Strong Morphisms of Groupoids

Gh. Ivan

Abstract

We reffer to the groupoids in the sense of Ehresmann. The aim of this paper is to give some various topics of strong morphisms of groupoids.

Mathematics Subject Classification: 20L13; 20L99: 18B40 Key words: groupoid morphism, normal subgroupoid, strong morphism of groupoids

Introduction

The concept of groupoid in the sense of Ehresmann is a natural generalization of the algebraic notion of groupoid introduced by H. Brandt in the paper: Uber eine Verallgemeinerung der Gruppenbegriffe, Math. Ann., 96 (1926), 360-366.

The notions of topological and differentiable groupoids has been introduced by Ehresmann in 1950 in his paper on connections ( cf. [4] )

Many authors have investigated the Lie groupoids ( in particular, symplectic groupoids) in connection with their applications in differential geometry, symplectic geometry, Poisson geometry, quantum mechanics ergodic theory , geometric quantization and gauge theories ( cf. [1], [11] - [15], [17] - [20] ).

Recent applications of Lie groupoids endowed with supplementary structures have also contributed to a renewed interest in these studies.

In this paper we study a special case of groupoid morphism, namely: strong morphism of Ehresmann groupoids.

Other special morphisms of groupoids are the following:

- similar morphisms of Brandt groupoids ( these morphisms are used in [7] for construct a cohomology theory of Brandt groupoids which extends the usual cohomology theory of groups;

- pullback, fibrewise injective ( resp., surjective, bijective ) and piecewise injective ( resp., surjective, bijective ) morphism of groupoids ( for various topics concerning these special morphisms see [9] ).

Balkan Journal of Geometry and Its Applications, Vol.4, No.1, 1999, pp. 91-102 \begin{matrix}c \\ ◯ \end{matrix}Balkan Society of Geometers, Geometry Balkan Press

1 Morphisms of groupoids

In this section we construct the category of Ehresmann groupoids and some important properties concerning the morphisms of groupoids are given. Definition 1.1. ([15]) A groupoid (in the sense of Ehresmann) Γ over Γ_{0} or groupoid with the base Γ_{0}, is a pair (Γ; Γ_{0}) of sets equipped with:

(i) two surjections α, β : Γ ⟶ Γ_{0}, called the source and the target map; (ii) a (partial) composition law μ : Γ_{(2)} ⟶ Γ, (x, y) ⟶ μ(x, y) = x · y = xy, with

domain Γ_{(2)} = {(x, y) ∈ Γ × Γ | β(x) = α(y)}; (iii) an injection ϵ : Γ_{0} ⟶ Γ, u ⟶ ϵ(u) = ũ, called the inclusion map; (iv) a map i : Γ ⟶ Γ, x ⟶ i(x) = x^{−1}, called the inversion map. These maps must satysfy the following algebraic axioms generalizing those of

groups:

(G1) (associative law) For arbitrary x, y, z ∈ Γ the triple product (xy)z is defined iff x(yz) is defined. In case either is defined, we have (xy)z = x(yz) ; hence, the triple xyz is defined whenever β(x) = α(y) and β(y) = α(z).

(G2) (identities) For each x ∈ Γ we have (ϵ(α(x)), x) ∈ Γ_{(2)}; (x, ϵ(β(x)) ∈ Γ_{(2)} and ϵ(α(x)) · x = x · ϵ(β(x)) = x

(G3) (inverses) For each x ∈ Γ we have (x, i(x)) ∈ Γ_{(2)}; (i(x), x) ∈ Γ_{(2)} and x · i(x) = ϵ(β(x)), i(x) · x = ϵ(α(x)) △

Every group G with e as unity, is a groupoid over G_{0} = {e}. We denote a groupoid Γ over Γ_{0} by (Γ, α, β, ϵ, i, μ; Γ_{0}) or (Γ, α, β; Γ_{0}) or (Γ; Γ_{0}). For each u ∈ Γ_{0}, the set Γ_{u} = α^{−1}(u) (resp. Γ^{u} = β^{−1}(u)) is called the α-fibre

(resp. β-fibre) of Γ over u ∈ Γ_{0} and if u, v ∈ Γ, we will write Γ_{u}^{v} = Γ_{u} ∩ Γ^{v}.

A groupoid Γ over Γ_{0} such that Γ_{0} is a subset of Γ is called Γ_{0}-groupoid or Brandt groupoid.

We summarize some properties of these mappings obtained from definitions. Proposition 1.1. Let Γ be a groupoid over Γ_{0}. The following assertions hold:

(i) α ∘ ϵ = β ∘ ϵ = Id_{Γ_{0}}. (ii) α(xy) = α(x) and β(xy) = β(y) for all (x, y) ∈ Γ_{(2)}. (iii) ϵ(u) · ϵ(u) = ϵ(u) for each u ∈ Γ_{0}. (iv) Let u, v ∈ Γ_{0}. We have:

(a) if (x, ϵ(u)) ∈ Γ_{(2)} such that x · ϵ(u) = x then ϵ(u) = ϵ(β(x)).

(b) if (ϵ(v), x) ∈ Γ_{(2)} such that ϵ(v) · x = x then ϵ(v) = ϵ(α(x)).

(v) For all x ∈ Γ we have β(x^{−1}) = α(x) and α(x^{−1}) = β(x) (vi) For u ∈ Γ_{0} we have (ϵ(u))^{−1} = ϵ(u). (vii) α ∘ i = β, β ∘ i = α and i ∘ i = Id_{Γ}. (viii) For each u ∈ Γ_{0}, the set Γ(u) = α^{−1}(u) ∩ β^{−1}(u) is a group under the

restriction of the partial multiplication (this group is be called the isotropy group at u of the groupoid Γ).

(ix) In the case Γ_{0} ⊆ Γ, we have:

(a) ϵ(Γ_{0}) = Γ_{0}.

(b) ϵ(u) = u, for each u ∈ Γ_{0}. △ In view of Proposition 1.1., the element ϵ(α(x)) (resp. ϵ(β(x)) ) is the left unit

(resp., right unit) of x ∈ Γ. The subset ϵ(Γ_{0}) is be called the unity set of Γ. Definition 1.2. (a) A groupoid Γ over Γ_{0} is said to be transitive if the map α × β : Γ ⟶ Γ_{0} × Γ_{0}, given by (α × β)(x) = (α(x), β(x)), (∀)x ∈ Γ is surjective.

(b) By group bundle we mean a groupoid Γ over Γ_{0} such that α(x) = β(x) for each x ∈ Γ. Moreover, a group bundle is the union of its isotropy groups Γ(u) = α^{−1}(u), u ∈ Γ_{0} (here two elements may be composed iff they lie in the same fiber.) △

If (Γ, α, β; Γ_{0}) is a groupoid over Γ_{0}, then Is(Γ) = {x ∈ Γ | α(x) = β(x)} is a group bundle, called the isotropy group bundle associated to Γ. It is easy to see that ϵ(Γ_{0}) ⊆ Is(Γ). Proposition 1.2. If (Γ, α, β; Γ_{0}) is a groupoid, then the following assertions hold:

(i) (cancellation law) If x · z_{1} = x · z_{2} (resp., z_{1} · x = z_{2} · x) then z_{1} = z_{2}. (ii) If (x, y) ∈ Γ_{(2)} then (y^{−1}, x^{−1}) ∈ Γ_{(2)} and (x · y)^{−1} = y^{−1} · x^{−1}. (iii) The isotropy groups Γ(α(x)) and Γ(β(x)) are isomorphic. (iv) If Γ is transitive, then the isotropy groups of Γ are groups isomorphes.

Proof. (i) and (ii) These assertions follows from definitions.

(iii) We prove that the map φ : Γ(α(x)) ⟶ Γ(β(x)), a ⟶ φ(a) = x · a · x^{−1} is a isomorphism of groups.

(iv) It follows from (iii) and the fact that α × β : Γ ⟶ Γ_{0} × Γ_{0} is surjective. △ Example 1.1. (a) Nul groupoid. Any set B is a groupoid on itself with Γ = Γ_{0} := B, α = β = ϵ := id_{B} and every element is a unity, called it the nul groupoid. The multiplication is given by x · x := x for all x ∈ B.

(b) Coarse groupoid. If B is any non-empty set, then B × B is a groupoid over B with the rules:

α(x, y) := x; β(x, y) := y; ϵ(x) := (x, x), i(x, y) := (y, x)

and

μ((x, y), (y′, z)) := (x, z) iff y = y′.

The unit set of this groupoid, called it the coarse groupoid associated to B, is the diagonal Δ_{B} of the cartesian product B × B.

(c) Trivial groupoid. Let B be any non- empty set and G be a multiplicative group with e as unity. Construct a transitive groupoid Γ over B, called the trivial groupoid on B with group G, in the following way:

Γ := B × B × G; Γ_{0} := B; α(a, b, x) := a; β(a, b, x) := b; ϵ(b) := (b, b, e);

i(a, b, x) := (b, a, x^{−1}) and μ((a, b, x), (b′, c, y)) := (a, c, xy) iff b = b′.

For this groupoid we have

ϵ(Γ_{0}) = {(b, b, e) | b ∈ B} and Γ(b) = {(b, b, x) | x ∈ G},

which are identified with B resp. G.

If G = {e}, then we can identify B × B × G with the coarse groupoid associated to B.

(d) A vector bundle E ⟶^{π} M is a group bundle on M. Here Γ := E is the total space, Γ_{0} := M is the base space, α = β := π so that Γ_{(2)} := ⊎_{x∈M}E_{x} × E_{x} ( E_{x} is the fibre at x ) and the composition law is fibrewise addition.△

Other examples of groupoids are the following: the fundamental groupoid of a topological space ( see [6]), the disjoint union of a disjoint family of groupoids ( see [9] ) and the action groupoid ( see [13] ).

Definition 1.3. Let (Γ, α, β, ϵ, i, μ; Γ_{0}) and (Γ′, α′, β′, ϵ′, i′, μ′; Γ′_{0}) be two groupoids. A morphism of groupoids or groupoid morphism is a pair (f, f_{0}) of maps f : Γ ⟶ Γ′ and f_{0} : Γ_{0} ⟶ Γ′_{0} such that the following two conditions are satisfied:

(1)  f(μ(x, y)) = μ′(f(x), f(y)) for every (x, y) ∈ Γ_{(2)}
(2)  α′ ∘ f = f_{0} ∘ α and β′ ∘ f = f_{0} ∘ β.△

If Γ_{0} = Γ′_{0} and f_{0} = Id_{Γ_{0}}, we say that f is a Γ_{0}- morphism.△

Note that the condition (1) ensure that (f(x), f(y)) ∈ Γ′_{(2)}, i.e. μ′(f(x), f(y)) is defined whenever μ(x, y) is defined.

Applying Propositions 1.1 and 1.2 we obtain: Proposition 1.3. The groupoids morphisms preserve unities and inverses, i.e. f(ũ) = f̃_{0}(u), (∀)u ∈ Γ_{0} and f(x^{−1}) = (f(x))^{−1}, (∀)x ∈ Γ; in other words, we have: f ∘ ϵ = ϵ′ ∘ f_{0} and f ∘ i = i′ ∘ f △ Proposition 1.4. A pair (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid morphism iff the following condition holds:

(3) (∀)(x, y) ∈ Γ_{(2)} ⟹ (f(x), f(y)) ∈ Γ′_{(2)} and f(μ(x, y)) = μ′(f(x), f(y))

Proof. The condition (3) is a consequence of Definition 1.4. and Prop.1.3.

Conversely, let f : Γ ⟶ Γ′ which satisfy (3) and we define the map f_{0} : Γ_{0} ⟶ Γ′_{0} by f_{0}(u) = α′(f(ϵ(u))), (∀)u ∈ Γ_{0}. We prove that α′∘f = f_{0}∘α and β′∘f = f_{0}∘β.

Indeed, since (x, ϵ(β(x))) ∈ Γ_{(2)} it follows that (f(x), f(ϵ(β(x)))) ∈ Γ′_{(2)} and

f(x) · f(ϵ(β(x)))) = f(x · ϵ(β(x))) = f(x); but f(x) · ϵ′(β′(f(x))) = f(x);

⟹ ϵ′(β′(f(x))) = f(ϵ(β(x))) ⟹ α′(ϵ′(β′(f(x)))) = α′(f(ϵ(β(x))))

and applying Prop.1.1. we obtain succesivelly

β′(f(x)) = (f_{0} ∘ α)(ϵ(β(x))) ⟹ β′(f(x)) = f_{0}(β(x))

i.e. β′ ∘ f = f_{0} ∘ β. Similarly we prove that α′ ∘ f = f_{o} ∘ α.△ Example 1.2. (a) If (Γ, α, β, ϵ; Γ_{0}) is a groupoid, then (Id_{Γ}, Id_{Γ_{0}}) is a groupoid morphism.

(b) If (f, f_{0}) : (Γ, Γ_{0}) → (Γ′; Γ′_{0}) and (g, g_{0}) : (Γ′, Γ′_{0}) → (Γ′′, Γ′′_{0}) are groupoid morphisms, then the composition (g, g_{0})o(f, f_{0}) : (Γ, Γ_{0}) → (Γ′′, Γ′′_{0}) defined by (g, g_{0}) ∘ (f, f_{0}) = (g ∘ f, g_{0} ∘ f_{0}) is a groupoid morphism.△

If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid morphism, then for every u, v ∈ Γ_{0} we have:

f(Γ_{u}) ⊆ Γ′_{f_{0}(u)}; f(Γ^{v}) ⊆ (Γ′)^{f_{0}(v)} and f(Γ_{u}^{v}) ⊆ (Γ′)_{f_{0}(u)}^{f_{0}(v)}.

Then the restriction of f to Γ_{u}, Γ^{v}, Γ_{u}^{v} respectively, defines the groupoid morphisms

Γ_{u} ⟶ Γ′_{f_{0}(u)}; Γ^{v} ⟶ (Γ′)^{f_{0}(v)}, Γ_{u}^{v} ⟶ (Γ′)_{f_{0}(u)}^{f_{0}(v)},

denoted by f_{u}, f^{v} and f_{u}^{v}. Definition 1.4. A groupoid morphism (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is said to be isomorphism of groupoids if there exists a groupoid morphism (g, g_{0}) : (Γ′; Γ′_{0}) → (Γ, Γ_{0}) with (g, g_{0})o(f, f_{0}) = (id_{Γ}, id_{Γ_{0}}) and (f, f_{0})o(g, g_{0}) = (id_{Γ′}, id_{Γ′_{0}}). Two groupoids (Γ; Γ_{0}) and (Γ′; Γ′_{0}) are said to be isomorphic if there exists an isomorphism (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}).△ Proposition 1.5. Let (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) be a groupoid morphism. Then the following assertions hold:

(i) If f is injective ( resp., surjective ), then also is f_{0}.

(ii) (f, f_{0}) is an isomorphism iff the map f is bijective.

(iii) f(Is(Γ)) ⊆ Is(Γ′).

(iv) A groupoid morphism (f, f_{0}) such that f is surjective and f_{0} is injective ( in particular, every surjective Γ_{0}- morphism of groupoids ) preserve the isotropy group bundles, i.e. f(Is(Γ)) = Is(Γ′). Proof. (i) This follows imediately from Definition 1.3. and Proposition 1.4.

(ii) It is a consequence of Definition 1.4. and of the assertion (i).

(iii) Let x′ ∈ f(Is(Γ)). Then x′ = f(x) with x ∈ Is(Γ) and we have

α′(x′) = α′(f(x)) = f_{0}(α(x)) = f_{0}(β(x)) = β′(f(x)) = β′(x′),

since α(x) = β(x); hence x′ ∈ Is(Γ′). Therefore, f(Is(Γ)) ⊆ Is(Γ′).

(iv) It suffices to prove that Is(Γ′) ⊆ f(Is(Γ)). Let x′ ∈ Is(Γ′) i.e. x′ ∈ Γ′ such that α(x′) = β′(x′). For x′ ∈ Γ′ there exists x ∈ Γ such that x′ = f(x), since f is surjective. Then α′(f(x)) = β′(f(x)) and we obtain that f_{0}(α(x)) = f_{0}(β(x)). Hence, α(x) = β(x), since f_{0} is injective. Thus, x ∈ Is(Γ) and x′ ∈ f(Is(Γ)). Therefore, Is(Γ′) ⊆ f(Is(Γ)).△ Example 1.3. (a) Let (Γ, α, β, ϵ; Γ_{0}) be a groupoid and (Γ_{0} × Γ_{0}, α′, β′, ϵ′; Γ_{0}) the coarse groupoid associated to Γ_{0}.Then α × β : Γ ⟶ Γ_{0} × Γ_{0}, (α × β)(x) = (α(x), β(x)) is a Γ_{0}- morphism of the groupoid Γ into the coarse groupoid Γ_{0} × Γ_{0}.

b) Let (Γ, α, β, ϵ, i, μ; Γ_{0}) be a groupoid over Γ_{0} and X a set with the same cardinal as Γ_{0}, i.e. there exists a bijection φ from Γ_{0} to X. Then Γ has a canonical structure of a groupoid over X, that is (Γ, α′, β′, ϵ′, i′, μ′; X) is a groupoid over X where α′ := φoα; β′ := φoβ; ϵ′ := ϵoφ^{−1}; i′ := φoi; μ′ := μ.Moreover, (id_{Γ}, φ) : (Γ; Γ_{0}) → (Γ; X) is an isomorphism of groupoids.△ Example 1.4. (the induced groupoid) Let (Γ, α, β, ϵ; Γ_{0}) be a groupoid, X an abstract set and f : X ⟶ Γ_{0} a map from X to Γ_{0}. Then the set:

f^{∗}(Γ) = {(x, y, a) ∈ X × X × Γ | f(x) = α(a), f(y) = β(a)}

has a canonical structure of groupoid over X with respect to the following rules:

α^{∗}(x, y, a) := x; β^{∗}(x, y, a) := y; ϵ^{∗}(x) := (x, x, ϵ(f(x))); i^{∗}(x, y, a) := (y, x, i(a)),

and

μ^{∗}((x, y, a), (y′, z, b)) := (x, z, μ(a, b))

iff y = y′ and (a, b) ∈ Γ_{(2)}.

The groupoid (f^{∗}(Γ), α^{∗}, β^{∗}, ϵ^{∗}, μ^{∗}; Γ_{0}) is called the induced groupoid or the inverse image of Γ under f; it is denoted sometimes by f^{∗}(Γ).

If f^{∗}(Γ) is the induced groupoid of Γ under f : X ⟶ Γ_{0} then f_{Γ}^{∗} : f^{∗}(Γ) ⟶ Γ defined by f_{Γ}^{∗}(x, y, a) = a together with f define a groupoid morphism (f_{Γ}^{∗}, f) : (f^{∗}(Γ); X) ⟶ (Γ; Γ_{0}) and it is called the canonical morphism of an induced groupoid.△

2 Strong morphisms of groupoids

This section is dedicated to study of a particular type of groupoid morphisms, namely: the strong morphisms of groupoids. One of the most important results of strong morphisms is the correspondence theorem for subgroupoids ( resp., for normal subgroupoids ). Definition 2.1. A subgroupoid of a groupoid (Γ; Γ_{0}) is a pair (Γ′; Γ′_{0}) of subsets, where Γ′ ⊆ Γ, Γ′_{0} ⊆ Γ_{0} such that the following conditions are verified:

(i) α(Γ′) ⊆ Γ′_{0}; β(Γ′) ⊆ Γ′_{0}

(ii) for every x, y ∈ Γ′ such that the product x · y is defined implies that x · y ∈ Γ′, i.e. Γ′ is closed under the partial multiplication.

(iii) (∀)u ∈ Γ′_{0} ⟹ ϵ(u) ∈ Γ′

(iv) (∀)x ∈ Γ′ ⟹ x^{−1} ∈ Γ′.

A subgroupoid (Γ′; Γ′_{0}) of (Γ; Γ_{0}) is wide if Γ′_{0} = Γ_{0}. △ Definition 2.2. A normal subgroupoid of a groupoid (Γ; Γ_{0}) is a wide subgroupoid N of Γ such that: for any λ ∈ N and any x ∈ Γ such that β(x) = α(λ) = β(λ) we have x · λ · x^{−1} ∈ N.△ Example 2.1. (a) If (Γ; Γ_{0}) is a groupoid, then ϵ(Γ_{0}) = {ũ | u ∈ Γ_{0}} is a normal subgroupoid of Γ over Γ_{0}, called the nul subgroupoid of Γ.

(b) If (Γ; Γ_{0}) is a groupoid, then Is(Γ) = ⋃_{u∈Γ_{0}} Γ_{u}^{u} is a normal subgroupoid of Γ over Γ_{0}, called the inner subgroupoid of Γ.

(c) The kernel of a groupoid morphism (f.f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) defined by: Kerf = {x ∈ Γ | f(x) ∈ ϵ′(Γ′_{0})} is a normal subgroupoid of Γ over Γ_{0}.△ Proposition 2.1. Let (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) be a groupoid morphism. Then the following assertions hold:

(i) If (Ω′; Ω′_{0}) is a subgroupoid of (Γ′; Γ′_{0}), then (f^{−1}(Ω′); f_{0}^{−1}(Ω′_{0})) is a subgroupoid of (Γ; Γ_{0}).

(ii) If Ω′ is a normal subgroupoid of Γ′, then f^{−1}(Ω′) is a normal subgroupoid of Γ such that Kerf ⊆ f^{−1}(Ω). Proof. (i) We prove that (f^{−1}(Ω′); f_{0}^{−1}(Ω′_{0})) satisfies the conditions of Definition 2.1.

- α(f^{−1}(Ω′)) ⊆ f_{0}^{−1}(Ω′_{0}). Indeed, if u ∈ α(f^{−1}(Ω′)) it follows that u = α(x) with x ∈ f^{−1}(Ω′). Then f_{0}(u) = f_{0}(α(x)) = α′(f(x)) ∈ Ω′_{0}, since f(x) ∈ Ω′ and α′(Ω′) ⊆ Ω′_{0}. Hence, u ∈ f_{0}^{−1}(Ω′_{0}). Similarly, β(f^{−1}(Ω′)) ⊆ f_{0}^{−1}(Ω′_{0}).

- Let x, y ∈ f^{−1}(Ω′) such that x · y is defined, i.e. β(x) = α(y). It follows that f(x), f(y) ∈ Ω′ and β′(f(x)) = f_{0}(β(x)) = f_{0}(α(y)) = α′(f(y)); hence, f(x) · f(y) is defined in Γ′. Then f(x) · f(y) ∈ Ω′, since Ω′ is subgroupoid. Then f(x · y) ∈ Ω′, i.e. x · y ∈ f^{−1}(Ω′).Therefore, the condition (ii) of Definition 2.1. is verified.

- For every u ∈ f_{0}^{−1}(Ω′_{0}) we have ϵ(u) ∈ f^{−1}(Ω′). Indeed, f_{0}(u) ∈ Ω′_{0} and we have ϵ′(f_{0}(u)) ∈ Ω′, since Ω′ is subgroupoid. Then f(ϵ(u)) ∈ Ω′, i.e. ϵ(u) ∈ f^{−1}(Ω′).

- For every x ∈ f^{−1}(Ω′), we have x^{−1} ∈ f^{−1}(Ω′). Indeed, from f(x) ∈ Ω′ follows (f(x))^{−1} ∈ Ω′, since Ω′ is subgroupoid. Then f(x^{−1}) ∈ Ω′, i.e. x^{−1} ∈ f^{−1}(Ω′).

(ii) In view of (i) follows that f^{−1}(Ω′; Γ_{0}) is a subgroupoid of (Γ; Γ_{0}).

Let λ ∈ f^{−1}(Ω′) and x ∈ Γ such that β(x) = α(λ) = β(λ) and we prove that x · λ · x^{−1} ∈ f^{−1}(Ω′).

Indeed, we have f(λ) ∈ Ω′ and β′(f(x)) = f_{0}(β(x)) = f_{0}(α(λ)) = α′(f(λ)) and β′(f(x)) = f_{0}(β(x)) = f_{0}(β(λ)) = β′(f(λ)). From f(λ) ∈ Ω′, β′(f(x)) = α′(f(λ)) = β′(f(λ)) and the fact that Ω′ is normal in Γ′ follows f(x)·f(λ)·(f(x))^{−1} ∈ Ω′. Hence, f(x · λ · x^{−1}) ∈ Ω′, i.e. x · λ · x^{−1} ∈ f^{−1}(Ω′). Therefore, f^{−1}(Ω′) is normal.

- We have Kerf ⊆ f^{−1}(Ω′). Indeed, for x ∈ Kerf, we have f(x) = ϵ′(u′) with u′ ∈ Γ′_{0} and by the condition (iii) of Definition 2.1. follows ϵ′(u′) ∈ Ω′. Then f(x) ∈ Ω′, i.e. x ∈ f^{−1}(Ω′). △ Corollary 2.1. Let f : Γ ⟶ Γ′ be a Γ_{0}- groupoid morphism. Then the following assertions hold:

(i) If (Ω′; Ω′_{0}) is a subgroupoid of (Γ′; Γ_{0}), then (f^{−1}(Ω′); f_{0}^{−1}(Ω′_{0})) is a subgroupoid of (Γ; Γ_{0}).

(ii) If Ω′ is a normal subgroupoid of Γ′, then f^{−1}(Ω′) is a normal subgroupoid of Γ such that Kerf ⊆ f^{−1}(Ω). Proof. We apply the Proposition 2.1.△ Remark 2.1. If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid, then not always, Imf = {f(x) | x ∈ Γ} is a subgroupoid of Γ′. For example, let

Γ = {(0, 0); (0, 1); (1, 0); (1, 1)} = B × B

the coarse groupoid associated to set B = {0, 1} and let the map f : Γ ⟶ Z defined by f(0, 0) = 0; f(0, 1) = 1; f(1, 0) = −1; f(1, 1) = 0. W e denote by f_{0} : B ⟶ {0} the map defined by f_{0}(0) = 0 and f_{0}(1) = 0. We can prove easily the conditions of Definition 1.3. are satisfied for the pair (f, f_{0}) of the coarse groupoid Γ over B into the group additiv Z of entiers numbers over {0}, having Imf = {0, −1, 1} which is not a subgroup of Z. Hence Imf is not a subgroupoid. △ Definition 2.3. A strong morphism of groupoids or groupoid strong morphism is a groupoid morphism (f, f_{0}) : (Γ; Γ′) ⟶ (Γ′; Γ′_{0}) such that the following condition holds:

(4)  for every (f(x), f(y)) ∈ Γ′_{(2)} we have (x, y) ∈ Γ_{(2)}.△

Remark 2.2. The concept of strong morphism has considered by A. Ramsay ( cf.

[18] ) in the case of Brandt groupoids, called it true morphism of groupoids. △ Remark 2.3. If (f, f_{0}) is a strong morphism of groupoids, then

f_{u} : Γ_{u} ⟶ Γ′_{f_{0}(u)}; f^{v} : Γ^{v} ⟶ (Γ′)^{f_{0}(v)} and f_{u}^{v} : Γ_{u}^{v} ⟶ (Γ′)_{f_{0}(u)}^{f_{0}(v)}

are also strong morphisms of groupoids. △ Theorem 2.1. (i) If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid morphism such that the map f_{0} is injective, then (f, f_{0}) is a groupoid strong morphism.

(ii) Every Γ_{0}- morphism of groupoids f : Γ ⟶ Γ′ is a groupoid strong morphism. Proof. (i) We suppose that (f(x), f(y)) ∈ Γ′_{(2)}, with x, y ∈ Γ. Then

β′(f(x)) = α′(f(y))⟹(β′ ∘ f)(x) = (α′ ∘ f)(y)⟹(f_{0} ∘ β)(x) = (f_{0} ∘ α)(y)⟹

⟹f_{0}(β(x)) = f_{0}(α(y))⟹β(x) = α(y)(since f_{0} is injective)⟹(x, y) ∈ Γ_{(2)}.

Hence (f, f_{0}) is a groupoid strong morphism.

(ii) This is a consequence of (ii), since f_{0} = Id_{Γ_{0}}. △ Example 2.2. (i) The morphism α × β : Γ ⟶ Γ_{0} × Γ_{0}, given in Definition 1.2., is a groupoid strong morphism.

(ii) The canonical morphism (f_{Γ}^{∗}, f) of induced groupoid f^{∗}(Γ) of Γ by f : X ⟶ Γ_{0} is not a groupoid strong morphism. △ Proposition 2.2. Let (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) be a groupoid strong morphism. Then the following assertions hold:

(i) If (Ω; Ω_{0}) is a subgroupoid of (Γ; Γ_{0}), then (f(Ω); f_{0}(Ω_{0})) is a subgroupoid of (Γ′; Γ′_{0}). In particular, Imf is a subgroupoid of Γ′ over Imf_{0}.

(ii) If f is surjective and Ω is a normal subgroupoid of Γ, then f(Ω) is a normal subgroupoid of Γ′. Proof. (i) We have (α′(f(Ω)) ⊆ f_{0}(Ω_{0}). Indeed, for any u′ ∈ α′(f(Ω)) exists y′ ∈ f(Ω) such that u′ = α′(y′). For y′ ∈ f(Ω) exists y ∈ Ω such that f(y) = y′. Then u′ = α′(y′) = α′(f(y)) = f_{0}(α(y)); hence u′ ∈ f_{0}(Ω), since α(y) ∈ Ω_{0}. Similarly, β′(f(Ω)) ⊆ f_{0}(Ω_{0}).

- We have ϵ′(u′) ⊆ f(Ω), for all u′ ∈ f_{0}(Ω_{0}).Indeed, for u′ ∈ f_{0}(Ω_{0}) exists u ∈ Ω_{0} such that u′ = f_{0}(u) ⟹ ϵ′(u′) = ϵ′(f_{0}(u)) = f(ϵ(u)) ∈ f(Ω), since ϵ(u) ∈ Ω.

- Let x′, y′ ∈ f(Ω) such that x′ · y′ is defined.We prove that x′ · y′ ∈ f(Ω). Indeed, x′ = f(x), y′ = f(y) with x, y ∈ Ω. Since, x′ · y′ is defined it implies that (f(x), f(y)) ∈ Γ′_{(2)}, and we have (x, y) ∈ Γ_{(2)}, since f is a groupoid strong morphism. Hence x · y is defined. We have x · y ∈ Ω, since Ω is subgroupoid of Γ,and therefore x′ · y′ = f(x) · f(y) = f(x · y) ∈ f(Ω).

- For any x′ ∈ f(Ω) we have (x′)^{−1} ∈ f(Ω). Indeed, x′ = f(x), with x ∈ Ω ⟹ (x′)^{−1} = (f(x))^{−1} = f(x^{−1}) ∈ f(Ω), since x^{−1} ∈ Ω.

Therefore (f(Ω); f_{0}(Ω_{0})) is a subgroupoid of (Γ′; Γ′_{0}).

(ii) By (i) (f(Ω); Γ′_{0}) is a subgroupoid of (Γ′; Γ′_{0}), since f_{0} is surjective.

Let λ′ ∈ f(Ω) and x′ ∈ Γ′ such that β′(x′) = α′(λ) = β′(λ). We prove that x′ · λ′ · (x′)^{−1} ∈ f(Ω).

Indeed, λ′ = f(λ) with λ ∈ Ω and x′ = f(x) with x ∈ Γ, since f is surjective. From (f(x), f(λ)), (f(λ), (f(x))^{−1}) ∈ Γ′_{(2)}, it follows that (x, λ), (λ, x^{−1}) ∈ Γ_{(2)}, since f is a groupoid strong morphism. It follows that x · λ · x^{−1} is defined and x · λ · x^{−1} ∈ Ω, since Ω is normal in Γ. Hence, f(x · λ · x^{−1}) ∈ f(Ω) and

f(x) · f(λ) · f(x^{−1}) = f(x) · f(λ) · (f(x))^{−1} = x′ · λ′ · (x′)^{−1} ∈ f(Ω).

Thus, f(Ω) is a normal subgroupoid of Γ′. △ Corollary 2.2. Let f : Γ ⟶ Γ′ be a Γ_{0}- morphism of groupoids. Then the following assertions hold:

(i) If (Ω; Ω_{0}) is a subgroupoid of (Γ; Γ_{0}), then (f(Ω); f_{0}(Ω_{0})) is a subgroupoid of (Γ′; Γ_{0}). In particular, Imf is a subgroupoid of Γ′ over Imf_{0}.

(ii) If f is surjective and Ω is a normal subgroupoid of Γ, then f(Ω) is a normal subgroupoid of Γ′. Proof. We apply Theorem 2.1.(ii) and Proposition 2.2.△

If (Γ; Γ_{0}) is a groupoid, we denote by S(Γ; Γ_{0}) ( resp., N (Γ) ) the set of the subgroupoids ( resp., the normal subgroupoids ) of (Γ; Γ_{0}).

If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid morphism, we denote by S̃(Γ; Γ_{0}) ( resp., Ñ(Γ) ) the set of the subgroupoids ( resp., the normal subgroupoids ) of (Γ; Γ_{0}), which contains the kernel of f, i.e.: S(Γ; Γ_{0}) = {Ω | Ω is a subgroupoid of (Γ; Γ_{0}) such that Kerf ⊆ Ω} N (Γ) = {Ω | Ω is a normal subgroupoid of Γ such that Kerf ⊆ Ω}.

In view of Example 2.1.(a),(b),(c) we have that S(Γ; Γ_{0}), N (Γ) ), S̃(Γ; Γ_{0}) and Ñ(Γ) ) are nonempty sets. Theorem 2.2. (the correspondence theorem for subgroupoids) For any surjective strong morphism of groupoids (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}), there exists a bijection from the set S(Γ′; Γ′_{0}) of the subgroupoids of (Γ′; Γ′_{0}) to the set S(Γ; Γ_{0}) of the subgroupoids of (Γ; Γ_{0}). Proof. We take the maps

φ : S̃(Γ; Γ_{0}) ⟶ S(Γ′; Γ′_{0})

and

ψ : S(Γ′; Γ′_{0}) ⟶ S̃(Γ; Γ_{0}),

given by:

(5)  φ(Ω) = f(Ω), (∀)Ω ∈ S̃(Γ);

(6)  ψ(Ω′) = f^{−1}(Ω′), (∀)Ω′ ∈ S(Γ′).

By Proposition 2.2.(i), it follows that f(Ω) is a subgroupoid of Γ′, for all Ω ∈ S̃(Γ). Hence, φ is well-defined. Also, by Proposition 2.1.(i),it follows that f^{−1}(Ω′) is a subgroupoid of Γ, for all Ω′ ∈ S(Γ′). Hence, ψ is well-defined.

The maps φ and ψ given by (5) and (6) have the following properties:

(7)  ψ ∘ φ = Id_{_{S}(Γ)} and φ ∘ ψ = Id_{S(Γ′)}.

The equalities (7) are equivalently with:

f^{−1}(f(Ω)) = Ω, (∀)Ω ∈ S̃(Γ) and f(f^{−1}(Ω′)) = Ω′, (∀)Ω′ ∈ S(Γ′).

- (a) If x ∈ Ω, then f(x) ∈ f(Ω) and we have x ∈ f^{−1}(f(Ω)). Hence, Ω ⊆ f^{−1}(f(Ω)).

- (b) If x ∈ f^{−1}(f(Ω)), then f(x) ∈ f(Ω) and exists y ∈ Ω such that f(x) = f(y). We have f(x) · (f(y))^{−1} = ϵ′(f(y)). Therefore, f(x · y^{−1}) = ϵ′(f(y)) and we obtain that x · y^{−1} ∈ Kerf. Thus, x · y^{−1} = z, with z ∈ Kerf ⊆ Ω. Hence, x = z · y with y, z ∈ Ω and we have x ∈ Ω. Therefore, f^{−1}(f(Ω)) ⊆ Ω.

From (a) and (b), it follows the first equality of (7′).

- (c) If x′ ∈ f(f^{−1}(Ω′)), then x′ = f(x) ∈ f(Ω) with x ∈ f^{−1}(Ω′) and follows f(x) ∈ Ω′. Hence x′ ∈ Ω′. Therefore, f(f^{−1}(Ω′)) ⊆ Ω′.

- (d) If x′ ∈ Ω′, exists x ∈ Γ such that x′ = f(x), since f is surjective. Then x ∈ f^{−1}(Ω′), since f(x) ∈ Ω′.Therefore, x′ ∈ f(f^{−1}(Ω′)). Hence, Ω′ ⊆ f(f^{−1}(Ω′)).

From (c) and (d), it follows the second equality of (7′).

From (7), it follows that ψ is invertible. Hence, ψ is a bijection.△ Corollary 2.3. (the correspondence theorem for subgroupoids via a Γ_{0}- morphism) For any surjective Γ_{0}-morphism of groupoids f : Γ ⟶ Γ′, there exists a bijection from the set S(Γ′; Γ_{0}) of the subgroupoids of (Γ′; Γ_{0}) to set S(Γ; Γ_{0}) of the subgroupoids of (Γ; Γ_{0}). Proof. It is a consequence of Theorems 2.1.(ii) and 2.2. △

Applying the Propositions 2.1.(ii) and 2.2.(ii) we can prove similarly the following theorem. Theorem 2.3. (the correspondence theorem for normal subgroupoids) For any surjective strong morphism of groupoids (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}), there exists a bijection from the set N (Γ′) of the normal subgroupoids of (Γ′; Γ′_{0}) to the set Ñ(Γ) of the normal subgroupoids of (Γ; Γ_{0}) which contains Kerf.△ Corollary 2.4. (the correspondence theorem for normal subgroupoids via a Γ_{0}-morphism) For any surjective Γ_{0}-morphism of groupoids f : Γ ⟶ Γ′, there exists a bijection from the set N (Γ′) of the normal subgroupoids of (Γ′; Γ_{0}) to the set Ñ(Γ) of the normal subgroupoids of (Γ; Γ_{0}) which contains Kerf. Proof. It is a consequence of Theorems 2.1.(ii) and 2.3. △ Remark 2.3. (i) The Theorems 2.2 and 2.3. generalise the correspondence theorems for subgroups and normal subgroups by a surjective morphism of groups.

(ii) The Theorems 2.2 and 2.3. are not true for arbitrary surjective morphisms of groupoids.

(iii) If (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}) is a groupoid strong morphism, then (f̃, f̃_{0}) : (Γ; Γ_{0}) ⟶ (Imf; Imf_{0}) is a surjective strong morphism of groupoids, where f̃, f̃_{0} are given by f̃(x) = f(x), (∀)x ∈ Γ and f̃_{0}(u) = f_{0}(u), (∀)u ∈ Γ_{0}.△ Theorem 2.4. For any strong morphism of groupoids (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}), there exists a bijection from the set S(Imf; Imf_{0}) of the subgroupoids of (Imf; Imf_{0}) to the set S(Γ; Γ_{0}) of the subgroupoids of (Γ; Γ_{0}). Proof. We apply the Theorem 2.2. of the strong morphism of groupoids (f̃, f̃_{0}) : (Γ; Γ_{0}) ⟶ (Imf; Imf_{0}) associated to (f, f_{0}). △

Similarly, we can prove the following theorem. Theorem 2.5. For any strong morphism of groupoids (f, f_{0}) : (Γ; Γ_{0}) ⟶ (Γ′; Γ′_{0}), there exists a bijection from the set N (Γ′) of the normal subgroupoids of (Γ′; Γ′_{0}) to the set Ñ(Γ) of the normal subgroupoids of (Γ; Γ_{0}) which contains Kerf. △ Corollary 2.5. Let f : Γ ⟶ Γ′ a Γ_{0}- morphism of groupoids. Then the following assertions hold:

(i) There exists a bijection from the set S(Imf; Imf_{0}) of the subgroupoids of (Imf; Imf_{0}) to the set S(Γ; Γ_{0}) of the subgroupoids of (Γ; Γ_{0}).

(ii) There exists a bijection from the set N (Imf) of the normal subgroupoids of (Imf; Imf_{0}) to the set Ñ(Γ) of the normal subgroupoids of (Γ; Γ_{0}) which contains Kerf. Proof. This is a consequence of Theorems 2.1.(ii), 2.4 and 2.5. △ Remark 2.4. We conclude that the strong morphisms of groupoids have the same properties as the morphisms of groups. △

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University of Timişoara Department of Mathematics 4, Bd. V. Pârvan 1900, Timişoara, Romania