ϕ

E′

p′

• G0

that respects the G-action, ϕ(g · x) = g · ϕ(x).

• The category of G-sheaves forms a Grothendieck topos

Sh(G). We also write ShG(X) for Sh(G ⋉ X).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etale Complete Groupoids

Which groupoids can be obtained in this fashion?

• The source and target maps must be open or closed

surjections.

• There is geometric morphism πG0 : Sh(G0) → Sh(G) with

π∗

G0 : ShG(X) → Sh(G0) the forgetful functor (that forgets

the action).

• The groupoid G is étale complete if the following square of

toposes is a pullback: Sh(G1)

s

• t

Sh(G0)

πG0

• Sh(G0)

πG0

Sh(G)

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etale Groupoids

• A groupoid is étale if both source and target maps are local

homeomorphisms.

• For an étale groupoid, t : G1 → G0 is an element of Sh(G).
• We have that Sh(G)/(G1

t

→ G0) ≃ Sh(G0).

• Every étale groupoid is étale complete, that is, the

following square is a weak pullback: Sh(G0)/π∗

G0(G1) ≃Sh(G1) t

• s
• Sh(G0)≃ Sh(G)/G1

πG0

• Sh(G0)

πG0

Sh(G)

.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

A Site for Sh(G)

• A bisection of G consists of an open subset U ⊆ G0 with a

section σ: U → G1 of s such that V = t ◦ σ(U) ⊆ G0 is

• pen and t ◦ σ : U

∼

→ V.

• The objects of C(G) are the domains of all possible

bisections of G.

• An arrow (U, σ): U → U′ is a bisection σ: U → G1 such

that t ◦ σ(U) ⊆ U′.

• A family of arrows (Ui, σi): Ui → U) is a covering if
• i∈I

t ◦ σi(Ui) = U.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etendues

Definition

A Grothendieck topos E is an étendue if it contains an epimorphism U

1, such that E/U is a topos of sheaves on

a topological space, i.e., E/U ≃ Sh(X).

Remarks

• In our example above, we had

(U

1)

=

(G1

t

• t

G0)

• G0

.

• We can also describe étendues as toposes with a site with

monic maps.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

The 2-Category of Topological Groupoids

• A homomorphism ϕ: G → H of topological groupoids is an

internal functor in the category of topological spaces. It consists of continuous maps ϕ0 : G0 → H0, ϕ1 : G1 → H1, which commute with all the structure maps.

• A 2-cell α: ϕ ⇒ ψ: G ⇒ H is represented by a continuous

map α: G0 → H1, such that s ◦ α = ϕ and t ◦ α = ψ and m(ψ1(g), α(s(g))) = m(α(t(g)), ϕ1(g)).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Essential Equivalences of Groupoids

Definition

A morphism ϕ: G → H of topological groupoids is an essential equivalence when it satisfies the following two conditions:

• ϕ is essentially surjective on objects in the sense that

t ◦ π2 is an open surjection: G0 ×H0 H1

π1 π2

H1

s

• t

H0

G0

ϕ0

H0

;

• F is fully faithful in the sense that the following diagram is

a pullback: G1

ϕ1

• (s,t)

H1

(s,t)

• G0 × G0

ϕ0×ϕ0 H0 × H0

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Equivalences of Toposes

Proposition

An essential equivalence ϕ: G → H of groupoids gives rise to an equivalence of categories ϕ: Sh(G)

∼

→ Sh(H).

Remark

This gives us one of the implications for our proposed Morita equivalence theorem.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Fibered Products

The 2-category of topological groupoids has both strong and weak fibered products. Let ϕ: G → K and ψ: H → K.

• The strong fibered product G ×K H has space of objects

G0 ×K0 H0 and space of arrows G1 ×K1 H1.

• The weak fibered product G ×K H has space of objects

G0 ×K0 K1 ×K0 H0 = {(x, k, y)| ϕ(x)

k

→ ψ(y)} and the elements of the space of arrows with source (x, k, y) and target (x′, k′, y′) are determined by pairs of arrows g ∈ G1 and h ∈ H1 such that k′ϕ(g) = ψ(h)k in K.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Properties of Essential Equivalences

• The weak pullback of an essential equivalence along an

arbitrary morphism of topological groupoids is again an essential equivalence.

• If θ is an essential equivalence and there is a 2-cell

α: θ ◦ ϕ ⇒ θ ◦ ψ then there is a 2-cell α′ : ϕ ⇒ ψ such that θ ◦ α′ = α.

• The class of essential equivalences is closed under

2-isomorphisms.

• The class of essential equivalences admits a right calculus
• f fractions.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Localic Groupoid Representations

• (Moerdijk, 1988) For open étale complete localic

groupoids, there is an equivalence of categories with isomorphism classes of maps: [OEC-LocGrpds][W −1] ≃ [Toposes]

• Essential step in the proof: show that for each geometric

morphism ϕ: Sh(G) → Sh(H) there exist an essential equivalence w : K → G and a groupoid homomorphism f : G → H such that Sh(f) ϕ ◦ Sh(w). Sh(K0)

w0

• Sh(H0)

πH0

• K1

(s,t)

• w1

G1

(s,t)

• Sh(G0)

πG0

Sh(G)

ϕ

Sh(H)

K0 × K0 w0×w0

G0 × G0

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Comments

• Morphisms between open étale complete localic groupoids

should be equivalence classes of spans G K

w

• f

H

where w is an essential equivalence.

• For open étale complete localic groupoids G and H,

Sh(G) ≃ Sh(H) if and only if there exists a localic groupoid K with essential equivalences G K

ϕ

• ψ

H

• In this case we call the two groupoids G and H Morita

equivalent.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

We have some issues left:

• This result is about categories, not about 2-categories.
• Spacial groupoids and spacial toposes?

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etendues

Theorem

There is an equivalence of bicategories SpEtendues2-iso ≃ EtaleGrpd[W −1], But we can do a bit better...

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

• Proposition

An étale complete open topological groupoid is Morita equivalent to an étale groupoid if and only if all its isotropy groups are discrete.

• We will call such groupoids topological foliation groupoids,

and denote their category by TopFolGrpd.

• Theorem

There is an equivalence of bicategories SpEtendues2-iso ≃ TopFolGrpd[W −1].

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Consequences

• For any two topological foliation groupoids G and H,

Sh(G) ≃ Sh(H) if and only if there is a third such groupoid K with essential equivalences G K

ϕ

• ψ

H .

• In this case we will call G and H Morita equivalent.
• A geometric morphism G → H corresponds to a span of

groupoid homomorphisms G K

ψ

• ϕ

H ,

where ψ is an essential equivalence. We call such a span a generalized morphism.

• Generalized morphisms are composed using chosen weak

pullbacks and the identities are represented by spans of identity arrows.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

A 2-cell (ψ, ϕ) ⇒ (ψ′, ϕ′) is an equivalence class of diagrams of the form K

ψ

• ϕ
• G

α⇓

L

θ

• θ′
• β⇓

H K′

ψ′

• ϕ′
• where θ and θ′ are essential equivalences.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Further Comments

• Any homotopy invariants defined for geometric objects

represented by these topological groupoids needs to be invariant under Morita equivalence.

• There is a different description of the arrows in this

bicategory in terms of groupoid bibundles (Hilsum-Skandalis maps) with isomorphisms of bibundles as 2-cells. This fits in the rest of our theory as expressed in the following theorem:

Theorem

For topological foliation groupoids G and H the following are equivalent:

• 1. Sh(G) ≃ Sh(H);
• 2. G and H are Morita equivalent;
• 3. there is a left principal G-, right principal H-bibundle

G

|

H .

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Lie Groupoids

Definition

• A Lie groupoid G is a diagram

G1 ×G0 G1

m

• π1
• π2

G1

i

G1

s

• t

G0

,

u

• in the category of manifolds and smooth maps.
• The source and target maps need to be surjective

submersions.

• Note that the source and target maps are effective descent

morphisms.

• G0 needs to be Hausdorff.
• We call G Hausdorff if G1 is.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

The 2-Category of Lie Groupoids

• A homomorphism ϕ: G → H between Lie groupoids is an

internal functor, that is, it is given smooth maps ϕ0 : G0 → H0 and ϕ1 : G1 → H1, which commute with all the structure maps.

• A 2-cell α: ϕ → ψ: G ⇒ H is an internal natural

transformation, that is, it is given by a smooth map α: G0 → H1 such that s ◦ α = ϕ0, t ◦ α = ψ0, and the following square commutes for each g ∈ G1, ϕ0(s(g))

ϕ1(g)

• α(s(g)) ψ0(s(g))

ψ1(g)

• ϕ0(t(g))

α(t(g))

ψ0(t(g))

.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• For a manifold M the fundamental groupoid of homotopy

classes of paths in M is a Lie groupoid.

• For a manifold M with atlas U there is a groupoid

homomorphism G(U) → M.

• For any manifold M, the space Γ(M) is the groupoid of

germs of diffeomorphisms between open subsets of M.

• For an étale Lie groupoid (i.e., a Lie groupoid with étale

source and target maps) there is a homomorphism G → Γ(G0). If this homomorphism is injective, we call G

• effective. Note that G ⋉ M is effective as a groupoid

precisely when the action of G on M is effective.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Gauge Groupoids and Bisections, I

• For a Lie group G and a principal G-bundle π: P → M, the

associated gauge groupoid Gauge(P) has

• space of objects M;
• space of arrows (P × P)/G, where G acts diagonally.

Source and target maps are the projections composed with π.

• Moreover, the quotient map of the action by G induces a

homomorphism Pair(P) → Gauge(P).

• A global bisection σ: G0 → G1 of a Lie groupoid G is a

section of s, such that t ◦ σ: G0 → G0 is a diffeomorphism.

• The set of global bisections forms a group, this is called the

gauge group of G. The product of two sections σ and σ′ is defined by σ′σ(x) = σ′(t(σ(x)))σ(x).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Gauge Groupoids and Bisections, II

• Consider the pullback

P × P

• π1
• (P × P)/G = Gauge(P)

s

• P

π

M

A section σ of s corresponds to a map (1, σ): P → P × P, where σ is a G-equivariant diffeomorphism of P.

• The gauge group of Gauge(P) is isomorphic to DiffG(P).
• A local bisection (U, σ) of a Lie groupoid G consists of
• U ⊆ G0 open;
• σ is a section of s on U, such that
• t ◦ σ is an open embedding.

The germs of such bisections form the manifold of arrows

• f the groupoid Bis(G) over G0. Bis(G) is an étale groupoid

with a surjective groupoid homomorphism Bis(G) ։ G.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Essential Equivalences

Definition

An essential equivalence ϕ: G → H between Lie groupoids is a homomorphism with the following properties.

• 1. It is essentially surjective, i.e., the map

t ◦ π2 : G0 ×H0 H1 → H0 from the manifold G0 ×H0 H1 = {(x, h) | φ0(x) = t(h)} is a surjective submersion.

• 2. It is fully faithful, i.e., the diagram

G1

ϕ1

• (s,t)

H1

(s,t)

• G0 × G0 ϕ0×ϕ0

H0 × H0

is a pullback of manifolds.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• 1. For a manifold M the homomorphism Pair(M) → 1, is an

essential equivalence.

• 2. A surjective submersion p: N → M between manifolds

induces a weak equivalence Ker(p) → M.

• 3. In particular, take N =

i Ui for an open cover {Ui} of M,

and p is the evident map. Then Ker(p) is the atlas groupoid, and we see that G(U) → M is an essential equivalence.

• 4. A Lie groupoid G is transitive if the map

(s, t): G1 → G0 × G0 is a surjective submersion. For any object x of a transitive Lie groupoid G, the inclusion Gx → G is an essential equivalence. In particular, for a principal G-bundle P → M this yields an essential equivalence G → Gauge(P).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Morita Equivalence

• Essential equivalences of Lie groupoids are stable under

weak pullbacks.

• Two Lie groupoids G and H are called Morita equivalent if

there exists a third Lie groupoid K with essential equivalences G K

ϕ

• ψ

H.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Smooth Structure for Toposes

• The category Sh(G) for a Lie groupoid is defined in exactly

the same way as if G were just a topological groupoid.

• For each manifold M, there is a structure sheaf OM of

germs of smooth functions on M.

• The structure sheaf OG0 can be made into a G-sheaf where

the action is by composition with germs of bisections.

• The sheaf OG is a sheaf of rings, so Sh(G), OG) a ringed

topos.

• A morphism of ringed toposes (ϕ, f): (E, R) → (E′, R′)

consists of a geometric morphism ϕ: E → E′ together with an arrow f : ϕ∗R′ → R in E.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etale Groupoids

Proposition

For a Lie groupoid G the following are equivalent:

• 1. The source and target maps are local diffeomorphisms

(étale maps).

• 2. dim G0 = dim G1.

Theorem

For a smooth groupoid G, the following are equivalent:

• G is Morita equivalent to a smooth étale groupoid;
• All isotropy Lie groups of G are discrete.
• G is a foliation groupoid.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Smooth Etendues

Definition

A ringed Grothendieck topos (E, R) is a smooth étendue if it contains an epimorphism U

1, such that E/U is a topos of

sheaves on a Hausdorff manifold, i.e., E/U ≃ Sh(M), and π∗

M(R) OM.

Theorem

There are equivalences of bicategories SmoothEtendues ≃ Etale Lie Grpds[W −1], Smooth Etendues ≃ Foliation Grpds[W −1]

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Definition of a Foliation,I

• let M be a manifold of dimension n. A foliation on M is

given by a foliation atlas of codimension q, with 0 ≤ q ≤ n, with foliation charts (ϕi : Ui → Rn = Rn−q × Rq)i∈I and change-of-charts diffeomorphisms that are locally of the form ϕi,j(x, y) = (gi,j(x, y), hi,j(y)), with respect to the decomposition Rn = Rn−q × Rq.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Definition of a Foliation, II

• Each chart Ui is divided into plaques, the connected

components of ϕ−1

i (Rn−q × {y}) for y ∈ Rq.

• These plaques amalgamate globally into leaves. These

leaves are (n − q)-dimensional manifolds that are injectively immersed into M.

• Two foliation atlases define the same foliation if they give

the same decomposition of M into leaves. A foliated manifold (M, F ) is a manifold with a maximal foliation atlas F .

• The space of leaves is the quotient space M/F .
• The dimension of F is n − q.
• A map f : (M, F ) → (M′, F ′) between foliated manifolds is

a smooth map f : M → M′ which sends leaves in F to leaves in F ′.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• (Trivial Foliation) Rn → Rn−q × Rq gives the trivial foliation
• n Rn.
• (Submersions) A submersion f : M → N induces a foliation

F (f) on M whose leaves are the connected components of the fibers f −1(x) of f.

• Foliations derived from submersions are called simple
• foliations. The foliation is strictly simple if the fibers are

all connected. A foliation is strictly simple precisely when its space of leaves is Hausdorff.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• (Kronecker Foliation) Let a be an irrational real number and

define the submersion p: R2 → R by p(x, y) = x − ay. This induces a foliation F (p) of R2 Consider π: R2 → T 2 = S1 × S1, π(x, y) = (e2πix, e2πiy). Then F (p) induces a foliation F on T 2 as follows. If (ϕ, U) ∈ F (p) is a foliation chart such that π|U is injective, then ϕ ◦ (π|U)−1 is a foliation charts for F on T 2. Note that each leaf of F is diffeomorphic to R and lies dense in T 2.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Holonomy of Paths in Leaves

• For a point x in a leaf L, a transversal T at x is a

submanifold of M of dimension q which contains x and is transversal to the leaves of F .

• For a path α in a leaf L from x to y, we may define its

holonomy germ hol(α) = holS,T(α) with respect to transversals T at x and S at y, as the germ of the induced homeomorphism from a neighbourhood of x in T to a neighbourhood of y inS.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Remarks

• If two paths are homotopic in L they give rise to the same

holonomy germ.

• For x = y, we obtain a group homomorphism

π1(L, x) → Diffx(T) DiffO(Rq) We call the image of this homomorphism the holonomy group at x.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Foliation Groupoids, I

The monodromy groupoid Mon(M, F ) is the groupoid with

• bject space M and the space of arrows is defined as follows:
• If x, y ∈ M are points in the same leaf L, then

Mon(M, F )(x, y) is the space of homotopy classes of paths from x to y in L (with homotopies in L);

• If x, y ∈ M lie in different leaves then Mon(M, F )(x, y) = ∅.

Remark

Note that Mon(M, F )x = π1(L, x) where x ∈ L.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Foliation Groupoids, II

The holonomy groupoid Hol(M, F ) is the groupoid with object space M and the space of arrows is defined as follows:

• If x, y ∈ M are points in the same leaf L, then

Hol(M, F )(x, y) is the space of holonomy classes of paths from x to y in L;

• If x, y ∈ M lie in different leaves then Hol(M, F )(x, y) = ∅.

Remarks

• Note that Hol(M, F )x is the holonomy group at x.
• Since homotopic paths have the same germ in the

holonomy group, there is a quotient homomorphism Mon(M, F ) ։ Hol(M, F ).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• 1. For a surjective submersion f : M → N with connected

fibers, the leaves of F (f) have all trivial holonomy and Hol(M, F (f)) = Ker(f). If the fibers of f are also simply connected, then Mon(M, F (f)) = Ker(f).

• 2. Let F be a foliation on M, whose leaves are invariant under

a free properly discontinuous action by a discrete group G. This gives rise to an isomorphism of Lie groupoids Mon(M, F )/G Mon(M/G, F /G) and a surjective homomorphism Hol(M, F )/G → Hol(M/G, F /G), which is generally not an isomorphism.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Proposition

Let F be a foliation of a manifold M.

• The orbits of Mon(M, F ) and Hol(M, F ) are the leaves of

F .

• The isotropy groups of Mon(M, F ) and Hol(M, F ) are

discrete.

• For x ∈ L, the target map of Mon(M, F ) restricts to a

universal covering t : Mon(M, F )(x, −) → L,

• t : Hol(M, F )(x, −) → L is the covering projection

corresponding to the kernel of the holonomy homomorphism π(L, x) → Hol(L, x).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etale Groupoid and Foliations

• Any étale Lie groupoid induces a foliation by its orbits on

the space of objects G0.

• For any foliation F on a manifold M, a groupoid G over M

is said to integrate F if its orbits coincide with the leaves of the foliation. If moreover s has connected fibers, there are local diffeomorphisms Mon(M, F ) → G → Hol(M, F ).

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Etale Holonomy and Monodromy Groupoids

• Let F be a foliation of a manifold M. Choose a complete

transversal section T of (M, F ), i.e., an immersed (not necessarily connected) submanifold of M of dimension q, which is transversal to the leaves of F and intersects each leaf in at least one point.

• Define the Lie groupoid MonT(M, F ) over T as the

restriction of the (full) monodromy groupoid Mon(M, F ) to T.

• dim MonT(M, F )1 = dim T.
• The inclusion MonT(M, F ) → Mon(M, F ) is a weak

equivalence.

• Analogously, define the étale holonomy groupoid

HolT(M, F ) over T.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Examples

• Let F be the standard foliation of the Möbius band M. The

étale holonomy groupoid of (M, F ) is isomorphic to the translation groupoid Z/2 ⋉ (−1, 1).

• The étale holonomy groupoid of the Kronecker foliation F
• f the torus T 2 is Z ⋉ S1.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

Some References

• 1. M. Artin, A. Grothendieck, J.L. Verdier, Théorie des Topos

et Cohomologie Etale des Schémas, SGA, Lecture Notes in Math. 269, 270, and 305, Springer-Verlag, Berlin, 1972.

• 2. Francis Borceux, Handbook of Categorical Algebra 1,

Cambridge University Press, Cambridge, 1994.

• 3. Carsten Butz and Ieke Moerdijk, Representing topoi by

topological groupoids, J.P .A.A. 130 (1998), pp. 223–235.

• 4. Marius Crainic and Ieke Moerdijk, Foliation groupoids and

their cyclic homology, Advances in Mathematics 157 (2001), pp. 177–197.

• 5. Peter T. Johnstone, Sketches of an Elephant, A Topos

Theory Compendium, Volumes 1 and 2, Clarendon Press, Oxford, 2002.

Groupoid Representations Morphisms and Morita Equivalence Lie Groupoids Foliations

References - Continued

• 1. I. Moerdijk, J. Mrcun, Lie groupoids, sheaves and

cohomology, in: Poisson Geometry, Deformation Quantisation and Group Representations, pp. 145–272, London Math. Soc. Lecture Note Ser. 323, Cambridge

• Univ. Press, Cambridge, 2005.
• 2. I. Moerdijk, J. Mrcun, Introduction to foliations and Lie

groupoids, Cambridge Studies in Advanced Mathematics 91, Cambridge Univ. Press, Cambridge, 2003.

• 3. Dorette A. Pronk, Etendues and stacks as bicategories of

fractions, Compositio Math., 102 (1996), pp. 243–303.