Novi Sad J. Math. 23 Vol. 31, No. 2, 2002, 23-36 SPECIAL MORPHISMS OF GROUPOIDS Gheorghe Ivan1 Abstract. The aim of this paper is to give the most important algebraic properties of special morphisms in the category of groupoids. Also some various topics of their are established. AMS Mathematics Subject Classification (2000): 20L13, 20L99, 18B40 Key words and phrases: induced groupoid, pullback, fibrewise injective (surjective, bijective) morpfism, piecewise injective (surjective, bijective) morphism. Introduction Groupoids were introduced and named by H. Brandt ([2]) in 1926, in a paper on the composition of quadratic forms in four variables. A groupoid is, roughly speaking, a set with a not everywhere defined binary operation which would a group if the operation were defined everywhere. There are various equivalent definitions for groupoids (see [6], [10],[11], [14]) and various ways of thinking of them. In 1950, in his paper ([4]) on connections, C. Ehresmann added further struc- tures (topological and differentiable) to groupoids, thereby introducing them as a tool in differentiable topology and geometry. The differentiable groupoids endowed with supplementary structures (for example: Lie groupoids, symplectic groupoids, contact groupoids, Riemannian groupoids, measure groupoids) has used by C. Albert,P. Dazord, M. V. Karasev, P. Libermann, K. Mackenzie, G. W. Mackey, J. Pradines, J. Renault, A. Wein- stein, in a series of papers for applications to differential topology and geometry, symplectic geometry, Poisson geometry, quantum mechanics, quantization the- ory, ergodic theory. 1. The category of groupoids The purpose of this section is to construct the category of groupoids and give several properties characterizing them. Definition 1.1. ([11]) A groupoid Γ over Γ0 or groupoid with the base Γ0 is a 7-tuple (Γ; ®; ¯; ²; i; ¹;Γ0) formed by: the sets Γ and Γ0, the surjec- tions ®; ¯ :Γ ¡! Γ0, called respectively the source and the target map, an 1Facultatea de Matematic˘a,Universitatea de Vest din Timi¸soara,Bd. V. Pˆarvan, 4, 1900, Timi¸soara,Romania 24 Gh. Ivan injection ² :Γ0 ¡! Γ; u ¡! ²(u) = ue, called the inclusion map, a map i :Γ ¡! Γ; x ¡! i(x) = x¡1, called the inversion map and a (partial) composition law ¹ :Γ(2) ¡! Γ; (x; y) ¡! ¹(x; y) = x ¢ y = xy, with the do- main Γ(2) = f(x; y) 2 Γ £ Γ j ¯(x) = ®(y)g, such that the following axioms are satisfied: (G1) (associative law) For arbitrary x; y; z 2 Γ the triple product (xy)z is defined iff x(yz) is defined. In case either is defined, we have (xy)z = x(yz) (G2) (identities) For each x 2 Γ we have (²(®(x)); x) 2 Γ(2);(x; ²(¯(x))) 2 Γ(2) and ²(®(x)) ¢ x = x ¢ ²(¯(x)) = x (G3) (inverses) For each x 2 Γ we have (x; i(x)) 2 Γ(2);(i(x); x) 2 Γ(2) and x ¢ i(x) = ²(®(x)); i(x) ¢ x = ²(¯(x)) :: We will denote sometimes a groupoid Γ over Γ0 by (Γ; ®; ¯;Γ0) or (Γ; Γ0). A groupoid Γ over Γ0 such that Γ0 is a subset of Γ is called Γ0-groupoid or Brandt groupoid. ¡1 u ¡1 For each u 2 Γ0, the set Γu = ® (u) (resp. Γ = ¯ (u)), called the ®-fibre v v (resp. ¯-fibre) of Γ over u 2 Γ0 and if u; v 2 Γ, we will write Γu = Γu \ Γ . Definition 1.2 A groupoid Γ over Γ0 is said to be transitive if the map ®£¯ : Γ ¡! Γ0 £ Γ0, given by (® £ ¯)(x) = (®(x); ¯(x)); (8)x 2 Γ is surjective.: We summarize some properties of groupoids obtained from the definitions. Proposition 1.1 Let Γ be a groupoid over Γ0. The following assertions hold: (i) For all x 2 Γ we have ¯(x¡1) = ®(x) and ®(x¡1) = ¯(x) ¡1 ¡1 (ii) If (x; y) 2 Γ(2) then x (xy) = y and (xy)y = x. (iii) If (x; y) 2 Γ(2) then ®(xy) = ®(x) and ¯(xy) = ¯(y): (iv) (cancellation law) If xz1 = xz2 ( resp., z1x = z2x) then z1 = z2. ¡1 ¡1 ¡1 ¡1 ¡1 (v) If (x; y) 2 Γ(2) then (y ; x ) 2 Γ(2) and (x ¢ y) = y ¢ x . (vi) (x¡1)¡1 = x; (8) x 2 Γ: (vii) ®(²(u)) = u and ¯(²(u)) = u; (8) u 2 Γ0. ¡1 (viii) ²(u) ¢ ²(u) = ²(u) and (²(u)) = ²(u) for each u 2 Γ0. Proof. (i) This assertion follows from the axiom (G.3). For example , for each ¡1 ¡1 x 2 Γ we have (x ; x) 2 Γ(2): Then ¯(x ) = ®(x): (ii) Let (x; y) 2 Γ(2) . By the axiom (G.2), for y 2 Γ we have ²(®(y))y = y: Hence ²(¯(x))y = y , since ¯(x) = ®(y): But, using (G.3), we have that ²(¯(x)) = x¡1x and we obtain (x¡1x)y = y; i.e. the triple product (x¡1x)y is defined. Applying now (G.1), imply that the triple x¡1(xy) is defined and we have (x¡1x)y = x¡1(xy): Therefore, x¡1(xy) = y: ¡1 ¡1 (iii) Let (x; y) 2 Γ(2): Then x (xy) = y; by (ii). Hence, (x ; xy) 2 Γ(2) and it follows that ¯(x¡1) = ®(xy): We have ®(xy) = ®(x); since ¯(x¡1) = ®(x); by (i). Special morphisms of groupoids 25 ¡1 (iv) It is a consequence of (ii). Indeed, if xz1 = xz2; then x (xz1) = ¡1 x (xz2) , i.e. z1 = z2: ¡1 ¡1 (v) Let (x; y) 2 Γ(2): By (ii), ¯(y ) = ®(y) and ®(x ) = ¯(x): Then, ¡1 ¡1 ¡1 ¡1 ¯(y ) = ®(x ); since ¯(x) = ®(y): Hence , (y ; x ) 2 Γ(2): We have (xy)(xy)¡1 = ²(®(xy)) = ²(®(x)); and (xy)(y¡1x¡1) = x(yy¡1)x¡1 = x²(®(y))x¡1 = x²(¯(x))x¡1 = xx¡1 = ²(®(x)): Hence (xy)(xy)¡1 = (xy)(y¡1x¡1) and by (iv) we obtain (xy)¡1 = y¡1x¡1: (vi) Applying (G.3) for the elements x; x¡1 2 Γ and (i), we have x¡1x = ²(¯(x)) and x¡1(x¡1)¡1 = ²(®(x¡1)) = ²(¯(x)): Hence x¡1x = x¡1(x¡1)¡1 and using (iv), we obtain (x¡1)¡1 = x: (vii) Let u 2 Γ0: We denote ²(u) = x: From (G.2) follows ²(®(x))x = x and ²(®(²(u))x = x: We have ²(®(²(u)))x = ²(®(x))x; and by (iv), we obtain ²(®(²(u))) = ²(®(x)): Hence, ®(²(u)) = ®(x); since ² is injective . Then, ®(²(u)) = u; since ®(x) = u: (viii) Let u 2 Γ0: We denote ²(u) = x: From (G.2) follows ²(®(x))x = x; i.e. ²(®(²(u)))²(u) = ²(u); i.e. ²(u)²(u) = ²(u); since ®(²(u)) = u , cf. (vii). We have ²(u)(²(u))¡1 = ²(®(²(u))) = ²(u) and ²(u)²(u) = ²(u): Hence, ²(u)(²(u))¡1 = ²(u)²(u) and by (iv) we have (²(u))¡1 = ²(u):: The element ²(®(x)) (resp. ²(¯(x)) ) is the left unit (resp., right unit) of x 2 Γ, and the subset ²(Γ0) of Γ is called the unity set of Γ. Applying (i), (vi) and (vii) from Prop. 1.1, respectively , we obtain: Remark 1.1 Let (Γ; ®; ¯; ²; i;Γ0) be a groupoid. Then the maps ®; ¯; ²; i satisfy the following relations : (i) ® ± i = ¯ and ¯ ± i = ®: (ii) i ± i = IdΓ: (iii) ® ± ² = ¯ ± ² = IdΓ0 :: Proposition 1.2 Let Γ be a groupoid over Γ0. The following assertions hold: ¡1 ¡1 (i) For each u 2 Γ0, the set Γ(u) = ® (u) \ ¯ (u) is a group under the restriction of the partial multiplication, called the isotropy group at u of Γ. (ii) If ®(x) = u and ¯(x) = v; then the map ! : Γ(u) ! Γ(v); z ! !(z) = x¡1zx is an isomorphism of groups. (iii) If Γ is transitive, then the isotropy groups of Γ are isomorphes. Proof. (i) For any x; y 2 Γ(u) we have ®(x) = ¯(x) = ®(y) = ¯(y) = u: Hence ,the product xy is defined. We have ®(xy) = ®(x) = u; ¯(xy) = ¯(y) = u and imply that xy 2 Γ(u): Therefore, the restriction of the partial multiplication defined on Γ is a binary operation on Γ(u): It is easy to verify that Γ(u) is a group. The unity of Γ(u) is the element ²(u); since by (G.2) we have ²(®(x))x = x²(¯(x)) = x; i.e. ²(u)x = x²(u) = x; (8)x 2 Γ(u): (ii) Let z 2 Γ(u): Then ®(z) = ¯(z) = u: Then the map ! is well-defined, since ®(x¡1zx) = ®(x¡1) = ¯(x) = v ¯(x¡1zx) = ¯(x) = v and hence ¡1 x zx 2 Γ(v): It is easy to verify that !(z1z2) = !(z1)!(z2); (8) z1; z2 2 Γ(u): 26 Gh. Ivan Applying the cancellation law we obtain that the map ! is injective. If y 2 Γ(v) then there exists z = xyx¡1 2 Γ(u) such that !(z) = y; i.e. the map ! is surjective. Therefore, ! is a bijective morphism of groups. Hence, the groups Γ(u) and Γ(v) are isomorphes. (iii) Since Γ is transitive, then the map (®£¯):Γ ! Γ0 £Γ0 is surjective, i.e. for any pair (u; v) 2 Γ0 £ Γ0 there exists x 2 Γ such that ®(x) = u and ¯(x) = v: Applying now (ii) we obtain that the isotropy groups Γ(u) and Γ(v) are isomorphes, for all u; v 2 Γ0:: Definition 1.3 By group bundle we mean a groupoid Γ over Γ0 such that ®(x) = ¯(x) for each x 2 Γ. Moreover, a group bundle is the union of its ¡1 isotropy groups Γ(u) = ® (u); u 2 Γ0 (here two elements may be composed iff they lie in the same fiber.): If (Γ; ®; ¯;Γ0) is a groupoid, then Is(Γ) = fx 2 Γ j ®(x) = ¯(x)g is a group bundle called the isotropy group bundle associated to Γ.