Proceedings of the Algebra Symposium
“Babeş-Bolyai” University Cluj, 2002; pp. 69-90.
ALGEBRAIC CONSTRUCTIONS OF BRANDT
GROUPOIDS
GHEORGHE IVAN
Abstract. The aim of this paper is to present some important
constructions of groupoids from the category of Brandt groupoids.
Introduction
The algebraic notion of groupoid was introduced and named by H.
Brandt [1] in a 1926 paper on the composition of quadratic forms in four
variables. A groupoid (in the sense of Brandt) can be thought of as a
generalized group in which only certain multiplications are possible and it
contains several neutral elements.
Groupoids also appeared in Galois theory in the description of relations between subfields of a field K via morphisms of K in a paper of A.
Loewy [21] around 1927 (the isotropy groups of the constructed groupoid
turn out to be the Galois groups).
A generalization of Brandt groupoid has appeared in the work of
C. Ehresmann [7] around 1950. C. Ehresmann added further structures
(topological and differentiable as well as algebraic) to groupoids, thereby
introducing them as a tool in topology and differential geometry.
Groupoids occur also as generalizations of equivalence relations in the
paper of A. Grothendieck [10] on moduli spaces.
The topological and differentiable groupoids endowed with supplementary structures (measure groupoids, Lie groupoids, symplectic
groupoids, Poisson groupoids, Riemannian groupoids) play an essential
role by their applications in measure theory , harmonic analysis , differential geometry, symplectic geometry, Poisson geometry , gauge theory and
quantum mechanics. For details in these areas , see [3],[4], [6],[9],[11],[16],
[20], [22] – [26], [28] – [30], [32], [33].
1991 Mathematics Subject Classification. 20L13, 20L99, 18B40.
Key words and phrases. groupoid, symmetric groupoid, general linear groupoid,
disjoint union of groupoids, projective and inductive system of groupoids.
69
70
GHEORGHE IVAN
1. The Brandt groupoid as universal algebra
The purpose of this section is to define groupoids and to give several
useful properties of them. For more details about groupoids we refer the
reader to [2], [4], [22], [32].
The definition of the Brandt groupoid is essentially the same as the
one given by A. Coste, P. Dazord and A. Weinstein in [4].
Let (Γ, α, β, µ, i) an universal algebra , where Γ is a nonempty set
endowed with the maps α (source) and β (target) α, β : Γ −→ Γ, the multiplication map µ : Γ(2) −→ Γ, (x, y) −→ µ(x, y), where Γ(2) = {(x, y) ∈
Γ × Γ|β(x) = α(y)} and the inversion map i : Γ −→ Γ, x −→ i(x). We
write sometimes x · y or xy for µ(x, y) and x−1 for i(x). The elements of
Γ(2) are called composable pairs of Γ.
Definition 1.1. A (Brandt) groupoid is an universal algebra (Γ, α, β, µ, i)
which satisfy the following conditions :
(1) (associativity)(xy)z = x(yz), in the sense that, if one side of the equation is defined so is the other and then they are equal;
(2) (identities) (α(x), x), (x, β(x)) ∈ Γ(2) and α(x)x = xβ(x) = x;
(3) (inverses) (x−1 , x), (x, x−1 ) ∈ Γ(2) and x−1 x = β(x), xx−1 = α(x).
The element α(x) (resp., β(x) ) is the left unit (resp., right unit) of
x ∈ Γ. The subset Γ0 = α(Γ) = β(Γ) of Γ, denoted sometimes by UΓ , is
called the unit set of Γ and we say that Γ is a Γ0 - groupoid or that Γ is a
groupoid over Γ0 .
A Γ0 -groupoid Γ will be denoted by (Γ, α, β, µ, i; Γ0 ) or (Γ, α, β; Γ0 )
or (Γ; Γ0 ). The maps α, β, µ and i of a groupoid (Γ; Γ0 ) are called the
structural functions of Γ.
For each u ∈ Γ0 , the set Γu = α−1 (u) (resp., Γv = β −1 (v) ) is called
the α - fibre (resp., β - fibre) of Γ over u ∈ Γ0 and if u, v ∈ Γ0 we will
write Γuv = α−1 (u) ∩ β −1 (v).
A Γ0 -groupoid Γ is said to be transitive (resp., principal), if the map
α × β : Γ −→ Γ0 × Γ0 , x → (α × β)(x) = (α(x), β(x)) is surjective (resp.,
injective). The map α × β is called the anchor of Γ.
Convention. Whenever we write a product in a given groupoid, we are
assuming that it is defined.
In the following proposition we summarize some properties of the
structural functions of a groupoid obtained directly from definitions.
Proposition 1.1. For any groupoid (Γ, α, β, µ, i; Γ0 ) we have:
(1.1) α(u) = β(u) = u and u · u = u for all u ∈ Γ0 ;
(1.2) α(xy) = α(x) and β(xy) = β(y), (∀)(x, y) ∈ Γ(2) ;
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
(1.3)
(1.4)
(1.5)
(1.6)
(1.7)
from
71
α ◦ i = β, β ◦ i = α and i ◦ i = IdΓ ;
(cancellation law) If xz1 = xz2 (resp., z1 x = z2 x ), then z1 = z2 ;
x−1 (xy) = y, (∀)(x, y) ∈ Γ(2) and (zx)x−1 = z, (∀)(z, x) ∈ Γ(2) ;
(x, y) ∈ Γ(2) =⇒ (y −1 , x−1 ) ∈ Γ(2) and (xy)−1 = y −1 x−1 ;
For x ∈ Γ the left translation Lx defined by Lx (y) = xy is a bijection
α−1 (β(x)) onto α−1 (α(x)) and (Lx )−1 = Lx−1 .
Proposition 1.2. Let Γ be a groupoid,x ∈ Γ and u ∈ Γ0 . Then:
(1.8) Γ(u) = α−1 (u) ∩ β −1 (u) = {x ∈ Γ|α(x) = β(x) = u} is a group
under the restriction of µ to Γ(u), called the isotropy group at u of Γ;
(1.9) the isotropy groups Γ(α(x)) and Γ(β(x)) are isomorphic;
(1.10) If Γ is transitive, then the groups Γ(u), u ∈ Γ0 are isomorphic.
Proof. The assertion (1.8) follows from definitions and Prop.1.1. For (1.9),
we prove that φ : Γ(α(x)) → Γ(β(x)), z → φ(z) = xzx−1 is a isomorphism
of groups. The assertion (1.10) follows from (1.9), since the anchor map
is surjective.
In a groupoid (Γ, α, β; Γ0 ) the relation defined on Γ0 by :
(1.11) u ∼Γ v ⇐⇒ (∃)x ∈ Γwithα(x) = uandβ(x) = v is an equivalence
relation. Its equivalence classes are called orbits and the orbit of u ∈ Γ0 is
denoted [u]. The quotient set Γ0 /Γ determined by this equivalence relation
is called the orbit space.
A groupoid (Γ, Γ0 ) is transitive iff Γ0 /Γ is a singleton.
There is a natural decomposition of the unit space Γ0 of a groupoid Γ
into orbits. Over each orbit there is a transitive groupoid and the disjoint
union of these transitive groupoids is the original groupoid Γ.
The following definition of the groupoid is essentially the same as the
one given by P. Hahn [11].
Definition 1.2. Let (G, m, i) an universal algebra , where G is a set
endowed with the multiplication map m : G(2) −→ G, (x, y) −→ m(x, y) =
xy, G(2) is the set of composable elements of G and the inversion map
i : G −→ G, x −→ i(x) = x−1 .
A groupoid is an universal algebra (G, m, i) such that the following
conditions are satisfied:
(i) (x−1 )−1 = x, (∀)x ∈ G;
(ii) (x, y), (y, z) ∈ G(2) ⇒ (xy, z), (x, yz) ∈ G(2) and (xy)z = x(yz);
(iii) (x−1 , x) ∈ G(2) and if (x, y) ∈ G(2) , then x−1 (xy) = y;
(iv) (x, x−1 ) ∈ G(2) and if (z, x) ∈ G(2) , then (zx)x−1 = z.
If (G, m, i) is a groupoid in the sense of Def.1.2 and if x ∈ G, then
λ(x) = xx−1 is the source of x and ρ(x) = x−1 x is the (target) of x ∈ G.
72
GHEORGHE IVAN
Also, G0 = λ(G) = ρ(G) is the unit space of G; its elements are units in
the sense that λ(x)x = x and xρ(x) = x.
Remark 1.1. It is easy to prove that the Def.1.1 and Def.1.2 of the notion
of groupoid are equivalent .
Remark 1.2. There are various definitions for Brandt groupoids , see
[4], [26], [29], [31]. In the paper [6] it has proved that these definitions are
equivalent. In [22] are disscused the groupoids in the sense of Ehresmann.
In the other words, an Ehresmann groupoid is a groupoid Γ for which the
unit space Γ0 is not a subset of Γ.
By group bundle we mean a Γ0 -groupoid Γ such that α(x) = β(x)
for all x ∈ Γ. Moreover,a group bundle is the union of its isotropy groups
Γ(u) = α−1 (u), u ∈ Γ0 (here, two elements may be composed iff they lie
in the same fiber α−1 (u)).
If (Γ, α, β; Γ0 ) is a groupoid then Is(Γ) = {x ∈ Γ|α(x) = β(x)} is a
group bundle, called the isotropy group bundle of Γ.
Definition 1.3. A nonempty subset H of a G0 -groupoid G is called
subgroupoid of G if it is closed under multiplication (when defined) and
inversion, i.e. the following conditions hold :
(i) for all x, y ∈ H such that xy is defined, we have xy ∈ H;
(ii) for all x ∈ H, we have x−1 ∈ H.
Note that from the condition (ii) of Definition 1.3 , we deduce that
α(h) ∈ H and β(h) ∈ H, for all h ∈ H. If α(H) = β(H) = G0 , then H is
called a wide subgroupoid of G.
Example 1.1. (i) A group G having e as unity, is just a {e} - groupoid
and conversely, every groupoid with one unit is a group. The wide subgroupoids of G are the subgroups of G.
(ii) The nul groupoid over a set. We give on a nonempty set X may
the following groupoid structure : α = β = IdX , the elements x, y ∈ X
are composable iff x = y and we define xx = x. Any subset of X is a
subgroupoid and the only wide subgroupoid is X itself.
(iii) The pair groupoid over a set. Let X be a nonempty set. Then G =
X ×X is a groupoid with respect to rules: α(x, y) = (x, x), β(x, y) = (y, y),
the elements (x, y) and (y ′ , z) are composable in G iff y ′ = y and we take
(x, y)·(y, z) = (x, z) and the inverse of (x, y) is defined by (x, y)−1 = (y, x).
The unit space of the pair groupoid X × X is the diagonal ∆X =
{(x, x)|x ∈ X} which identify with X. For the pair groupoid X × X each
fibre can be identified with X. The isotropy group G(u) at u = (x, x) is
the nul group {(u, u)}. A subgroupoid H of the pair groupoid X × X is a
relation on X which is symmetric and transitive. A wide subgroupoid H
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
73
is an equivalence relation. Also, the subset ∆X is a wide subgroupoid of
the pair groupoid X × X, called the diagonal subgroupoid.
(iv) The trivial groupoid on a set X with group G. Let X be a set
and G be a group having e as unity. We give to Γ = X × X × G the
following groupoid structure: α(x, y, g) = (x, x, e), β(x, y, g) = (y, y, e),
the elements (x, y, g) and (y ′ , z, g ′ ) are composable in Γ iff y = y ′ and we
take (x, y, g) · (y, z, g ′ ) = (x, z, gg ′ ) and the inverse of (x, y, g) is defined
by (x, y, g)−1 = (y, x, g −1 ).
We have that UΓ = {(x, x, e)|x ∈ X} which identify with X and
X × X × G is a transitive groupoid. The isotropy group at u = (x, x, e) is
Γ(u) = {(x, x, g)|g ∈ G} which identify with G.
Example 1.2 (i) The groupoid associated to an equivalence relation. Let
R be the graph of an equivalence relation on a set X. We give on R ⊆
X × X the following groupoid structure : the elements (u, v) and (v ′ , w)
are composable in R iff v ′ = v and we define (u, v) · (v, w) = (u, w) and
the inverse of (u, v) in R is (u, v)−1 = (v, u). Then λ(u, v) = (u, u) and
ρ(u, v) = (v, v). The unit space UR is the diagonal and may be identified
with X. We have that R is a principal groupoid. Conversely, if (G, λ, ρ; G0 )
is a principal groupoid, then the anchor map λ × ρ identifies G with the
graph of the equivalence relation.
The principal groupoids play an important role in theory of measure
groupoids and harmonic analysis, see [25], [29], [30].
(ii) A vector bundle ξ = (E, π, B) is a group bundle. Here Γ = E is
the total space of ξ, Ex is the fibre over x, α(x) = β(x) = the nul vector
of the vector space Ex , Γ(2) = ⊎x∈B (Ex × Ex ) and the multiplication map
is fiberwise addition.
2. Morphisms of groupoids
Definition 2.1. (a) Let (Γ, α, β, µ, i; Γ0 ) and (Γ′ , α′ , β ′ , µ′ , i′ ; Γ′0 ) be two
groupoids. A morphism of groupoids or groupoid morphism is a pair (f, fe)
of maps f : Γ −→ Γ′ and fe : Γ0 → Γ′0 such that the following two
conditions are satisfied:
(i) f (µ(x, y)) = µ′ (f (x), f (y)), for all (x, y) ∈ Γ(2) ;
(ii) α′ ◦ f = fe ◦ α and β ′ ◦ f = fe ◦ β.
If Γ0 = Γ′0 and fe = IdΓ0 , we say that f : Γ → Γ′ is a Γ0 - morphism
of Γ0 - groupoids.
(b) A groupoid morphism (f, fe) : (Γ; Γ0 ) → (Γ′ ; Γ′0 ) is said to be
isomorphism of groupoids, if f and fe are bijective maps.
74
GHEORGHE IVAN
From the condition (i) of Def. 2.1 follows that µ′ (f (x), f (y)) is defined
whenever µ(x, y) is defined.
If (Γ; Γ0 ) is a groupoid, then the set of the automorphisms of Γ endowed with the composition law of maps is a group, and denoted by Aut(Γ)
is called the group of automorphisms of Γ.
Applying Proposition 1.1 and the definitions, we obtain
Proposition 2.1. If (f, fe) : (Γ, α, β, µ, i; Γ0 ) → (Γ′ , α′ , β ′ , µ′ , i′ ; Γ′0 ) is a
groupoid morphism, then:
(2.1) f (u) = fe(u), (∀)u ∈ Γ0 and f ◦ i = i′ ◦ f.
This implies that fe is the restriction of f to unit spaces.
Proposition 2.2 Let (Γ; Γ0 ) and (Γ′ ; Γ′0 ) be two groupoids. A pair of
maps (f, fe) : (Γ; Γ0 ) → (Γ′ ; Γ′0 ) is a groupoid morphism iff the following
two conditions hold:
(2.2) (∀)(x, y) ∈ Γ(2) =⇒ (f (x), f (y)) ∈ Γ′(2) ;
(2.3) f (µ(x, y)) = µ′ (f (x), f (y)), (∀)(x, y) ∈ Γ(2) .
Proof. The conditions (2.2) and (2.3) are consequences of Definition 2.1
and Proposition 2.1. Conversely, let f : Γ → Γ′ which satisfy (2.2) and
(2.3). We define fe : Γ0 → Γ′0 by fe(u) = α′ (f (u)), (∀)u ∈ Γ0 and verify the
relations (ii) from Definition 2.1.
Proposition 2.3 (a) Let (Γ; Γ0 ) be a groupoid. Then Γ0 is a wide subgroupoid of Γ, called the nul subgroupoid of Γ.
(b) Let (f, fe) : (Γ; Γ0 ) → (Γ′ ; Γ′0 ) be a groupoid morphism. Then the
following assertions hold:
(2.4) If (H ′ ; H0′ ) is a subgroupoid of (Γ′ ; Γ′0 ), then (f −1 (H ′ ); fe−1 (H0′ )) is
a subgroupoid of (Γ; Γ0 );
(2.5) The kernel of the groupoid morphism (f, fe) : (Γ; Γ0 ) → (Γ′ ; Γ′0 )
defined by Kerf = {x ∈ Γ|f (x) ∈ Γ′0 } is a wide subgroupoid of Γ.
Proof. The assertions (a) and (2.4) follows from definitions and Proposition 1.1, 2.1 and 2.2, and (2.5) results from (2.4) taking H ′ = Γ′0 .
Remark 2.1. If (f, fe) : (Γ; Γ0 ) → (Γ′ ; Γ′0 ) is a groupoid morphism, then
not always Imf = {f (x)|x ∈ Γ} is a subgroupoid of Γ′ . For example, let
Γ = B × B the pair groupoid over B = {0, 1}. Let f : Γ → Z defined
by f (0, 0) = 0, f (0, 1) = 1, f (1, 0) = −1, f (1, 1) = 0 and fe : B → {0}
defined by fe(0) = fe(1) = 0. Then the pair (f, fe) is a groupoid morphism
from B × B into the additive group Z of integer numbers. We have that
Imf = {0, −1, 1} which is not a subgroup of Z.
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
75
Definition 2.2. A groupoid morphism (f, fe) : (Γ; Γ0 ) −→ (Γ′ ; Γ′0 ) satisfying the following condition:
(2.6) for all x, y ∈ Γ such that (f (x), f (y)) ∈ Γ′(2) we have (x, y) ∈ Γ(2)
will be called strong morphism of groupoids.
Remark 2.2. The concept of strong groupoid morphism has considered
by A. Ramsey [29] in the case of Brandt groupoids , called true morphism
of groupoids. We have that the image of a strong groupoid morphism is
a subgroupoid. For other properties concerning the subgroupoids and the
strong morphisms of groupoids, see [14], [15].
Example 2.1. (i) Let G be a group. The map δ : G × G → G given by
δ(x, y) = xy −1 is a groupoid morphism of the pair groupoid G × G into G.
It is not a strong groupoid morphism.
(ii) The anchor map α × β : Γ −→ Γ0 × Γ0 of the Γ0 -groupoid Γ
into the pair groupoid Γ0 × Γ0 is a strong morphism of groupoids. The
inverse image of the diagonal subgroupoid ∆ of the pair groupoid Γ0 × Γ0
by α×β, i.e. (α×β)−1 (∆) is a subgroupoid of Γ. We have (α×β)−1 (∆) =
{x ∈ Γ|α(x) = β(x)} = Is(Γ); it is named the isotropy subgroupoid of Γ.
(iii) Let Γ = X × X × G and Γ′ = X ′ × X ′ × G′ be two trivial
groupoids over the sets X and X ′ with the groups G and G′ , respectively.
Let φ : G → G′ be a homomorphism of groups and θ : X → G′ be an
arbitrary map. Then the map f : X × X × G → X ′ × X ′ × G′ defined
by f (x, y, g) = (f (x, x, e), f (y, y, e), θ(x) · φ(g) · (θ(y))−1 ) is a groupoid
morphism, where e is the unity of G.
Example 2.2. Let (Γ, α, β; Γ0 ) be a transitive groupoid. Since the anchor
map α × β : Γ → Γ0 × Γ0 is surjective, for u ∈ Γ0 the restriction αu of the
map α to Γu = β −1 (u) is surjective. There exists a right inverse σ : Γ0 →
Γu of αu : Γu → Γ0 . Then σ is an injective map such that αu ◦ σ = IdΓ0 .
The map f : Γ0 × Γ0 × Γ(u) → Γ, (u, v, x) → f (u, v, x) = σ(u) · x · (σ(v))−1
is an isomorphism of the trivial groupoid Γ0 × Γ0 × Γ(u) onto the groupoid
Γ. Therefore:
a transitive groupoid Γ is isomorphic with a trivial groupoid over the set
UΓ with the group G, where G is isomorphic with a isotropy group of Γ.
3. Some special examples of groupoids
A. The symmetric groupoid. Let M be a nonempty set. By quasipermutation of the set M we mean an injective map from a subset of M into
M. We denote by Γ = S(M ) =Inj(S) the set of all quasipermutations of
M ; i.e. S(M ) = {f |f : A → M, f is injective and ∅ ̸= A ⊆ M }.
76
GHEORGHE IVAN
For f ∈ S(M ), let D(f) be the domain of f and let R(f ) = f (D(f )).
Let Γ(2) = {(f, g)|R(f ) = D(g)} and for (f, g) ∈ Γ(2) we define µ(f, g) =
g ◦ f. If IdA denotes the identity on A, then Γ0 = {IdA |A ⊆ M } is the set
of units of Γ, denoted by S0 (M ) and f −1 is the inverse function from the
R(f ) to D(f ). The maps α and β are defined by α(f ) = IdR(f ) ; β(f ) =
IdD(f ) . Thus S(M ) is a groupoid over S0 (M ).S(M ) is called the symmetric
groupoid of the set M or the groupoid of all quasipermutations of M.
We have that if M and M ′ are equipotents sets, then the symmetric
groupoids S(M ) and S(M ′ ) are isomorphic groupoids.
When M = {1, 2, ..., n}, we write Sn for S(M ) and call Sn the symmetric groupoid of degree n.
( )
( )
∑n
In the paper [17] it was proved that |Sn | = k=1 k!( nk )2 , where nk
is the k-th binomial coefficient.
The structural
α
β - (fibres,
( functions,
)
( - fibres,
)
) the isotropy
(
)groups
1
2
1
2
of S2 = {u1 =
, u2 =
, f1 =
, f2 =
, u3 =
2
2
1
(
)
(2
)
1 2
1 2
, f3 =
} and the multiplication in the isotropy sub1 2
2 1
groupoid Is(S2 ) = Γ(u1 ) ∪ Γ(u2 ) ∪ Γ(u3 ) are given by the following tables:
◦
u1
u2
f1
f2
u3
f3
u1
u1
u2
f1
u2
f1
f2
u2
f2
u1
f1
u1
f1
u1
f1
u
α−1 (u)
β −1 (u)
Γ(u)
f2
f2
u3
f3
u2
f2
u2
u3
f3
f3
u3
f
α(f )
β(f )
i(f )
u1
u2
u3
{u1 , f2 } {u2 , f1 } {u3 , f3 }
{u1 , f1 } {u2 , f2 } {u3 , f3 }
{u1 }
{u2 }
{u3 , f3 }
u1
u1
u1
u1
u2
u2
u2
u2
f1
u2
u1
f2
◦ u1
u1 u 1
u2
u3
f3
f2
u1
u2
f1
u2
u3
u3
u3
u3
f3
u3
u3
f3
u3
f3
u3
f3
f3
u3
u2
Remark 3.1. We have that |S2 | = 6, |Is(S2 )| = 4 and the order of Is(S2 )
is not a divisor of |S2 |. Hence, the Lagrange’ s theorem for finite groups is
not valid for finite groupoids.
For a given groupoid (Γ; Γ0 ), let (S(Γ); S0 (Γ)) the symmetric groupoid
of the set Γ, where S0 (Γ) = {IdA |A ⊆ Γ}.
We consider now the set L(Γ) = {λa |a ∈ Γ} of all the left translations
La : Γ −→ Γ, x → La (x) = ax, whenever (a, x) ∈ Γ(2) . The domain of
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
77
La , i.e. D(La ) = {x ∈ Γ|(a, x) ∈ Γ(2) } is a nonempty set of Γ, since
(a, β(a)) ∈ Γ(2) and so La ∈ S(Γ). Therefore, L(Γ) is a subset of S(Γ). For
all a, b, x ∈ Γ such that β(a) = α(b) and β(b) = α(x) we have La (Lb (x)) =
La (bx) = a(bx) = (ab)x = Lab (x) and we note that La ◦ Lb = Lab if
(a, b) ∈ Γ(2) . Consequently, we have Lα(x) ◦Lx = Lx ◦Lβ(x) = Lx , (∀)x ∈ Γ.
For u ∈ Γ0 the domain of the translation Lu : Γ −→ Γ is D(Lu ) =
{x ∈ Γ|(u, x) ∈ Γ(2) }. From β(u) = α(x) it follows that u = α(x), since
α(u) = u, for any u ∈ Γ0 . Then Lu (x) = ux = α(x)x = x and it follows
that Lu = IdD(Lu ) . Hence Lu ∈ S0 (Γ) for all u ∈ Γ0 , and L0 (Γ) = {Lu |u ∈
Γ0 } is a subset of S0 (Γ). Since L(Γ) ⊆ S(Γ) and the conditions (i) and
(ii) from Def. 1.3 are satisfied, it follows that L(Γ) is a subgroupoid of
the symmetric groupoid S(Γ). This groupoid is called the groupoid of left
translations of Γ.
Theorem 3.1. (Cayley theorem for groupoids.) Every groupoid Γ is
isomorphic to a subgroupoid of the symmetric groupoid S(Γ).
Proof. Let (L(Γ); L0 (Γ)) the groupoid of left translations of the groupoid
Γ. We have that L(Γ) is a subgroupoid of S(Γ). We prove that φ : Γ −→
L(Γ) given by φ(a) = La , (∀)a ∈ Γ is an isomorphism of groupoids.
Remark 3.2. In view of Cayley’s theorem for groupoids, many groupoids
occur naturally as subgroupoids of some symmetric groupoid.
In the paper [13] it has proved the Cayley’ s theorem for monoidoids.
A groupoid is a monoidoid in which every element is invertible.
B. The action groupoid. Let r : X ×G → X, (x, g) → r(x, g) = x·g be a
right action of the group G on the set X. The set Γ = X ×G has a structure
of groupoid by the following rules : the elements (x, g) and (y, g ′ ) from Γ
are composable iff y = x · g and we define µ((x, g), (x · g, g ′ )) = (x, gg ′ );
the inverse of (x, g) ∈ Γ is (x, g)−1 = (x · g, g −1 ). We have that (Γ, µ, i) is
a groupoid (see, Def.1.2). Then, λ(x, g) = (x, e), ρ(x, g) = (x · g, e), where
e is the unity of G and UΓ = {(x, e)|(∀)x ∈ X}. This groupoid is called
the groupoid associated to action of the group G on the set X.
The groupoid X × G is transitive (resp., principal) iff the action of
the group G on the set X is transitive (resp., freely).
C. The groupoid associated to a surjective map. Let p : M → B
be a surjective map. For x ∈ B, the set p−1 (x) is the fibre over x and we
denote by Fx = p−1 (x), x ∈ B. Let F(M ) denote the set of all bijections
φ : Fx → Fy , for x, y ∈ B. Then F(M ) is a groupoid with respect to
following structure : for φ ∈ F(M ) we define α(φ) = IdFy , β(φ) = IDFx ,
the multiplication µ is the composition of maps and the inverse of φ : Fx →
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GHEORGHE IVAN
Fy is its inverse as map. The unit space is UF (M ) = {Idp−1 (x) |(∀)x ∈ B}.
The isotropy group at u = IdFx is the symmetric group of Fx .
In the paper [22], F(M ) is called the frame groupoid associated to
surjective map p : M → B.
D. The fundamental groupoid. Let B be a topological space and let
G = Π(B) be the collection of homotopy classes of paths in B with all
possible fixed endpoints. Specifically, if σ : [0, 1] → B is a path from
x = σ(0) to y = σ(1), let [σ] denote the homotopy class of σ relative to
the points x, y. Then Π(B) = {(x, [σ], y)|x, y ∈ B, σis a path from xtoy}.
We can define a structure of groupoid on Π(B) as follows. If σ
is a path from x to y, the source and the target are α(x, [σ], y) =
(x, [cx ], x), β(x, [σ], y) = (y, [cy ], y), where cx , cy are the constant paths at
x, y respectively. The multiplication map µ defined in Π(B) is µ([σ], [τ ]) =
[σ · τ ], where σ · τ is the concatenation of the paths σ and τ iff σ(1) = τ (0).
The inverse of (x, [σ], y) is (y, [σ], x), where σ is the inverse path of σ, i.e.
σ(t) = σ(1 − t), (∀)t ∈ [0, 1]. The groupoid Π(B) is called the fundamental
groupoid of B. The orbits of this groupoid are just the path components
of B. The isotropy group of Π(B) at x ∈ B is the fundamental group
π1 (B, x).
If B is an arcwise connected topological space , then Π(B) is a transitive groupoid.
Remark 3.3. In algebraic topology, the fundamental groupoid has been
exploited by P.J. Higgins [12] and R. Brown [3].
4. Universal constructions of groupoids
In this section we shall give some important ways of building up new
groupoids from old ones.
4.1. The inverse image of a groupoid by a bijective map
Let (G, α, β, µ, i; G0 ) be a groupoid and φ : G′ → G be a bijection.
We construct a structure of groupoid on G′ via φ as follows. We consider
the maps α′ = α ◦ φ, β ′ = β ◦ φ and i′ = i ◦ φ. For (x′ , y ′ ) ∈ G × G′ we
define µ′ (x′ , y ′ ) = φ−1 (µ(φ(x′ ), φ(y ′ )) iff (φ(x′ ), φ(y ′ ) ∈ G(2) , i.e. x′ · y ′ =
φ−1 (φ(x′ · φ(y ′ ) ⇐⇒ φ(x′ · φ(y ′ ) is defined in G.
If denote x′ = φ−1 (x) and y ′ = φ−1 (y), we have that (x, y) ∈ G(2) iff
′ ′
(x , y ) ∈ G′(2) . It is easy to verify that (G′ , α′ , β ′ , µ′ , i′ ; G′0 ) is a groupoid
and that φ : G′ → G is an isomorphism of groupoids.
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
79
The groupoid G′ is called the inverse image of the groupoid G by the
bijection φ : G′ → G. Also, we say that the groupoid G′ is obtained by
transport of structure from the groupoid G via the bijection φ.
4.2. The groupoid Γ(n) associated to a groupoid Γ
Let Γ be a Γ0 -groupoid. By Γ(n) (n ≥ 2) we denote the set of n-tuples
(x0 .x1 , ..., xn−1 ) of Γn such that (xi−1 , xi ) ∈ Γ(2) for i=1,2,...,n-1. We give
to Γ(n) the following groupoid structure:
α(n) (x0 , x1 , ..., xn−1 ) = (x0 , x1 , ..., xn−2 , α(xn−1 )),
β (n) (x0 , x1 , ..., xn−1 ) = (x0 x1 , ..., xn−2 xn−1 , β(xn−2 xn−1 )),
the elements (x0 , x1 , ..., xn−1 ) and (y0 , y1 , ..., yn−1 ) are composable iff y0 =
x0 x1 , y1 = x1 x2 , ..., yn−2 = xn−2 xn−1 ; we take
(x0 , . . . , xn−1 )(x0 x1 , ..., xn−2 xn−1 , yn−1 ) = (x0 , ..., xn−2 , xn−1 yn−1 );
the inverse is given by (x0 , x1 , ..., xn−1 )−1 = (x0 x1 , ..., xn−2 xn−1 , x−1
n−1 ).
If φ : Γ −→ Γ′ is a groupoid morphism, then φ(n) : Γ(n) −→ Γ′(n)
given by φ(n) (x0 , x1 , ..., xn−1 ) = (φ(x0 ), φ(x1 ), ..., φ(xn−1 )) is a morphism
of groupoids for any n ≥ 0 (here Γ(0) = Γ0 , Γ1 = Γ).
Remark 4.2.1. In the paper [6], the groupoid Γ(n) has considered in the
construction of a cohomology theory for Brandt groupoids. This extend
the usual cohomology theory for groups.
4.3. The direct product of a family of groupoids.
Let
∏ {Γi |i ∈ I} be a family of groupoids, where (Γi , αi , βi ; Γi,0 ). Let
Γ = i∈I Γi the direct product of the family of sets {Γi |i ∈ I}. We
give to Γ a structure of groupoid as follows. The elements x = (xi )i∈I
and y = (yi )i∈I from Γ are composable iff (xi , yi ) ∈ Γi,(2) and we take
(xi )i∈I · (yi )i∈I = (xi yi )i∈I . The maps α, β are given by α((xi )i∈I ) =
(αi (xi ))i∈I and β((yi )i∈I ) = (βi (yi ))i∈I . The inverse of x = (xi )i∈I is
−1
−1
x−1 =
∏ (xi )i∈I , where xi is the inverse of xi in Γi . It follows that
Γ0 = i∈I Γi,0 .
∏
The groupoid (Γ = i∈I Γi , α, β; Γ0 ) is called the direct product of
the groupoids
∏ Γi , i ∈ I.
If Γ = i∈I
∏ Γi is the direct product of the groupoids Γi , i ∈ I, then the
map pj : Γ = i∈I Γi → Γj , x = (xi )i∈I → pj ((xi )i∈I ) = xj is a surjective
morphism of groupoids, for all j ∈ I, called the canonical projection of
rank j from Γ to Γj .
In particular, if I = {1, 2} and (Γi , αi , βi ; Γi,0 ) for i = 1, 2 are
groupoids, then (Γ1 × Γ2 , α1 × α2 , β1 × β2 ; Γ1,0 × Γ2,0 ) is a groupoid, called
the direct product of the groupoids (Γ1 ; Γ1,0 ) and (Γ2 ; Γ2,0 ).
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GHEORGHE IVAN
The direct product of two transitive groupoids is also a transitive
groupoid.
4.4. The Whitney sum of two groupoids
Let (G, α, β, µ, i; G0 ) and (G′ , α′ , β ′ , µ′ , i′ ; G0 ) be two G0 - groupoids.
The set (G ⊕ G′ = {(x, x′ ) ∈ G × G′ |α(x) = α(x′ ), β(x) = β ′ (x′ )} has
a natural structure of groupoid with the following rules : α(x, x′ ) =
(α(x), α(x)), β(x, x′ ) = (β(x), β(x)), the elements (x, x′ ) and (y, y ′ ) are
composable in G ⊕ G′ iff (x, y) ∈ G(2) and (x′ , y ′ ) ∈ G′(2) and we define µ((x, x′ ), (y, y ′ )) = (µ(x, y), µ′ (x′ , y ′ )) and the inverse of (x, x′ ) is
i(x, x′ ) = (i(x), i′ (x′ )). Then (G ⊕ G′ , α, β, µ, i; ∆G0 ) is a groupoid, called
the Whitney sum of G and G′ .
If G ⊕ G′ is the Whitney sum of G0 - groupoids G and G′ , then
the projection maps p : G ⊕ G′ → G and p′ : G ⊕ G′ → G′ defined by
p(x, x′ ) = x and p′ (x, x′ ) = x′ are morphisms of groupoids.
The Whitney sum of two transitive G0 - groupoids is also a transitive
groupoid.
4.5. Semi-direct product.
Let (Γ; Γ0 ) be a groupoid, G a group with e as unity and ω : G −→
Aut(Γ) a homomorphism of groups (for every a ∈ G, ω(a) is an automorphism of the groupoid Γ). We denote by x.a = [ω(a−1 )](x), , where a ∈ G
and x ∈ Γ.
We give to Γ×G the following groupoid structure: (x, a) and (z, b) are
composable iff z = y.a with (x, y) ∈ Γ(2) and we take µ
b((x, a), (y.a, b)) =
(µ(x, y), ab), the inverse of (x, a) is defined by bi(x, a) = (i(x).a, a−1 ) , the
b a) = (β(x).a, e). We
maps α
b and βb are defined by α
b(x, a) = (α(x), e), β(x,
have UΓ×G = Γ0 × {e} and the unit set may be identified with Γ0 .
This groupoid, denoted by Γ ×ω G, is called the semi-direct product of
the groupoid Γ with the group G, via the homomorphism ω : G −→ Aut(Γ).
In particular, if Γ is a group, then Γ ×ω G is the usual semi-direct
product of two groups.
4.6. Congruences on groupoids
Definition 4.6.1. Let (Γ, α, β, µ, i; Γ0 ) be a groupoid and let ρ be an
equivalence relation defined on Γ. We say that the equivalence relation ρ
is compatible with the structure of groupoid of Γ, or that ρ is a congruence
on the groupoid Γ, if the
{ following conditions hold :
x · zρy · t, for allx, y, z, t ∈ Γ
(4.6.1) xρyandzρt =⇒
such that(x, z), (y, t) ∈ Γ(2)
(4.6.2) xρy =⇒ i(x)ρi(y).
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
81
From definitions and the properties of structural functions of a
groupoid it follows that, if ρ is a congruence on the groupoid Γ, then
ρ is compatible with the functions α and β, that is,
(4.6.3) xρy =⇒ α(x)ρα(y)andβ(x)ρβ(y).
If ρ is a congruence defined on the groupoid Γ, we denote by [x]
the equivalence class of x ∈ Γ relative to equivalence relation ρ and let
Γ/ρ = {[x]|x ∈ Γ} the quotient set determined by ρ in Γ.
Then the quotient set Γ/ρ determined by the congruence ρ in the
groupoid Γ has a natural structure of groupoid with respect to the following rules : the maps α, β : Γ/ρ → Γ/ρ are defined by α([x]) =
[α(x)], β([x]) = [β(x)], (∀)x ∈ Γ, the elements [x], [y] ∈ Γ/ρ are composable in Γ/ρ iff (x, y) ∈ Γ(2) and we define µ([x], [y]) = [µ(x, y)] and the
inverse of [x] in Γ/ρ is given by i([x]) = [i(x)].
We have that the unit space of Γ/ρ is α(Γ/ρ) = {[u]|u ∈ Γ0 } and we
denote this by Γ0 /ρ.
The groupoid (Γ/ρ, α, β, µ, i; Γ0 /ρ) will be called the quotient groupoid
of the groupoid Γ relative to congruence ρ. The canonical map p : Γ →
Γ/ρ, x → p(x) = [x] is a surjective morphism of groupoids, called the
canonical morphism of groupoid Γ onto the quotient groupoid Γ/ρ.
Another manner for construction of a quotient groupoid is that based
on the normal subgroupoids.
Let H be a subgroupoid of the groupoid (Γ, α, β; Γ0 ). The set Γα(H) =
β −1 (α(H)) = {x ∈ Γ|(∃)h ∈ H, α(h) = β(x)} is called the set of left
composable elements with H.
For all x ∈ Γα(H) , we denote by xH = {xh|h ∈ H}, i.e. the set of
product xh for all h ∈ H such that xh is defined; xH is called the left coset
of Γ determined by x. The cardinal of the set Γα(H) /H = {xH|x ∈ Γα(H) }
is called the index of H in Γ.
In the paper [27] it has proved the index theorem for the subgroupoids
of a finite groupoid. From the index theorem is obtained the Lagrange’s
theorem for finite groupoids with a single unit (i.e., for finite groups).
Definition 4.6.2. By normal subgroupoid of a groupoid Γ, we mean a
wide subgroupoid H of Γ having the following property: for all x ∈ Γ and
h ∈ H such that xhx−1 is defined, we have xhx−1 ∈ H.
Proposition 4.6.1. ([6]) A wide subgroupoid H of the Γ0 -groupoid Γ is
normal iff xH(β(x)) = H(α(x))x, for all x ∈ Γ, where H(u) denotes the
isotropy group of the groupoid H at u.
Example 4.6.1. (i) If Γ is a Γ0 - groupoid, then H = Γ0 and K = Is(Γ)
are normal subgroupoids of Γ.
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GHEORGHE IVAN
(ii) If f : Γ −→ Γ′ is a morphism of groupoids, then Kerf is a normal
subgroupoid of Γ.
Let H a wide subgroupoid of the Γ0 -groupoid Γ. We define on Γ the
following relation by:
{
there existh ∈ H(α(x))andh′ ∈ H(β(x))
(4.6.4) x ≡ y(modH) ⇐⇒
such thaty = hxh′
The relation ” ≡ ” defined by (4.6.4) is an equivalence relation. We
denote by x
b the equivalence class of x ∈ Γ relative to equivalence relation
” ≡ ” and let Γ/ ≡= {b
x|x ∈ Γ} the set of the equivalence classes defined
on Γ by ” ≡ ”.
Proposition 4.6.2. ([6]) If H is a normal subgroupoid of Γ, then :
(4.6.5) x
b = xH(β(x)) = H(α(x))x, (∀)x ∈ Γ.
If H is a normal subgroupoid of Γ, then:
(4.6.6) Γ/ ≡= {xH(β(x))|x ∈ Γ} = {H(α(x))x|x ∈ Γ} and
(4.6.7) Γ0 / ≡= {b
u|u ∈ Γ0 } = {H(u)|u ∈ Γ0 }.
Proposition 4.6.3. ([6]) If H is a normal subgroupoid of Γ, then the
relation” ≡ ” defined on Γ by (4.6.4) is a congruence on the groupoid Γ
and we have [x] = x
b, (∀)x ∈ Γ.
Proof. Applying the definitions and Prop. 4.6.2, it easy to prove that the
equivalence relation ” ≡ ” verify (4.6.1) and (4.6.2) from Definition 4.6.1.
Then the set Γ/ ≡ has a natural structure of groupoid having
Γ0 / ≡ as unit set with respect to the following rules: α
b, βb : Γ/ ≡−→
b
Γ/ ≡ are defined by α
b(b
x) = H(α(x)), β(b
x) = H(β(x)), x
b · yb =
(xH(β(x))).(yH(β(y))) = x
cy = (xy)H(β(xy)) ⇐⇒ β(x) = α(y) and
the inverse of x
b = xH(β(x)) is defined by (b
x)−1 = x−1 .H(β(x−1 )) =
−1
x .H(α(x)).
b Γ0 / ≡) is be called the quotient groupoid
The groupoid (Γ/ ≡, α
b, β;
of the groupoid Γ relative to normal subgroupoid H and will be denoted
by Γ/H. The canonical map p : Γ −→ Γ/H defined by p(x) = x
b for all
x ∈ Γ is a morphism of groupoids, called the canonical morphism of the
groupoid Γ onto the quotient groupoid Γ/H.
For more details concerning the quotient groupoids, see [6], [19], [22].
4.7. The disjoint union of a family of groupoids.
If {Γi |i ∈ I} is a disjoint family of groupoids, let Γ = ∪i∈I Γi and
Γ(2) = ∪i∈I Γi,(2) . Here, two elements x, y ∈ Γ may be composed iff they
lie in the same groupoid Γi and they are composable in Γi . This groupoid
is called the disjoint union of the groupoids Γi , i ∈ I and is denoted by
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
83
⨿
i∈I Γi . The unit set of this groupoid is Γ0 = ∪i∈I Γi,0 , where Γi,0 is the
unit set of Γi .
In particular, the disjoint union of the groups Gi , i ∈ I is a groupoid,
which be called the groupoid associated to family of groups Gi , i ∈ I (for
this groupoid, the isotropy groups over ei ∈ Gi is the group Gi and G0 =
{ei |i ∈ I} where ei is the unity of Gi .
For example, let GL(n; R) the general linear group of order n over
R. We consider the family of groups {GL(k; R)|1 ≤ k ≤ n}. Then the
n
∪
disjoint union
GL(k; R) is a groupoid (in fact, a group bundle), called
k=1
the general linear groupoid of order n over R and denoted by GL(n; R).
Let the map f : GL(n; R) → R∗ defined by f (A) = det(A), for all
matrix A ∈ GL(n; R). We have that f is a surjective groupoid morphism
from the groupoid GL(n; R) onto the multiplicative group of reel numbers
different from zero. Then Kerf = {A ∈ GL(n; R)|det(A) = 1} is a wide
subgroupoid of GL(n; R), called the special linear groupoid of order n over
R and denoted by SL(n; R).
4.8. Projective limit of a projective system of groupoids
A partially ordered set (I, ≤) is be called directed on the right, if for
all i, j ∈ I there exists k ∈ I such that i ≤ k and j ≤ k.
Definition 4.8.1. A projective system is a triple (I, (Gi )i∈I , (fij )i,j∈I ),
where (I, ≤) is a partially ordered set directed on the right, (Gi )i∈I is a
family of groupoids and (fij )i,j∈I is a family of groupoid morphisms such
that the following conditions hold :
(i) fij : Gj → Gi is a groupoid morphism for all i ≤ j;
(ii) fii = 1Gi ;
(iii) if i ≤ j ≤ k, then fik = fij ◦ fjk , i.e. the diagrams :
fij
→
Gj
(4.8.1)
fjk
↖
Gk
Gi
↗
fik
are comutative.
A projective system of groupoids will be denoted by (Gi , fij ), i, j ∈ I.
Example 4.8.1. On a set I, we introduce the equality relation and (I, =)
becomes a ordered set directed on the right. If (Gi )i∈I is a family of
groupoids , then (Gi , IdGi )i∈I is a projective system of groupoids.
Definition 4.8.2. By the projective limit of a projective system of
groupoids (Gi , fij ), i, j ∈ I we mean a pair (P, (πi )i∈I ), where P is a
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GHEORGHE IVAN
groupoid and (πi )i∈I is a family of groupoid morphisms πi : P → Gi such
that for all i, j ∈ I, i ≤ j the folloving diagrams :
Gj
(4.8.2)
πj
↖
fij
→
P
Gi
↗ πi
are commutative, i.e. fij ◦ πj = πi .
We denote P = lim(Gi , fij ) or P = lim.proj.(Gi , fij ).
←−
∏ For a projective system of groupoids (Gi , fij ), i, j ∈ I, let G =
i∈I Gi the direct
∏ product of the groupoids Gi , i ∈ I with the projections pj : G = i∈I Gi → Gj , x = (xi∏
)i∈I → pj (x) = xj , j ∈ I. Let
(4.8.3)
P = {x = (xi )i∈I ∈ i∈I Gi |pi (x) = fij (pj (x)), (∀)i, j ∈
I, i ≤ j}.
Proposition 4.8.1. If (Gi , fij ), i, j ∈ I is a projective system of
groupoids, then the set P given by (4.8.3) is a groupoid and the map
πj : P → Gj , x = (xi )i∈I → πj (x) = xj is a morphism of groupoids,
for all j ∈ I.
∏
Proof. It suffices to prove that P is a subgoupoid of G = i∈I Gi . Let
x = (xi )i∈I and y = (yi )i∈I two elements from P and suppose that the
product xy is defined in G. Then the products xi yi is defined in Gi , for
each i ∈ I and for all i, j ∈ I such that i ≤ j we have pi (x) = fij (pj (x))
and pi (y) = fij (pj (y)), i.e. xi = fij (xj ) and yi = fij (yj ). It follows
xi yi = fij (xj )fij (yj ) = fij (xj yj ), since fij is a groupoid morphism.
Hence, pi (xy) = fij (pj (xy)), i.e. xy ∈ P.
−1
Also, if x = (xi )i∈I ∈ P, then x−1 = (x−1
) =
i )i∈I , pi (x
−1
−1
−1
−1
−1
pi ((xi )i∈I ) = xi and fij (pj (x )) = fij (xj ) = (fij (xj )) . From
xi = fij (xj ) results x−1
= (fij (xj )−1 . Then pi (x−1 ) = fij (pj (x−1 )) and
i
−1
x ∈ P. Therefore, the conditions (i) and (ii) from Def.1.3 are verified.
Hence P is a subgroupoid of G. . It is easy to verify that πj : P → Gj is
a groupoid morphism.
Theorem 4.8.1. Let (Gi , fij ), i, j ∈ I be a projective
system of
∏
G
groupoids. Let the groupoid P = {x = (xi )i∈I ∈
i |pi (x) =
i∈I
fij (pj (x)), (∀)i, j ∈ I, i ≤ j} and for all i ∈ I the groupoid morphism
πi : P → Gi , πi ((xi )i∈I ) = xi . Then lim.proj.(Gi , fij ) = (P, (πi )i∈I ).
Proof. It remains to verify that the diagrams (4.8.2) are commutative.
Let i, j ∈ I, i ≤ j and x = (xi )i∈I ∈ P. Then (fij ◦ πj )(x) = fij (πj (x)) =
fij (xj ) = fij (pj (x)) and πi (x) = xi = pi (x). On the other hand , pi (x) =
fij (pj (x)), since x ∈ P and we have πi (x) = (fij ◦ πj )(x), (∀)x ∈ P,
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
85
i.e. fij ◦ πj = πi . Hence , (P, (πi )i∈I ) is the projective limit of the given
projective system of groupoids.
Theorem 4.8.2. Let (P, (πi )i∈I ) the projective limit of the system
(Gi , fij ), i, j ∈ I. Then the pair (P, (πi )i∈I ) has the property (PU ), called
the universal property of the projective limit, where :
(PU )
for each pair(E, (ui )i∈I ) formed by a groupoid E and a family of groupoid morphisms ui : E → Gi , i ∈ I such that fij ◦ uj =
ui , (∀)i, j ∈ I, i ≤ j there exists an unique morphism of groupoids
φ : E → P such that the following diagrams :
φ
→
E
(4.8.4)
ui
↘
Gi
P
↙ πi
are commutative, i.e. πi ◦ φ = ui , for all i ∈ I.
Proof. We define the map φ : E → P by φ(a) = (ui (a))i∈I , (∀)a ∈ E.
We have pi (φ(a)) = pi ((ui (a))i∈I ) = ui (a) and fij (pj (φ(a))) = fij (uj (a)).
Then pi (φ(a)) = fij (pj (φ(a)), since ui = fij ◦ uj . Hence φ(a) ∈ P and φ is
well-defined. It easy to verify that φ is a morphism. We have (πi ◦ φ)(a) =
πi (φ(a)) = πi ((ui (a))i∈I ) = ui (a), (∀)a ∈ E i.e. πi ◦ φ = ui , for all
i ∈ I. Hence, the diagram (4.8.4) is commutative. If there exists another
morphism φ′ : E → P such that πi ◦ φ′ = ui , (∀)i ∈ I, then φ′ = φ.
Proposition 4.8.2. Let (Gi , IdGi ), i ∈ I the projective
∏system associated
to the groupoids Gi , i ∈ I. Then lim.proj.(Gi , IdGi ) = i∈I Gi .
Theorem 4.8.3. Let (Gi , fij ), i, j ∈ I be a projective system of groupoids
and (P, (πi )i∈I ) its projective limit. Let P ′ be a groupoid and the groupoid
morphisms πi′ : P ′ → Gi , i ∈ I. If the pair (P ′ , (πi′ )i∈I ) verify the universal
property of the projective limit, then P ′ and P are groupoids isomorphes.
Proof. Since (P, (πi )i∈I ) has the property (PU ), it follows that there exists
an unique morphism φ : P ′ → P such that πi ◦ φ = πi′ , i ∈ I.
Similarly, from the fact that (P ′ , (πi′ )i∈I ) has the property (PU ), it
follows that there exists an unique morphism ψ : P → P ′ such that
πi′ ◦ ψ = πi , i ∈ I.
From πi ◦ φ = πi′ and πi′ ◦ ψ = πi we obtain that πi ◦ (φ ◦ ψ) = πi and
′
πi ◦ (ψ ◦ φ) = πi′ . On the other hand , πi′ ◦ IdP = πi′ and πi ◦ IdP ′ = πi .
Using the unicity of the morphisms from the diagrams (4.8.4), results
φ ◦ ψ = IdP and ψ ◦ φ = IdP ′ . Then φ is an isomorphism of groupoids.
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GHEORGHE IVAN
Remark 4.8.1. From the Theorems 4.8.1 and 4.8.3 follows that the
projective limit of a projective system of groupoids is unique, except an
isomorphism of groupoids.
4.9. Inductive limit of a inductive system of groupoids
Definition 4.9.1. An inductive system of groupoids is defined to be a
triple (I, (Gi )i∈I , (fij )i,j∈I ), where (I, ≤) is a partially ordered set directed
on the right , (Gi )i∈I is a family of groupoids and (ij )i,j∈I is a family of
groupoids morphisms such that the following conditions hold:
(i) fij : Gi → Gj is a groupoid morphism for all i ≤ j;
(ii) fii = 1Gi ;
(iii) if i ≤ j ≤ k then fik = fjk ◦ fij , i.e. the diagrams:
fik
→
Gi
(4.9.1)
fij
↘
Gj
Gk
↗
fjk
are comutative.
An inductive system of groupoids will be denoted (Gi , fij ), i, j ∈ I.
Example 4.9.1. If (Gi )i∈I is a family of groupoids , then (Gi , IdGi )i∈I
is an inductive system of groupoids.
Example 4.9.2. Let the family of groups (Gi )i∈N∗ , where Gi = GL(i; R)
for 1 ≤ i ≤ n − 1 and Gi = GL(n; R) for i ≥ n.
For each 1 ≤ i < j ≤ n, let the map fij : Gi = GL(i; R) → Gj =
GL(j; R) given by
( A ∈ GL(i;
) R) → fij (A) = B ∈ GL(j; R), where B is
A
O1
the block-matrix
, with O1 the zero matrix of type i × (j − i),
O2 Ij−i
O2 the zero matrix of type (j − i) × i and Ij−i the identity matrix of order
j − i. We have that fij , i, j ∈ N∗ is a morphism of groups. For each i ∈ N∗
we take fii = IdGL(i;R) .
We obtain the sequence of groups and morphisms of groups:
f12
f23
GL(1; R) −→ GL(2; R) −→ ...
fn−2,n−1
−→
GL(n − 1; R)
fn−1,n
−→
Id
GL(n; R) −→ ...
Then (GL(i; R), fij ), i, j ∈ N∗ is an inductive system of groupoids.
Example 4.9.3. Let (Hi )i∈N∗ an increasing sequence of G0 - subgroupoids of a G0 - groupoid G, i.e. Hn ⊆ Hn+1 , (∀)n ≥ 1. Then
the inclusion maps ωi,j : Hi → Hj are morphisms of groupoids and
(Hi , ωij ), i, j ∈ N∗ is an injective system of groupoids.
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
87
Definition 4.9.2. By the inductive limit of an inductive system of
groupoids (Gi , fij ), i, j ∈ I, we mean a pair (E, (ui )i∈I ), where E is a
groupoid and (ui )i∈I is a family of groupoid morphisms ui : Gi → E such
that for all i, j ∈ I, i ≤ j the following diagrams :
Gi
(4.9.2)
ui
↘
fij
→
E
Gj
↙ uj
are commutative, i.e. uj ◦ fij = ui .
We denote E = lim(Gi , fij ) or E = lim.ind.(Gi , fij ).
−→
For an inductive
⨿ system of groupoids (Gi , fij ), i, j ∈ I consider the
disjoint union G = i∈I Gi of the family (Gi )i∈I . Then G is a groupoid
(we assume that the Gi are pairwise disjoint) ; if this is not the case, then
∪
we take to form the disjoint union
({i} × Gi ) . Also, the canonical
i∈I
⨿
injections σi : Gi → G =
Gi , xi → σi (xi ) = xi , i ∈ I are injective
i∈I
morphisms of groupoids. ⨿
On the groupoid
{ G = i∈I Gi we define a binary relation ”∼” by:
x ∈ Gi , y ∈ Gj for somei, j ∈ Iand
(4.9.3) x∼y ⇐⇒
(∃)k ∈ Isuch thati ≤ k, j ≤ kandfik (x) = fjk (y)
Proposition 4.9.1.([5]) The binary relation ”∼” defined on groupoid G =
⨿
Gi is a congruence on G.
i∈I
(⨿
)
We denote by E =
i∈I Gi /∼ the quotient groupoid determined
by the congruence ”∼” on G. Then p : G → E, x → p(x) = [x] is a
morphism of groupoids.
⨿
p
σi
We have Gi −→
G = i∈i Gi −→ E = G/ ∼ and ui = p◦σi : Gi → E
is a groupoid morphism, for all i ∈ I.
Theorem 4.9.1. Let (Gi , fij ), i, j ∈ I be an inductive system of groupoids,
⨿
Gi / ∼ and the groupoid morphisms ui : Gi → E,
the groupoid E =
i∈I
where ui = p ◦ σi , (∀)i ∈ I. Then lim.ind.(Gi , fij ) = (E, (ui )i∈I ).
Proof. It remains to verify that the diagrams (4.9.2) are commutative.
Let i, j ∈ I, i ≤ j and xi ∈ Gi . Then ui (xi ) = (p ◦ σi )(xi ) = p(σi (xi )) =
[σi (xi )] = [xi ]. In the other hand , (uj ◦ fij )(xi ) = (p ◦ σj )(fij (xi )) =
p(σj (fij (xi ))) = [σj (fij (xi ))] = [fij (xi )]. We prove that xi ∼ fij (xi ).
From hypothesis, xi ∈ Gi , fij (xi ∈ Gj and for i, j ∈ I there exists k ∈ I
88
GHEORGHE IVAN
such that i ≤ k, j ≤ k. We have fik (xi ) = fjk (fij (xi )), since fik = fjk ◦fij .
It follows that xi ∼ fij (xi ) and hence [xi ] = [fij (xi )]. Therefore, ui (xi ) =
(uj ◦ fij )(xi ), for all xi ∈ Gi . Thus, uj ◦ fij = ui . Hence , (E, (ui )i∈I ) is
the inductive limit of the given inductive system of groupoids.
⨿
Theorem 4.9.2. Let (E = i∈I Gi | ∼, (ui )i∈I ) the inductive limit of the
inductive system of groupoids (Gi , fij ), i, j ∈ I. Then (E, (ui )i∈I ) has the
property (PU ), called the universal property of the inductive limit,
where:
(PU )
for each pair (F, (vi )i∈I ) formed by a groupoid F and a
family of groupoids morphisms vi : Gi → F, i ∈ I such that vj ◦ fij =
vi , (∀)i, j ∈ I, i ≤ j there exists an unique morphism of groupoids
ω : E → F such that the following diagrams :
ω
→
E
(4.9.4)
ui
↖
Gi
F
↗ vi
are commutative , i.e. ω ◦ ui = vi , for all i ∈ I.
Proof. For [x] ∈ E there exists i ∈ I such that x ∈ Gi and we consider the
morphism vi : Gi → F. We define the map ω : E → F by ω([x]) = vi (x),
where x ∈ Gi . The map ω is well- defined. Indeed, if y ∈ [x] then y ∈ Gj
and there exist k ∈ I, i ≤ k, j ≤ k such that fik (x) = fjk (y). We prove that
vi (x) = vj (y). From hypothesis, we have vk ◦ fik = vi and vk ◦ fjk = vj .
Then vi (x) = (vk ◦ fik )(x) = vk (fik (x)) = vk (fjk (y)) = (vk ◦ fjk )(y) =
vj (y). It easy to verify that ω is a morphism. For xi ∈ Gi we have
(ω ◦ui )(xi ) = ω(ui (xi )) = ω((p◦σi )(xi )) = ω([xi ]) = vi (xi ), i.e. ω ◦ui = vi
for all i ∈ I. Hence, the diagram (4.9.4) is commutative. If there exists
another morphism ω ′ : E → F such that ω ′ ◦ui = vi , (∀)i ∈ I, then ω ′ = ω.
Theorem 4.9.3. Let (Gi , fij ), i, j ∈ I be an injective system of groupoids
and (E, (ui )i∈I ) its injective limit. Let E ′ be a groupoid and u′i : Gi →
E ′ , i ∈ I be a family of groupoid morphisms. If the pair (E ′ , (u′i )i∈I ) verify
the universal property of the injective limit, then the groupoids E ′ and E
are isomorphic.
Proof. The proof is similar with that given of Theorem 4.8.3.
Remark 4.9.1. From the Theorems 4.9.1 and 4.9.3 follows that the
inductive limit of an injective system of groupoids is unique, except an
isomorphism of groupoids.
Proposition 4.9.2. Let (Gi , IdGi )i∈I the inductive system
associated to
⨿
the groupoids Gi∈I , i ∈ I. Then lim.ind.(Gi , IdGi ) = i∈I Gi .
ALGEBRAIC CONSTRUCTIONS OF BRANDT GROUPOIDS
89
Proposition 4.9.3. Let the inductive system of groupoids (GL(i; R), fij ),
i, j ∈ N∗ . Then lim.ind(GL(i; R), fij ) ∼
= GL(n; R).
∗
Proposition 4.9.4. Let (Hi )i∈N an increasing sequence of G0 subgroupoids of a G0 -groupoid G and ωi,j : Hi → Hj the inclusion maps.
∗
Then the inductive
⨿ limit of the inductive system (Hi , ωij ), i, j ∈ N is the
disjoint union i∈N∗ ({i} × Hi ) of the groupoids Hi , i ∈ N∗ .
References
[1] H. Brandt, Uber eine Verallgemeinerung der Gruppen-begriffes. Math. Ann.,
96, 1926, 360-366.
[2] R. Brown, From Groups to Groupoids ; a brief survey. Bull. London Math.
Soc., 19, 1987, 113-134.
[3] R. Brown, Topology : Geometric Account of General Topology, Homotopy
Types and the Fundamental Groupoid. Hal. Press, New York, 1988.
[4] A. Coste, P. Dazord and A. Weinstein, Groupoides symplectiques. Publ.
Dept. Math. Lyon, 2/A, 1987,1-62.
[5] M. Degeratu and Gh. Ivan, Inductive limits of Lie groupoids. Proceed.
oh the Fifth International Workshop on Differential Geometry and Its Appl.,
Timişoara - Romania, September 18- 22, 2001, to appear.
[6] B. Dumons and Gh. Ivan, Introduction à la théorie des groupoides.
Dept.Math. Univ. Poitiers(France),URA,C.N.R.S. D1322,nr.86, 1994.
[7] C. Ehresmann, Oéuvres complétes . Parties I.1, I.2. Topologie algébrique et
géométrie différentielle .Dunod, Paris,1950.
[8] C. Ehresmann, Oéuvres complétes et commentées. A. Ehresmann, ed., Suppl.
Cahiers Top. Géometrie Diff., Amiens, 1980-1984.
[9] G. Galego, L. Gualandri, G. Hector and A. Reventos, Groupoides riemanniens. Publicaciones Math., Vol.33, 1989.
[10] A. Grothendieck, Techniques de construction et théorèmes d’ existence en
géométrie algébrique III: préschemas quotients. Séminaire Bourbaki , 13-e année
1960/61, no. 212, 1961.
[11] P. Hahn, Haar measure for measure groupoids. Trans. Amer. Math. Soc.,
242, 1978, 1-33.
[12] P. J. Higgins, Notes on categories and groupoids. Von Nostrand Reinhold,
London,1971.
[13] Gh. Ivan, Cayley Theorem for Monoidoids. Glasnick Matematicki, Vol.
31(51), 1996, 73-82.
[14] Gh. Ivan, Strong Morphisms of Groupoids. Balkan Journal Geometry and
Its Appl., Vol. 4, No. 1, 1999, 91-102.
[15] Gh. Ivan, Lattices of Ehresmann Subgroupoids. Anal. Univ. Timişoara,
Ser. Matematica-Informatica, Vol. 37, 2, 1999, 69-82.
[16] Gh. Ivan and M. Puta, Groupoids. Lie-Poisson Structures and Quantization. Proceed. International Conf. on Group Theory , Timişoara, September
17-21, 1992. Anal. Univ. Timişoara, Seria Matematică , 1993, 117-123.
90
GHEORGHE IVAN
[17] M. Ivan, On the Symmetric Groupoid. to appear.
[18] M. V. Karasev, Analogues of objects of Lie groups theory for nonlinear
Poisson brackets. Math. U.S.S.R. Izv., 26, 1987, 497-527.
[19] S. Keel and S. Mori, Quotients by Groupoids. Ann. of Math. (2), 145,
1997, 193-213.
[20] P. Liebermann , Lie Algebroids and Mechanics. Archivum Math. (BRNO),
T. 32, 1996, 147-192.
[21] A. Loewy, Neue elementare Begrundung und Erweiterung der Galoisschen
Theorie. S.-B. Heidelberger Akad. Wiss. Math. Nat. Kl.1925, 1927, Abh. 7.
[22] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry.
London Math. Soc., Lectures Notes Series, 124, Cambridge Univ.Press.,1987.
[23] K. Mackenzie, Classification of principal bundles and Lie groupoids with
prescribed gauge group bundle. J. Pure Appl. Algebra, 58, 1989, 181-208.
[24] G. W. Mackey, Ergodic theory, groups theory and differential geometry.
Proc. Nat. Acad. Sci., USA,50, 1963, 1184-1191.
[25] G. W. Mackey, Ergodic theory and virtual groups. Math. Ann., t. 166,
1966, 187-207.
[26] K. Mikami and A. Weinstein, Moments and reduction for symplectic
groupoid actions. Publ. RIMS Kyoto Univ., 42, 1988, 121-140.
[27] V. Popuţa and A. Popuţa , Cosets in Brandt Groupoids. Proc. 9-th Symposium of Math. and Its Appl., ” Politehnica ” University of Timişoara, November
2-3, 2001, to appear.
[28] M. Puta, Hamiltonian mechanics and geometric quantization. Kluwer,1993.
[29] A. Ramsey, Virtual groups and group actions. Advances in Math., 6, 1971,
253-322.
[30] J. Renault, A groupoid approach to C ∗ -algebras. Lectures Notes Series, 793,
Springer,1980.
[31] A. Weinstein, Symplectic groupoids and Poisson manifolds. Bull. Amer.
Math. Soc., 16, 1987, 101-104.
[32] A. Weinstein, Groupoids: Unifying Internal and External Symmetries. Notices Amer. Math. Soc., 43, 1996, 744 - 752.
[33] J. J. Westman, Harmonic analysis on groupoids. Pacific J. Math., 27, 1968,
621-632.
West University of Timişoara, Department of Mathematics , 4, Bd. V.
Pârvan, 1900, Timişoara, Romania
E-mail address: [email protected]