THE GROUPOID STRUCTURE OF GROUPOID MORPHISMS 1. INTRODUCTION

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1 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS BOHUI CHEN, CHEN-YON DU, AND RUI WAN ABSTRACT. In this paper we construct two groupoids from morphisms of groupoids, with one from a categorical viewpoint and the other from a geometric viewpoint. We show that for each pair of groupoids, the two kinds of groupoids of morphisms are equivalent. Then we study the automorphism groupoid of a groupoid. 1. INTRODUCTION In this paper we study morphisms and automorphisms of groupoids. Our motivation comes from the study of morphisms of orbifold groupoids (i.e. proper étale Lie groupoids) and group actions on orbifold groupoids. There are various kinds of morphisms between groupoids, such as strict morphisms (i.e. functors), generalized morphisms (cf. [ALR]), and etc. It is wellknown that orbifolds are Morita equivalent classes of orbifold groupoids. Hence it is not suitable to use strict morphisms to study maps between orbifold groupoids. Motived by the study of space of morphisms in orbifold romov Witten theory [CR1, CR2], it is important to build a groupoid structure over morphisms of groupoids. This is the main issue discussed in this paper. We now outline our approach. iven a pair of groupoids and H, by a morphism we mean a pair of strict morphisms ψ K u H with ψ being an equivalence of groupoids. We denote the morphism by (ψ, K, u) : H. An arrow between two such morphisms is captured by the following diagram, with α being a natural transformation for the diagram on the right. We denote the morphism by (ψ 1, K 1, u 1 ) α (ψ 1, K 2, u 2 ). (1.1) ψ 1 K 1 π 1 K 1 K 2 α H. π 2 ψ 2 u K 2 2 u 1 We define the (vertical) composition of arrows in 3.1 (cf. Construction 3.3). The main ingredient in the construction is fiber product. Then we get a groupoid of morphisms (cf. Theorem 3.7) Mor(, H) = (Mor 1 (, H) Mor 0 (, H)). In 3.2, we replace fiber product by strict product to simplify the constructions. For this purpose, we focus on full morphisms (a morphism (ψ, K, u) is called full if ψ 0 is surjective). With the 1

2 2 BOHUI CHEN, CHEN-YON DU, AND RUI WAN same procedure as in 3.1, we get a groupoid of full-morphisms (cf. Theorem 3.14) FMor(, H) = (FMor 1 (, H) FMor 0 (, H)). We show in Theorem 3.15 that there is a natural equivalence (1.2) i : FMor(, H) Mor(, H). In 4, we consider the composition functors: and similarly, : Mor(, H) Mor(H, N) : FMor(, H) FMor(H, N) Mor(, N), FMor(, N). For instance, the composition of denoted by (φ, L, v) (ψ, K, u), is given by (ψ, K, u) : H, and (φ, L, v) : H N, (1.3) ψπ 1 K H L vπ 1 N. Furthermore, we study the horizontal composition of arrows in (1.1) to get the functor. Moreover, we show that these compositions are associative (under certain canonical isomorphisms). As an application, we consider the automorphisms of in 5. An automorphism of is defined as a morphism (ψ, K, u) :, such that after composing with a morphism (φ, L, v) : there are arrows (ψ, K, u) (φ, L, v) α 1, and (φ, L, v) (ψ, K, u) β 1, where id 1 = (id,, id ) : id. Denote by Aut 0 () the set of all automorphisms of. We restrict Mor(, ) to Aut 0 () to get a groupoid of automorphisms Aut() of. We show that the coarse space Aut() is a group, and the automorphism groupoid Aut() is a K()-gerbe over Aut() (cf. Theorem 5.6). As an application we give a definition of group actions on a groupoid with trivial K() (see Definition 5.9). All discussion can be easily generalized to the cases of topological groupoids and Lie groupoids. One only has to add some continuous conditions or smooth conditions on various maps involved. In fact in application, we could also consider a more restrictive full-morphisms. That is, for example when we consider topological groupoids, we could require that in every fullmorphism (ψ, K, u) : H the ψ 0 : K 0 0 is an open covering of 0, and K is the pull-back groupoid over K 0 via ψ BASIC CONCEPTS OF ROUPOIDS For basic concepts about groupoids we refer readers to [M, MM, ALR].

3 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS roupoids. Let be a small category with the set of objects denoted by 0 and the set of morphisms denoted by 1. Here 1 is the collection of morphisms 1 = 1 (a, b), (a,b) 0 0 where 1 (a, b) 1 is the set of morphisms from a to b. We call a morphism x 1 (a, b) an arrow from a to b, and call a and b to be the source and the target of α respectively. We write where s, t : 1 a = s(x), b = t(x), and x : a b (or a x b), 0 are called the source map and the target map of the category respectively. Denote the composition of arrows 2 by : 1 (a, b) 1 (b, c) 1 (a, c), (x, y) x y. Definition 2.1. We say that is a groupoid if (1) for any a 0, there exists a unit 1 a 1 (a, a) with respect to the composition, i.e, 1 a x = x and x 1 a = x; (2) for any x 1 (a, b), there exists a unique inverse y 1 (b, a) such that x y = 1 a and y x = 1 b. We denote y by x 1. Define two maps: the unit map u : 0 1, a 1 a ; the inverse map i : 1 1, x x 1. Therefore a groupoid is a pair of sets ( 0, 1 ) with structure maps (, s, t, u, i). We may denote by ( 1 0 ), where the double arrows denote the source and target maps s and t. If we assume that 0 and 1 are topological space and the structure maps are continuous, we call a topological groupoid. 1 defines an equivalence relation on 0 : we say that a b 1 (a, b). We call the quotient space 0 / 1 to be the coarse space of and denote it by. The projection map from 0 to is denoted by : 0. Definition 2.2. By a strict morphism from a groupoid = ( 1 0 ) to a groupoid H = (H 1 H 0 ), we mean a functor from the category to H. We denote a strict morphism by f = (f 0, f 1 ) with f 0 : 0 H 0, f 1 : 1 H 1. A strict morphism f : H is an isomorphism if it has an inverse strict morphism. For a groupoid we denote by id = (id 0, id 1) : the identity strict morphism. 1 In literatures on category, this is denoted by Hom(a, b). In this paper, we use this notation to emphasis the groupoid structure. 2 In this paper we use the convention that the composition of arrows of a groupoid is going from left to right, not the usual notation of composition of maps.

4 4 BOHUI CHEN, CHEN-YON DU, AND RUI WAN Definition 2.3. Let f, g : H be two strict morphisms. A natural transformation from f to α α g, denoted by f g : H or simply by f g, is a natural transformation between the two functors. A strict morphism from f : H induces a map f : H on coarse spaces. If there α is an f g : H, then f = g Equivalence of groupoids. Definition 2.4. Let and H be two groupoids. A strict morphism f : H is called an equivalence if (1) the map t proj 2 : 0 f 0,H 0,s H 1 proj 2 H 1 (2) the square t H 0 is surjective; 1 f 1 H 1 is a fiber product. s t s t 0 0 f 0 f 0 H 0 H 0. Remark 2.5. (1) The first condition means that f is essentially surjective. (2) The second condition means that f is full and faithful, that is for any a, b 0, f 1 induces a bijection (2.1) f 1 : 1 (a, b) H 1 (f 0 (a), f 0 (b)). Consequently, consider three arrows x, y, z H 1 which fit into a commutative diagram a x b z c, y i.e. z = x y, and a, b, c 0 such that f 0 (a ) = a, f 0 (b ) = b, f 0 (c ) = c. Then from (2.1) we get a commutative diagram in 1 (2.2) a (f 1 ) 1 (x) b (f 1 ) 1 (z) We state two simple facts without proofs. c. (f 1 ) 1 (y) Lemma 2.6. iven a natural transformation f then so is the other one. α g : H, if one of f and g is an equivalence Lemma 2.7. iven a pair of equivalences ψ H φ N, the composition φ ψ : N is also an equivalence.

5 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 5 In the following we use ψ, φ, ϕ,... to denote equivalences, and use f, g, h, u, v, w,... to denote general strict morphisms. On the other hand we have the following obvious criterion for a special kind of subgroupoid 3 to be equivalence to the ambient one. Lemma 2.8. Suppose 0 H 0 and := H 0 is the restriction 4 of H on 0. If under the map : H 0 H we have 0 = H 0 = H, then the natural inclusion i : H is an equivalence Fiber product. Let f : F H and g : H be two strict morphisms. The fiber product F f,h,g (or simply F H ) is defined to be a groupoid as the following (cf. [ALR]): (1) The object space is (F H ) 0 = F 0 f 0,H 0,s H 1 t,h 0,g 0 0 = F 0 H 0 H 1 H 0 0. An object is a triple (a, x, b), with a F 0, b 0 and x H 1 (f 0 (a), g 0 (b)). We draw it as (2.3) a x b or a x H b. (2) iven two objects (a, x, b) and (a, x, b ), an arrow from (a, x, b) to (a, x, b ) consists of a pair of arrows (y, z) with y F 1 (a, a ), z 1 (b, b ), such that x g 1 (z) = f 1 (y) x, i.e. we have the following commutative diagrams (2.4) a y x b a x b after we transfer all arrows into H 1. Hence the arrow space is z (F H ) 1 = F 1 s,f 0,proj 1 ( F 0 H 0 H 1 H 0 0) proj3, 0,s 1. Denote an arrow by (y, (a, x, b), z). The source and target maps are obvious from the diagram (2.4). (3) All other structure maps are obvious. There are two natural strict morphisms, called projections: π 1 : F H F, (a, x, b) a, (y, (a, x, b), z) = y, π 2 : F H F, (a, x, b) b, (y, (a, x, b), z) = z. It is known that f π 1 and g π 2 are different up to a natural transformation. We have the following useful result, which can be verified straightforwardly. 3 Here we do not need the precise definition of sub-groupoid. One can think it as a subcategory. For explicit definition of subgroupoid see [M, Definition 2.4 in Chapter 1]. 4 This restriction groupoid := H 0 has object space 0 and morphism space a,b 0 H1 (a, b) H 1. The structures maps are inherited from H naturally.

6 6 BOHUI CHEN, CHEN-YON DU, AND RUI WAN Lemma 2.9. When g : H is an equivalence, π 1 : F H F is an equivalence. Similarly, when f : F H is an equivalence, π 2 : F H is an equivalence Strict fiber product. In this paper we will also consider a simpler version of fiber product which we call it strict fiber product. Lemma 2.13 explains that under certain conditions, we may replace fiber products by strict fiber products. Definition iven two strict morphisms f i : i = ( 1 i 0 i ) H = (H 1 H 0 ) for i = 1, 2, we define the strict fiber product 1 f1,h,f 2 2 (or simply 1 H 2 ) as where are fiber products of sets. 1 H 2 := ( 1 1 H H 0 0 2), 0 1 H = {(a, b) f 0 1 (a) = f 0 2 (b) H 0 }, 1 1 H = {(x, y) f 1 1 (x) = f 1 2 (y) H 1 } In the following, we will some times also write an object in 0 1 H in the way as (2.3). For example a 1 f 0 (a) a 0 H 1 H There are natural strict morphisms, also called projections: It is clear that π 1 : 1 H 2 1, (a, b) a, (x, y) x π 2 : 1 H 2 2, (a, b) b, (x, y) y. f π 1 = g π 2. There is an injective strict morphism connecting these two kinds of fiber products q : 1 H 2 1 H 2, (2.5) Set Via q we could view 1 (a, b) (a, 1 f 0 1 (a), b), (x, y) (x, (s(x), 1 f 0 1 (s(x)), s(y)), y). U 0 := Im q 0 ( 1 H 2 ) 0. H 2 as a sub-groupoid of 1 H 2. In fact we have Lemma q : 1 H 2 ( 1 H 2 ) U 0 is an isomorphism. Proof. Since both q 0 and q 1 are injective, we only need to show that q 1 : ( 1 H 2 ) 1 ((a, b), (a, b )) ( 1 H 2 ) 1 ((a, 1 f 0 1 (a), b), (a, 1 f 0 1 (a ), b )) is surjective. Suppose we have an arrow (x, (a, 1 f 0 1 (a), b), y) : (a, 1 f 0 1 (a), b) (a, 1 f 0 1 (a ), b )

7 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 7 in ( 1 H 2 ) 1. Then f 1 1 (x) 1 f 0 1 (a ) = 1 f 0 1 (a) f 1 2 (y), i.e. f 1 1 (x) = f 1 2 (y). Hence we get an arrow (x, y) in 1 above. H 2 which is a preimage of the arrow Definition An equivalence ψ : H is called a full equivalence if ψ 0 is surjective. Lemma When one of f 1 and f 2 is a full equivalence, q : 1 H 2 1 H 2 is an equivalence. In that case, ( 1 H 2 ) U 0 is equivalent to 1 H 2. Proof. By Lemma 2.8 we only need to show that every object (a, x, b) ( 1 H 2 ) 0 is connected to an object in U 0. Without loss of generality, we assume that f 1 is a full equivalence. Hence f1 0 is surjective. Take a pre-image a of f2 0 (b) under f1 0, i.e. f1 0 (a ) = f2 0 (b). Then (a, 1 f 0 1 (a ), b) U 0. Since x : f1 0 (a) f2 0 (b) = f1 0 (a ) and f 1 is an equivalence, we get a unique arrow (f1 1 ) 1 (x) : a a. Then one can see that ((f1 1 ) 1 (x), (a, x, b), 1 b ) : (a, x, b) (a, 1 f 0 1 (a ), b) is an arrow in ( 1 H 2 ) 1 that connects (a, x, b) with (a, 1 f 0 1 (a ), b) U 0. We also have analogues of Lemma 2.7 and Lemma 2.9. Lemma Let ψ : H and φ : H N both be full equivalences. Then the composition φ ψ : N is also a full equivalence. Lemma When f 2 is a full equivalence, π 1 : 1 H 2 1 is a full equivalence. When f 1 is a full equivalence, π 2 : 1 H 2 2 is a full equivalence Canonical isomorphisms for (strict) fiber products. Lemma iven four strict morphisms f : H, g : K, u : K L, and v : M L, we have the following two canonical isomorphisms and commutative diagram (H f,,g K) uπ2,l,v M = H f,,gπ1 (K u,l,v M) q (H f,,g K) uπ2,l,v M = H f,,gπ1 (K u,l,v M). q Lemma For a strict morphism f : H, there are canonical isomorphisms H idh,h,f = = f,h,idh H given by projections. 3. MORPHISM ROUPOIDS In this section for each pair (, H) of groupoids we construct two groupoids of morphisms, Mor(, H) and FMor(, H). Then we will show that these two groupoids are equivalent to each other.

8 8 BOHUI CHEN, CHEN-YON DU, AND RUI WAN 3.1. Morphism groupoids via fiber products. Definition 3.1. By a morphism 5 from to H, we mean two strict morphisms in the diagram ψ K u H with ψ being an equivalence. We denote such a morphism by the triple (ψ, K, u), and the set of morphisms from to H by Mor 0 (, H). Definition 3.2. iven two morphisms (ψ 1, K 1, u 1 ) : H and (ψ 2, K 2, u 2 ) : H, an arrow (ψ 1, K 1, u 1 ) α (ψ 2, K 2, u 2 ) is a natural transformation u 1 π α 1 u 2 π 2, i.e. ψ 1 K 1 π 1 K 1 K 2 α H. π 2 ψ 2 u K 2 2 u 1 Denote by Mor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )), the set of all arrows from (ψ 1, K 1, u 1 ) to (ψ 2, K 2, u 2 ), and by Mor 1 (, H) := Mor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )) (ψ i,k i,u i ) Mor 0 (,H),i=1,2 the set of all arrows between morphisms from to H. In the following we will define vertical 6 composition of arrows and show that Mor(, H) = (Mor 1 (, H) Mor 0 (, H)) is a groupoid. iven two arrows α i Mor 1 (, H)((ψ i, K i, u i ), (ψ i+1, K i+1, u i+1 )) for i = 1, 2, the vertical composition α 1 α 2 Mor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 3, K 3, u 3 )), is constructed as follow. Construction 3.3. Set K 12 := K 1 K 2, K 23 := K 2 K 3, K 13 := K 1 K 3, K 12,23 := K 12 K2 K Such a morphism between orbifold groupoids is called a generalized homomorphism (cf. [ALR]). 6 We use vertical composition to distinguish it from another composition of arrows constructed in the next section.

9 We have the following diagram (3.1) THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 9 K 12 K 1 Φ K 12,23 K 13 K 23 K 2 α 1 α 2 K 3 K 23 K 3 α 1 α 2? in which all unmarked strict morphisms are natural projections from fiber products to their factors. Objects and arrows in K 12,23 are of the form H (3.2) k 1 k 1 a x x k 2 k2 b z k K 2 2 c y z ỹ k K 2 2 k 3 d k3, with two rows being two objects and all four columns combine into an arrow in K12,23, 1 where k 1, k 1 K1, 0 k 2, k 2, k 2, k 2 K2, 0 k 3, k 3 K3, 0 and a K1, 1 b, c K2, 1 d K3, 1 and x, y, x, ỹ 1, z, z K2, 1 and ψ1(a) 1 x = x ψ2(b), 1 b z = z c, ψ2(c) 1 ỹ = y ψ3(d). 1 The strict morphism Φ : K 12,23 K 13 is given by Φ 0 (k 1, x, k 2, z, k 2, y, k 3 ) = (k 1, x φ 1 2(z) y, k 3 ), (3.3) Φ 1 (a, b, (k 1, x, k 2, z, k 2, y, k 3 ), c, d) = (a, (k 1, x φ 1 2(z) y, k 3 ), d). In the cube (3.1), the square with vertices {K 12,23, K 12, K 23, K 2 } has a natural transformation between the two composed strict morphisms from K 12,23 to K 2. By the definition of Φ and projections, the two squares with vertices {K 12,23, K 12, K 13, K 1 } and {K 12,23, K 23, K 13, K 3 } are commutative, i.e. K 12,23 = K 12 K1 K 13, K 12,23 = K 13 K3 K 23. Therefore five faces of the cube (3.1) have natural transformations except the face on the very right, which is the α 1 α 2 that we will define. The vertical composition α 1 α 2 : K13 0 H 1 is given by (3.4) α 1 α 2 (k 1, x, k 3 ) = α 1 (k 1, x 1, k 2 ) α 2 (k 2, x 2, k 3 ) for some splitting of x into ψ1(k 0 1 ) x 1 ψ2(k 0 2 ) x 2 ψ3(k 0 3 ) in 1 with k 2 K2 0 and x = x 1 x 2. It is direct to verify that this definition does not depend on the choices of the splitting of x and : (ψ 1, K 1, u 1 ) α 1 α 2 (ψ2, K 3, u 3 ). Lemma 3.4. The vertical composition of arrows is associative.

10 10 BOHUI CHEN, CHEN-YON DU, AND RUI WAN Proof. Take three arrows α i Mor 1 (, H)((ψ i, K i, u i ), (ψ i+1, K i+1, u i+1 )) for i = 1, 2, 3. First of all α 1 α 2 : (K 1 K 3 ) 0 H 1 is given by α 1 α 2 (k 1, x, k 3 ) = α 1 (k 1, x 1, k 2 ) α 2 (k 2, x 2, k 3 ) with x = x 1 x 2. Then (α 1 α 2 ) α 3 : (K 1 K 4 ) 0 H 1 is given by (α 1 α 2 ) α 3 (k 1, x, k 4 ) = (α 1 α 2 )(k 1, x 1, k 3 ) α 3 (k 3, x 2, k 4 ) = α 1 (k 1, x 11, k 2 ) α 2 (k 2, x 12, k 3 ) α 3 (k 3, x 2, k 4 ), with x = x 11 x 12 x 2 and x 1 = x 11 x 12. Similarly α 1 (α 2 α 3 ) : (K 1 K 4 ) 0 H 1 is given by α 1 (α 2 α 3 )(k 1, x, k 4 ) = α 1 (k 1, x 1, k 2 ) α 2 α 3 ( k 2, x 2, k 4 ) = α 1 (k 1, x 1, k 2 ) α 2 ( k 2, x 21, k 3 ) α 3 ( k 3, x 22, k 4 ). with x = x 1 x 21 x 22 and x 2 = x 21 x 22. We could take k i = k i for i = 2, 3, and x 11 = x 1, x 12 = x 21, x 2 = x 22. Therefore (α 1 α 2 ) α 3 = α 1 (α 2 α 3 ). There are also unit arrows with respect to vertical composition. Lemma 3.5. iven a morphism (ψ, K, u) Mor 0 (, H), there is an arrow 1 (ψ,k,u) serves as the unit arrow over (ψ, K, u) with respect to the vertical composition in Mor 1 (, H). Proof. 1 (ψ,k,u) is a natural transformation uπ 1 uπ 2, which as a map 1 (ψ,k,u) : (K K) 0 H 1 can be explicitly given by (3.5) 1 (ψ,k,u) (k, x, k ) := u 1 ((ψ 1 ) 1 (x)). The inverse arrow of an arrow also exists. Lemma 3.6. iven an arrow α Mor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )), there is a natural induced arrow α 1 Mor 1 (, H)((ψ 2, K 2, u 2 ), (ψ 1, K 1, u 1 )) satisfying α α 1 = 1 (ψ1,k 1,u 1 ), and α 1 α = 1 (ψ2,k 2,u 2 ). We call α 1 the inverse arrow of α with respective to the vertical composition. Proof. By definition α is a natural transformation u 1 π 1 α u 2 π 2 : K 1 K 2 H. Then u 2 π 1 α 1 u 1 π 2 : K 2 K 1 H can be explicitly given by α 1 (k 2, x, k 1 ) := α(k 1, x, 1, k 2 ) 1. Combining Lemma 3.4, Lemma 3.5 and Lemma 3.6 we get

11 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 11 Theorem 3.7. For each pair (, H) of groupoids, Mor(, H) is a groupoid Morphism groupoids via strict fiber products. Now we modify the construction in previous subsection by replacing all fiber products by strict fiber products to construct another morphism groupoid for each pair of groupoids. Definition 3.8. We call a morphism (ψ, K, u) : H a full-morphism if ψ is a full equivalence. We denote the set of full-morphisms from to H by FMor 0 (, H). Hence FMor 0 (, H) Mor 0 (, H). We could restrict the groupoid Mor(, H) to FMor 0 (, H) to get a groupoid. Instead, we use strict fiber products to define arrows between full-morphisms to get a new groupoid. Definition 3.9. For any two full-morphisms (ψ 1, K 1, u 1 ) : H and (ψ 2, K 2, u 2 ) : H, An arrow (ψ 1, K 1, u 1 ) α (ψ 2, K 2, u 2 ) is a natural transformation α from the strict morphism u 1 π 1 to the strict morphism u 2 π 2 in the following diagram ψ 1 K 1 π 1 K 1 K 2 α u 1 H ψ 2 π 2 K 2 u 2 where π i : K 1 K 2 K i, i = 1, 2, are the projections. Denote by FMor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )) the set of arrows from (ψ 1, K 1, u 1 ) to (ψ 2, K 2, u 2 ), and set FMor 1 (, H) := FMor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )) (ψ i,k i,u i ) FMor 0 (,H),i=1,2 For i = 1, 2, given two arrows between full-morphisms α i FMor 1 (, H)((ψ i, K i, u i ), (ψ i+1, K i+1, u i+1 )), the vertical composition α 1 α 2 FMor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 3, K 3, u 3 )) is constructed as follow. Construction Set K 12 := K 1 K 2, K23 := K 2 K 3, K 13 := K 1 K 3, K12,23 := K 12 K2 K 23.

12 12 BOHUI CHEN, CHEN-YON DU, AND RUI WAN We have the following diagram K 12,23 K12 12 K 12 K 1 Ψ K 13 K13 (3.6) α 1 α 1 α 2? K 2 H K 23 K 3 K 23 α 2 K 3 with unmarked strict morphisms being natural projections from strict fiber products to their factors. An arrow of K 12,23, denoted by (a, b, (k 1, k 2, k 2, k 3 ), b, c), can be illustrated in the following form (3.7) 1 ψ 0 1 (k 1 ) 1 1 k2 ψ 0 k 1 k2 3 (k 3 ) k 2 k3 a b k 1 1 ψ 0 1 ( k 1 ) k2 1 k2 k2 1 ψ 0 2 ( k 2 ) k3 b c with two rows being two objects and all four columns combine into an arrow of K 12,23, where k i, k i Ki 0 being objects of K i for i = 1, 2, 3, a K1, 1 b K2, 1 c K3 1 being arrows of K i, for i = 1, 2, 3, ψ1(a) 1 = ψ2(b), 1 and ψ2(b) 1 = ψ3(c). 1 The strict morphism Ψ : K 12,23 K 13 is given by (3.8) Ψ 0 (k 1, k 2, k 2, k 3 ) = (k 1, k 3 ), Ψ 1 (a, b, (k 1, k 2, k 2, k 3 ), b, c) = (a, c). Then from the definition of Ψ and natural projections we see that in the cube (3.6) the three squares with vertices { K 12,23, K 12, K 13, K 1 }, { K 12,23, K 23, K 13, K 3 } and { K 12,23, K 12, K 23, K 2 } are all commutative. The composition α β : K is give by (3.9) α β(k 1, k 3 ) = α(k 1, k 2 ) β(k 2, k 3 ) for a k 2 K2 0 satisfying ψ1(k 0 1 ) = ψ2(k 0 2 ) = ψ3(k 0 3 ). It is direct to verify that α β is a natural transformation α β : u 1 π 1 u 3 π 3, hence an arrow α β : u 1 u 3. By similar proof of Lemma 3.4, Lemma 3.5 and Lemma 3.6 we have Lemma The vertical compositions : FMor 1 (, H) FMor 1 (, H) FMor 1 (, H) is associative.

13 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 13 Lemma iven a full-morphism (ψ, K, u) FMor 0 (, H), there is an arrow 1 (ψ,k,u) serves as the unit arrow over (ψ, K, u) in FMor 1 (, H) with respect to the vertical composition, which is given by 1 (ψ,k,u) : (K K) 0 H 1, 1 (ψ,k,u) (k, k ) := u 1 ((ψ 1 ) 1 (1 ψ 0 (k))). Lemma iven an arrow α FMor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )) there is a natural induced arrow α 1 FMor 1 (, H)((ψ 2, K 2, u 2 ), (ψ 1, K 1, u 1 )), which is given by α 1 : (K 2 K 1 ) 0 H 1, α 1 (k 2, k 1 ) := α(k 1, k 2 ) 1, and it satisfies α α 1 = 1 (ψ1,k 1,u 1 ), and α 1 α = 1 (ψ2,k 2,u 2 ). Therefore Theorem For each pair (, H) of groupoids, FMor(, H) = (FMor 1 (, H) FMor 0 (, H)) is a groupoid Equivalence between Mor(, H) and FMor(, H). We have the following equivalence of morphism groupoids. Theorem There is a natural strict morphisms i = (i 0, i 1 ) : FMor(, H) Mor(, H). Moreover it is an equivalences between groupoids. Proof. We first construct the i. The i 0 : FMor 0 (, H) Mor 0 (, H) is the inclusion. We next define i 1. Take an arrow α FMor 1 (, H)((ψ, K, u), (φ, L, v)), then α is a natural transformation u π α 1 v π 2 : K L H. By Lemma 2.13, q : K L K L, is an injective equivalence. This q together with the equalities u π 1 = u π 1 q, v π 2 = v π 2 q, gives us a canonically induced natural transformation u π α 1 v π 2 : K ψ,,φ L H. We next describe α. Since q is an equivalence, for any object b (K L) 0, there is an object a (K L) 0 and an arrow x : q 0 (a) b in (K L) 1. Then we set (3.10) α(b) := [(u π 1 ) 1 (x)] 1 α(a) (v π 2 ) 1 (x). It is direct to verify that this definition of α does not depend on the choices of a and x and is a natural transformation by using the fact that α is a natural transformation. Then we set i 1 (α) = α. We next show that i = (i 0, i 1 ) is a strict morphism and an equivalence. By construction we have i 1 (α) : i 0 (ψ, K, u) i 0 (φ, L, v).

14 14 BOHUI CHEN, CHEN-YON DU, AND RUI WAN We next show that it also preserves the vertical composition. For two arrows (ψ, K, u) (φ, L, v), (ψ, L, v) β (φ, M, w) in FMor 1 (, H) we have α α β : (K M) 0 H 1, with α β(k, m) = α(k, l) β(l, m) for some l L 0 satisfying φ 0 (l) = ψ 0 (k) = ϕ 0 (m). Objects in (K M) 0 is of the form (k, x, m) with x : ψ 0 (k) ϕ 0 (m) in 1. Take an arrow in (K M) 1 (1 k, (ϕ 1 ) 1 (x)) : q 0 (k, m ) (k, x, m), where m satisfies φ 0 (m ) = ψ 0 (k) (See similar construction in the proof of Lemma 2.13). Then i 1 (α β)(k, x, m) = [(u π 1 ) 1 (1 k, (ϕ 1 ) 1 (x))] 1 α β(k, m ) (w π 2 ) 1 (1 k, (ϕ 1 ) 1 (x)) = u 1 (1 k ) α β(k, m ) w 1 ((ϕ 1 ) 1 (x)) = α(k, l ) β(l, m ) w 1 ((ϕ 1 ) 1 (x)) for some l L 0 such that ψ 0 (k) = φ 0 (l ) = ϕ 0 (m ). On the other hand, for this l, (k, 1 ψ 0 (k), l ) Im q 0 and is also an arrow in (L M) 1. Therefore i 1 (α) i 1 (β)(k, x, m) = i 1 (α)(k, 1 ψ 0 (k), l ) i 1 (β)(l, x, m) (1 l, (ϕ 1 ) 1 (x)) : (l, 1 ψ 0 (k), m ) (l, x, m) = i 1 (α)(k, 1 ψ 0 (k), l ) [(v π 1 ) 1 (1 l, (ϕ 1 ) 1 (x))] 1 β(l, m ) (w π 2 ) 1 (1 l, (ϕ 1 ) 1 (x)) = α(k, l ) v 1 (1 l ) β(l, m ) w 1 ((ϕ 1 ) 1 (x)) = α(k, l ) β(l, m ) w 1 ((ϕ 1 ) 1 (x)). Hence i 1 (α) i 1 (β) = i 1 (α β). Consequently i = (i 0, i 1 ) : FMor(, H) Mor(, H) is a strict morphism. We next show that i is an equivalence. First of all, for every morphism (ψ, K, u) Mor 0 (, H) there is another morphism (id π 1, K, u π 2 ) Mor 0 (, H). Obviously (id π 1, K, u π 2 ) is a full-morphism, hence belongs to Im i 0. We claim that there is an arrow α Mor 1 (, H)((ψ, K, u), (id π 1, K, u π 2 )). Now we construct the α. By definition α is a natural transformation u π 1 α u π 2 π 2 : K ( K) H. Set Q = (Q 1 Q 0 ) := K ( K). An object in Q 0 is of the form k x (g y k )

15 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 15 denoted by (k, x, g, y, k ). It is mapped by u π 1 and u π 2 respectively to u 0 (k), u 0 (k ). From (k, x, g, y, k ) we get an arrow x y : ψ 0 (k) ψ 0 (k ) in 1. Then since ψ is an equivalence we get a unique arrow (ψ 1 ) 1 (x y) : k k in K 1. We define α(k, x, g, y, k ) := u 1 ((ψ 1 ) 1 (x y)) = u 1 (ψ 1 ) 1 (x) u 1 (ψ 1 ) 1 (y). Then it is direct to check that this is the arrow we want. Hence i 0 is essentially surjective. Finally we show that i is full and faithful. In fact the inverse map (i 1 ) 1 of i 1 : FMor 1 (, H)((ψ 1, K 1, u 1 ), (ψ 2, K 2, u 2 )) Mor 1 (, H)(i 0 (ψ 1, K 1, u 1 ), i 0 (ψ 2, K 2, u 2 )) is given by (i 1 ) 1 (α) := α q 0 : (K 1 K 2 ) 0 (K 1 K 2 ) 0 H 1. Hence i is an equivalence. This i : FMor, H) Mor(, H) factors through i : FMor(, H) Mor(, H) FMor 0 (,H) Mor(, H). Hence all three groupoids are equivalent. Remark Now recall the definition of vertical composition and. We can give another definition of via the construction of i 1 and definition of. Suppose we have two arrows in α, β FMor 1 (, H)((ψ, K, u), (φ, L, v)), then via i 1 we get two arrows i 1 (α), i 1 (β) Mor 1 (, H)((ψ, K, u), (φ, L, v)). Then we have i 1 (α) i 1 (β), and α β = (i 1 ) 1 (i 1 (α) i 1 (β)) = (i 1 (α) i 1 (β)) q 0, with q 0 : K 13,23 0 K13,23. 0 In fact, the injective strict morphism q in (2.5) from strict fiber product to fiber product together with identity strict morphisms of K 1, K 2, K 3, H gives us a strict morphisms from the cube (3.6) to the cube (3.1), and Ψ is the composition of Φ and q 0 : K 13,23 0 K13, COMPOSITION OF MORPHISM ROUPOIDS In this section we show that their natural composition functors on morphism groupoids: : Mor(, H) Mor(H, N) Mor(, N), and : FMor(, H) FMor(H, N) FMor(, N).

16 16 BOHUI CHEN, CHEN-YON DU, AND RUI WAN 4.1. Composition functor. iven two morphisms (ψ, K, u) : H and (φ, L, v) : H N, let M := K H L be the fiber product of u : K H and φ : L H, π 1 : M K and π 2 : M L be the corresponding projections. By Lemma 2.9, π 1 is an equivalence. Hence by Lemma 2.7, ψ π 1 : M is also an equivalence. Definition 4.1. The composition of (ψ, K, u) and (φ, L, v) is defined to be (φ, L, v) (ψ, K, u) := (ψ π 1, M, v π 2 ) : N. This can be summarized in the following diagram M π 1 π 2 ψ K u H φ L v N. For a groupoid, we call id 1 := (id,, id ) : id the identity morphism of. We also denote it by 1. However it is not the unit for composition of morphisms since H H. Now we describe the horizontal composition of arrows. Take two arrows (ψ 1, K 1, u 1 ) α (ψ 2, K 2, u 2 ) : H, (φ 1, L 1, v 1 ) β (φ 2, L 2, v 2 ) : H N. The horizontal composition β α of α and β is an arrow (φ 1, L 1, v 1 ) (ψ 1, K 1, u 1 ) βα (φ 2, L 2, v 2 ) (ψ 2, K 2, u 2 ) : N. We describe the construction. Construction 4.2. Set K 12 := K 1 K 2, L := K 1 H J 1, J 12 := J 1 H J 2, M := K 2 H J 2, and U := L M. We have the following diagram π 1 v 1 π 2 (4.1) U ψ 1 ψ 2 K 1 π 1 K 12 π 2 K 2 π 1 u 1 α u 2 π 1 L H M π 2 φ 1 φ 2 π 2 J 1 π 1 J 12 π 2 J 2 v 1 β v 2 N π 2 v 2 π 2 The arrow β α we want is a natural transformation An objects in U 0 is of the form β α : v 1 π 2 π 1 v 2 π 2 π 2 : U N. x j 1 k 1 H z k 2 y N j 2,

17 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 17 denoted by (j 1, x, k 1, z, k 2, y, j 2 ). We define the horizontal composition by β α(j 1, x, k 1, z, k 2, y, j 2 ) := β(j 1, x 1 α(k 1, z, k 2 ) y, j 2 ). It is direct to verify that this is an arrow β α : (φ 1, L 1, v 1 ) (ψ 1, K 1, u 1 ) (φ 2, L 2, v 2 ) (ψ 2, K 2, u 2 ). Lemma 4.3. Combine with composition of morphisms we get a horizontal composition functor : Mor(H, N) Mor(, H) Mor(, N). So is a strict morphism of groupoids. Proof. For i = 1, 2, take α i : (ψ i, K i, u i ) (ψ i+1, K i+1, u i+1 ) : H β i : (φ i, L i, v i ) (φ i+1, L i+1, v i+1 ) : H N. Then we need to show that (β 1 α 1 ) (β 2 α 2 ) =? (β 1 β 2 ) (α 1 α 2 ). Note that they both define on the object space of Q = (Q 1 Q 0 ) := (K 1 H J 1 ) (K 2 H J 2 ). We first compute (β 1 α 1 ) (β 2 α 2 ). Take an object (j 1, x, k 1, z, k 3, y, j 3 ) Q 0. By definition (β 1 α 1 ) (β 2 α 2 )(j 1, x, k 1, z, k 3, y, j 3 ) =β 1 α 1 (j 1, x, k 1, z 1, k 2, w, j 2 ) β 2 α 2 (j 2, w, k 2, z 2, k 3, y, j 3 ) for some (k 2, w, j 2 ) (K 2 H J 2 ) 0 and z = z 1 z 2 in 1. Then we get (β 1 α 1 ) (α 2 β 2 )(j 1, x, k 1, z, k 3, y, j 3 ) =β 1 (j 1, x 1 α 1 (k 1, z 1, k 2 ) w, j 2 ) β 2 (j 2, w 1 α 2 (k 2, z 2, k 3 ) y, j 3 ) Similarly (β 1 β 2 ) (α 1 α 2 )(j 1, x, k 1, z, k 3, y, j 3 ) = β 1 β 2 (j 1, x 1 α 1 α 2 (k 1, z, k 3 ) y, j 3 ) = β 1 β 2 (j 1, x 1 α 1 (k 1, z 1, k 2 ) α 2 (k 2, z 2, k 3 ) y, j 3 ) = β 1 (j 1, x 1 α 1 (k 1, z 1, k 2 ) w, j 2 ) β 2 (j 2, w 1 α 1 (k 1, z 1, k 2 ) y, j 3 ), with k 2, z 1, z 2, j 2 being same as those above. This finishes the proof. Lemma 4.4. Under the canonical isomorphism of fiber products in Lemma 2.16, the horizontal composition functor is associative.

18 18 BOHUI CHEN, CHEN-YON DU, AND RUI WAN Proof. Take three arrows in Mor 1 as follow: ψ 1 I 1 J 1 K 1 I 12 α 1 H J 12 α 2 N K 12 α 3 M ψ 2 π 1 π 2 u 1 u 2 φ 1 φ 2 π 1 π 2 I 2 J 2 K 2 v 1 v 2 ϕ 1 ϕ 2 π 1 π 2 w 1 w 2 with I 12 = I 1 I 2, J 12 = J 1 H J 2, K 12 = K 1 N K 2. We first consider the compositions of (ψ 1, I 1, u 1 ), (φ 1, J 1, v 1 ) and (ϕ 1, K 1, w 1 ). We get two compositions (ψ π 1, I 1 H (J 1 N K), w π 2 π 2 ) and (ψ π 1 π 1, (I 1 H J 1 ) N K, w π 2 ). Then via the canonical isomorphism I 1 H (J 1 N K) = (I 1 H J 1 ) N K we could identify them. From this natural identification we could get a natural arrow 7 between them. We next show that via such canonical isomorphisms of fiber products, we can also identify (α 3 α 2 ) α 1 with α 3 (α 2 α 1 ). The arrow (α 3 α 2 ) α 1 is a natural transformation between strict morphisms over A := [(I 1 H J 1 ) N K 1 ] [(I 2 H J 2 ) N K 2 ] and α 3 (α 2 α 1 ) is a natural transformation between strict morphisms over B := [I 1 H (J 1 N K 1 )] [I 2 H (J 2 N K 2 )]. Under the canonical isomorphisms for fiber products given by Lemma 2.16 we get a canonical isomorphism A = B. In particular, the identification over objects is given by z (i x 1 H (i 2 x H j 1 ) j 2 ) y N ỹ N k 1 k 2, z i x 1 H i 2 x H (j 1 y (j 2 N ỹ N k 1 ) k 2 ). We write them both as ((i 1, x, j 1, y, k 1 ), z, (i 2, x, j 2, ỹ, k 2 )). Then by definition of horizontal composition of arrows we have (α 3 α 2 ) α 1 ((i 1, x, j 1, y, k 1 ), z, (i 2, x, j 2, ỹ, k 2 )) =α 3 (k 1, y 1 α 1 α 2 (i 1, x, j 1, z, i 2, x, j 2 ) ỹ, k 2 ) =α 3 (k 1, y 1 α 2 (j 1, x 1 α 1 (i 1, z, i 2 ) x, j 2 ) ỹ, k 2 ), 7 It is inverse since Mor(, H) is a groupoid.

19 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 19 and α 3 (α 2 α 1 )((i 1, x, j 1, y, k 1 ), z, (i 2, x, j 2, ỹ, k 2 )) =(α 3 α 2 )(j 1, y, k 1 ; x 1 α 1 (i 1, z, i 2 ) x; j 2, ỹ, k 2 ) =α 3 (k 1, y 1 α 2 (j 1, x 1 α 1 (i 1, z, i 2 ) x, j 2 ) ỹ, k 2 ). Hence we could identify the two kind of compositions of morphisms and arrows in the same time via the canonical isomorphisms of fiber products. This finishes the proof Composition functor. Definition 4.5. iven two full-morphisms (ψ, K, u) : H and (φ, L, v) : H N, the composition is defined to be (ψ, K, u) (φ, L, v) := (ψ π 1, K H L, v π 2 ) : N, which is summarized in the following diagram K H L π 1 π 2 ψ K u H φ L v N. Now suppose we have two arrows between full-morphisms (ψ 1, K 1, u 1 ) α (ψ 2, K 2, u 2 ) : H, (φ 1, L 1, v 1 ) β (φ 2, L 2, v 2 ) : H N. The horizontal composition α β should be an arrow (φ 1, L 1, v 1 ) (ψ 1, K 1, u 1 ) β α (φ 2, L 2, v 2 ) (ψ 2, K 2, u 2 ), i.e. an arrow (ψ 1 π 1, K 1 H L 1, v 1 π 2 ) β α (ψ 2 π 2, K 2 H L 2, v 2 π 2 ). Unlike the horizontal composition of arrows between morphisms in previous subsection, the construction of horizontal composition of arrows between full morphisms is slightly subtle. We now describe the construction. Construction 4.6. Set K 12 := K 1 K 2, L := K 1 H J 1, J 12 := J 1 H J 2, M := K 2 H J 2, and Ũ := L M. We have the following diagram (comparing with (4.1)) π 1 v 1 π 2 (4.2) Ũ ψ 1 ψ 2 K 1 π 1 K 12 π 2 K 2 π 1 u 1 α u 2 π 1 L H M π 2 φ 1 φ 2 π 2 J 1 π 1 J12 π 2 J 2 v 1 β v 2 N π 2 v 2 π 2 The arrow β α we want is a natural transformation v 1 π 2 π 1 α β = v 2 π 2 π 2 : Ũ N.

20 20 BOHUI CHEN, CHEN-YON DU, AND RUI WAN An object in Ũ 0 is of the form (k 1, j 1, k 2, j 2 ) with u 0 1(k 1 ) = φ 0 1(j 1 ), u 0 2(k 2 ) = φ 0 2(j 2 ) in H 0 and ψ 0 1(k 1 ) = ψ 0 2(k 2 ) in 0. It is mapped by v 1 π 2 π 1 and v 2 π 2 π 2 respectively to v 0 1(j 1 ) and v 0 2(j 2 ). From ψ 0 1(k 1 ) = ψ 0 2(k 2 ) we see that hence we get an arrow in H 1 from α This gives us an object (k 1, k 2 ) K 0 12, φ 0 1(j 1 ) = u 0 1(k 1 ) α(k 1,k 2 ) u 0 2(k 2 ) = φ 0 2(j 2 ). j 1 α(k 1,k 2 ) j 2 (J 1 φ1,h,φ 2 J 2 ) 0. Only if φ 0 1(j 1 ) = φ 0 2(j 2 ) and α(k 1, k 2 ) = 1 φ 0 1 (j 1 ) we get an object j 1 α(k 1,k 2 ) j 2 J 0 12 = (J 1 φ1,h,φ 2 J 2 ) 0. However since by Lemma 2.13, the natural strict morphism q : J 12 J 1 φ1,h,φ 2 J 2 is an equivalence, we could get an arrow in J 12 as follow. Since φ 1 and φ 2 are both full equivalences, there are j 1,2 J 0 2, and j 2,1 J 0 1 such that φ 0 1(j 2,1 ) = φ 0 2(j 2 ), and φ 0 2(j 1,2 ) = φ 0 1(j 1 ). Therefore (j 1, j 1,2 ), (j 2,1, j 2 ) J Via the equivalence φ 1 π 1, (by Lemma 2.7, φ 1 π 1 is an equivalence), these two objects in J 0 12 are mapped respectively to φ 0 1(j 1 ), φ 0 1(j 2,1 ) = φ 0 2(j 2 ), which are connected by α(k 1, k 2 ). Hence by Remark 2.5 there is a unique arrow in J 1 12 Denote by (x α, y α ) = [(φ 1 π 1 ) 1 ] 1 (α(k 1, k 2 )). Remark 4.7. In fact (j 1, j 1,2 ) [(φ 1 π 1 ) 1 ] 1 (α(k 1,k 2 )) (j 2,1, j 2 ). x α = (φ 1 1) 1 (α(k 1, k 2 )), y α = (φ 1 2) 1 (α(k 1, k 2 )). We now explain this. Since φ 1 and φ 2 are both full equivalence, then from (j 1, j 2,1 ) and (j 1,2, j 2 ) we get unique arrows (φ 1 1) 1 (α(k 1, k 2 )) : j 1 j 2,1, and (φ 1 2) 1 (α(k 1, k 2 )) : j 1,2 j 2. Then by the bijections of arrows under equivalence we have ((φ 1 1) 1 (α(k 1, k 2 )), (φ 1 2) 1 (α(k 1, k 2 ))) = [(φ 1 π 1 ) 1 ] 1 (α(k 1, k 2 )) = (x α, y α ). We can also use φ 2 π 2 to get [(φ 2 π 2 ) 1 ] 1 (α(k 1, k 2 )). However we get nothing else but [(φ 1 π 1 ) 1 ] 1 (α(k 1, k 2 )) = [(φ 2 π 2 ) 1 ] 1 (α(k 1, k 2 )) = (x α, y α ).

21 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 21 Let us continue the construction. By applying the natural transformation β to the arrow (x α, y α ) we get a commutative diagram in N 1 We define α β : Ũ 0 N 1 to be v 1 1 (xα) v 0 1(j 1 ) v 0 1(j 2,1 ) β(j 1,j 1,2 ) β(j 2,1,j 2 ) v 0 2(j 1,2 ) v 0 2(j 2 ) v 1 2 (yα) α β(k 1, j 1 ; k 2, j 2 ) := β(j 1, j 1,2 ) v 1 2(y α ) = v 1 1(x α ) β(j 2,1, j 2 ). It is direct to see that this definition does not depend on the choices of j 1,2 and j 2,1, and gives us an arrow (φ 1, L 1, v 1 )(ψ 1, K 1, u 1 ) α β (φ 2, L 2, v 2 )(ψ 2, K 2, u 2 ). This finishes the construction. Remark 4.8. We can also get α β via the horizontal composition in Mor 1 and i 1 in Theorem The procedure is similar to the way to get from in Remark iven α and β as above, we get i 1 (α) and i 1 (β). Hence we have i 1 (β) i 1 (α). Then we have with q 0 : Ũ 0 U 0. β α = (i 1 (β) i 1 (α)) q 0 Via this remark we have similar results for as Lemma 4.3. Lemma 4.9. Combining with composition of full-morphisms we also get a horizontal composition functor : FMor(, H) FMor(H, M) FMor(, N) i.e. the vertical and horizontal composition of arrows between full morphisms are compatible. Therefore is a strict groupoid morphism. Similar to Lemma 4.4 we have Lemma The horizontal composition functor is associative under the canonical isomorphism of strict fiber product of three groupoids in Lemma AUTOMORPHISM ROUPOIDS OF TOPOLOICAL ROUPOIDS In this section we study the morphism groupoid Mor(, ) of a groupoid. From now on we assume all the groupoids are topological groupoids Center of a topological groupoid. To study the automorphisms of groupoids we introduce a new concept of centers of groupoids. We first recall the concept of groupoid action on spaces. Definition 5.1. For a groupoid and a space X, a (left) -action on X consists of a map, called the anchor map, ρ : M 0,

22 22 BOHUI CHEN, CHEN-YON DU, AND RUI WAN an action map µ : 1 s, 0,ρ M M satisfying ρ(µ(x, p)) = t(x), µ(1 a, p) = p, and µ(x, µ(y, p)) = µ(y x, p) whenever the terms are well defined. iven an action of on M there is an induced groupoid M with ( M) 0 = M, ( M) 1 = 1 s, 0,ρ M, and source and target maps given by s(x, p) = p, t(x, p) = µ(x, p). Other structure maps are obvious. For a groupoid and an object a 0 the isotropy group of a in is Γ a := 1 (a, a). Denote by Z(Γ a ) the center of the isotropy group Γ a. Set Z 0 = a 0 Z(Γ a ) 1. There is a -action on Z 0, whose anchor map and action map are ρ : Z 0 0, x s(x) = t(x), µ : 1 s, 0,ρ Z 0 Z 0, (y, x) y 1 x y. Definition 5.2. We define the center groupoid of as Z := Z 0. There is a natural strict morphism π : Z with π 0 = ρ and π 1 given by π 1 : 1 s, 0,ρ Z 0 1, (y, x) y. Definition 5.3. By a section of π : Z we means a section σ : 0 Z 0 of the projection π 0 : Z 0 0 such that it is invariant under the -action in the meaning of that for every arrow x : a b in 1 (5.1) σ(b) = µ(x, σ(a)) = x 1 σ(a) x, i.e. x σ(b) = σ(a) x. We denote by K() the set of sections of π : Z. It is easy to see that Lemma 5.4. K() is a group. Proof. The multiplication is induced from the composition of arrows in 1. The identity for the multiplication is the unit section 1 : 0 Z 0, a (a, 1 a ). We call K() to be the center of the groupoid.

23 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS Automorphisms. Definition 5.5 (automorphism). Let (ψ, K, u) Mor 0 (, ). If there exists another morphism (φ, L, v) Mor 0 (, ) and two arrows (ψ, K, u) (φ, L, v) α 1, (φ, L, v) (ψ, K, u) β 1 in Mor 1 (, ), we call (ψ, K, u) an automorphism of. So is (φ, L, v). Let Aut 0 () be the set of automorphisms of and Aut 1 () be the induced arrows from Mor 1 (, ), i.e. we have the following groupoid Aut() = (Aut 1 () Aut 0 ()) = Mor(, ) Aut 0 (). The main theorem of this section is: Theorem 5.6. Aut() is a K()-gerbe over its coarse space Aut(). Moreover, Aut() is a topological group. The proof of this theorem consists of 5.4 (See Corollary 5.12) and roup action on trivial center topological groupoids. Motived by Theorem 5.6 we may consider group actions on topological groupoids. Definition 5.7. The automorphism group Aut() of is defined to be Aut(). Example 5.8 (Automorphism groupoid of classifying groupoid). Consider the classifying groupoid / := ( ) of a group. The automorphism groupoid is equivalent to the action groupoid Aut(), where Aut() is the group of automorphisms of and acts on it by conjugation. This is a Z()-gerbe over the coarse space Aut()/(/Z()) = Aut()/Inn() = Out(), where Inn() and Out() are the group of inner and outer automorphisms of. Now suppose that is a topological groupoid with trivial K(). Then the automorphism groupoid Aut() is equivalent to the topological group Aut(). This observation leads to the following definition. Definition 5.9. Let K be a group and be a topological groupoid with trivial K(). A K-action on is a morphism satisfying the following two conditions: (1) For every k K, the composition (ψ, H, Φ) : K {k} i k K (ψ,h,φ) induces an automorphism of, where i k is the natural embedding. This defines a map Φ : K Aut 0 ().

24 24 BOHUI CHEN, CHEN-YON DU, AND RUI WAN (2) Φ : K Aut() is a group homomorphism Isotropy groups of automorphisms. Proposition For a (ψ, K, u) : in Aut 0 (), there is a group isomorphism Ψ : K() = Γ (ψ,k,u), σ σ 1 (ψ,k,u), where σ 1 (ψ,k,u) is defined by (5.2) in the proof. Hence Γ (ψ,k,u) is canonically isomorphic to K(). First we find that automorphisms of have the following nice property. Lemma Suppose (ψ, K, u), (φ, L, v) Aut 0 (), and (ψ, K, u) (φ, L, v) α 1, (φ, L, v) (ψ, K, u) β 1. Then the strict morphisms u π 2 π 1 : (L K), and v π 2 π 1 : (K L) are both equivalences. Consequently u 1 : K 1 (k 1, k 2 ) 1 (u 0 (k 1 ), u 0 (k 2 )), and u : K are both surjective. Same properties holds for v. Now we proceed to prove the proposition. Proof of Proposition First of all we define the arrow Ψ(σ) = σ 1 (ψ,k,u) Γ (ψ,k,u). Since it is an arrow from (ψ, K, u) to it self, it is defined over (K K) 0 = {(k 1, x, k 2 ) x : ψ 0 (k 1 ) ψ 0 (k 2 ) in 1 }. It is given by (5.2) σ 1 (ψ,k,u) (k 1, x, k 2 ) : = σ(u 0 (k 1 )) 1 (ψ,k,u) (k 1, x, k 2 ) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x)) (Eq.(3.5)). = u 1 ((ψ 1 ) 1 (x)) σ(u 0 (k 2 )), (Eq.(5.1)). where (ψ 1 ) 1 : k 1 k 2 is the unique arrow in K 1 (k 1, k 2 ) that is mapped to x. We next show σ 1 (ψ,k,u) belongs to Γ (ψ,k,u), i.e. it is a natural transformation u π 1 u π 2 : K K. Take an arrow in (K K) 1 (5.3) (a, (k 1, x, k 2 ), b) : (k 1, x, k 2 ) ( k 1, x, k 2 ).

25 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 25 Hence ψ 1 (a) x = x ψ 1 (b) in 1, and consequently a (ψ 1 ) 1 ( x) = (ψ 1 ) 1 (x) b in K 1 and u 1 (a) : u 0 (k 1 ) u 0 ( k 1 ) in 1. Then we have σ 1 (ψ,k,u) (k 1, x, k 2 ) u 1 (b) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x)) u 1 (b) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x) b) = σ(u 0 (k 1 )) u 1 (a (ψ 1 ) 1 ( x)) = σ(u 0 (k 1 )) u 1 (a) u 1 ((ψ 1 ) 1 ( x)) = u 1 (a) σ(u 0 ( k 1 )) u 1 ((ψ 1 ) 1 ( x)) (Eq.(5.1)) = u 1 (a) σ 1 (ψ,k,u) ( k 1, x, k 2 ). Therefore σ 1 (ψ,k,u) Γ (ψ,k,u), and hence Ψ is well defined. We next show that Ψ is a group homomorphism. For two sections σ, δ K() we have (σ 1 (ψ,k,u) ) (δ 1 (ψ,k,u) )(k 1, x, k 2 ) = σ 1 (ψ,k,u) (k 1, x 1, k 2) δ 1 (ψ,k,u) (k 2, x 2, k 2 ) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x 1 ) u 1 ((ψ 1 ) 1 (x 2 )) δ(u 0 (k 2 )) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x 1 ) (ψ 1 ) 1 (x 2 )) δ(u 0 (k 2 )) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x 1 x 2 )) δ(u 0 (k 2 )) = σ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x)) δ(u 0 (k 2 )) = σ(u 0 (k 1 )) δ(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x)) = (σ δ)(u 0 (k 1 )) u 1 ((ψ 1 ) 1 (x)) = (σ δ) 1 (ψ,k,u) (k 1, x, k 2 ), where x = x y. Hence Ψ(σ) Ψ(δ) = Ψ(σ δ), and Ψ is a group homomorphism. We next construct the inverse map Φ of Ψ. iven an arrow (ψ, K, u) α (ψ, K, u), by applying both α and 1 (ψ,k,u) to the arrow (5.3) in (K K) 1, we get two commutative diagram u 0 (k 1 ) u 1 (a) u 0 ( k 1 ) α(k 1,x,k 2 ) α( k 1, x, k 2 ) u 0 (k 2 ) u 1 (b) u 0 ( k 2 ), and u 0 (k 1 ) 1 (ψ,k,u)(k 1,x,k 2 ) u 0 (k 2 ) u 1 (a) u 1 (b) u 0 ( k 1 ) 1 (ψ,k,u)( k 1, x, k 2 ) u 0 ( k 2 ). Consequently we have α(k 1, x, k 2 ) 1 (ψ,k,u) (k 1, x, k 2 ) 1 u 1 (a) = u 1 (a) α( k 1, x, k 2 ) 1 (ψ,k,u) ( k 1, x, k 2 ) 1. By Theorem 5.11, the map u 1 : K 1 (k 1, k 1 ) 1 (u 0 (k 1 ), u 0 ( k 1 )) is surjective. Therefore α(k 1, x, k 2 ) 1 (ψ,k,u) (k 1, x, k 2 ) 1 y = y α( k 1, x, k 2 ) 1 (ψ,k,u) ( k 1, x, k 2 ) 1

26 26 BOHUI CHEN, CHEN-YON DU, AND RUI WAN for every y 1 with s(y) = u 0 (k 1 ). In particular when k 1 = k 2 = k, x = 1 ψ 0 (k), k 1 = k 2 = k, x = 1 ψ 0 ( k) and any y : u0 (k) u 0 ( k) we have (5.4) α(k, 1 ψ 0 (k), k) y = y α( k, 1 ψ 0 (k), k). Therefore by taking k = k we see that for every k K 0 by α(k, 1 ψ 0 (k), k) Z(Γ u 0 (k)). For every a Im (u 0 ) take a pre-image k K 0 of a under u 0. We first define Φ(α) on Im u 0 Φ(α)(a) := α(k, 1 ψ 0 (k), k) Z(Γ a ). This is independent on the choices of k. Suppose there is another k K 0 satisfying u 0 (k ) = a. Then since u 1 : K 1 (k, k ) 1 (a, a) is surjective, there is an arrow x : k k in K 1 satisfying u 1 (x) = 1 a, which gives us an arrow (x, (k, 1 ψ 0 (k), k), x) : (k, 1 ψ 0 (k), k) (k, 1 ψ 0 (k ), k ) in (K K) 1. By applying α to this arrow we get i.e. α(k, 1 ψ 0 (k), k) = α(k, 1 ψ 0 (k ), k ). α(k, 1 ψ 0 (k), k) u 1 (x) = u 1 (x) α(k, 1 ψ 0 (k ), k ), We next extend Ψ(α) to the whole 0. Since by Theorem 5.11 u : K is surjective, every object b 0 is connected to an object a = u 0 (k) Im u 0 by an arrow x : a b. We then extend Φ(α) to the whole 0 by Φ(α)(b) := x 1 Φ(α)(a) x. One can see that this is similar to the construction of i 1 in the proof of Theorem It is also direct to check that the definition of Φ(α) does not depend on various choices and it is indeed a section in K(). It is direct to see that Φ is the inverse map of Ψ. This finishes the proof. This proposition implies that all objects of the groupoid Aut() have isomorphic isotropy groups. Therefore Corollary The groupoid Aut() is a K()-gerbe over its coarse space Aut() roup structure over Aut(). In this section we show that the coarse space Aut() of the automorphism groupoid of is a group. The proof consists of the following five lemmas. Lemma 5.13 (Multiplication). The composition of morphisms induced a multiplication over the coarse space Mor(, ). Proof. Let (ψ, K, u), (ψ, K, u ), (φ, L, v) Mor 0 (, ) and (ψ, K, u) α (ψ, K, u ) be an arrow. Then we have two arrows (see Construction 4.2) (ψ, K, u) (φ, L, v) α1 (φ,l,v) (ψ, K, u ) (φ, L, v), (φ, L, v) (ψ, K, u) 1 (φ,l,v)α (φ, L, v) (ψ, K, u ).

27 THE ROUPOID STRUCTURE OF ROUPOID MORPHISMS 27 Therefore we get a well-defined multiplication (ψ, K, u) (φ, L, v) = (ψ, K, u) (φ, L, v) on Mor(, ). This finishes the proof. Lemma 5.14 (Associativity). The induced multiplication over Mor(, H) is associative. Proof. For simplicity, here we denote morphisms by a single word. The associativity means that every triple A, B, C of morphisms in Mor 0 () satisfies ( A B ) C = A ( B C ). We have showed in Lemma 4.4 that via the isomorphism of fiber products of a triple of groupoids, the composition (A B) C is identified with A (B C). Then it is direct to construct an arrow (A B) C A (B C). The lemma follows. Lemma 5.15 (Identity). The identity in with respect to the multiplication over Mor(, ) is the image of 1 = (id,, id ) in Mor(, ). Proof. We construct two arrows 1 (ψ, K, u) α 1,(ψ,K,u) (ψ, K, u), (ψ, K, u) 1 for every automorphism (ψ, K, u) Aut 0 (). The composed morphism (ψ, K, u) 1 is α (ψ,k,u),1 (ψ, K, u) id π 1 W := id,,ψ K uπ 2 H. The arrow α (ψ,k,u),1 we want is a natural transformation in the diagram id π 1 ψ π 1 π 2 W L K uπ 2 α (ψ,k,u),1 u H with L = (L 1 L 0 ) := W id π 1,,ψ K. Elements in L 0 is of the form y k (g x k), with x : g ψ 0 (k), and y : g ψ 0 (k ). We denote it by (k, y, g, x, k). From this object we get y 1 x : ψ 0 (k ) ψ 0 (k). Since ψ is an equivalence, we get a unique arrow (ψ 1 ) 1 (y 1 x) : k k. We set α (ψ,k,u),1 (k, y, g, x, k) = u 1 ((ψ 1 ) 1 (y 1 x)). Then it is direct to check that this is the arrow we want. α 1,(ψ,K,u) is defined similarly. Lemma 5.16 (Closedness). Aut() is closed with respect to the multiplication on Mor(, ).

28 28 BOHUI CHEN, CHEN-YON DU, AND RUI WAN Proof. For simplicity, here we also denote morphisms by a single word. Suppose we have automorphisms A, B, A, B Aut 0 (, ) and arrows A A α 1, A A β 1, B B γ 1, B B δ 1. Then we have (A B) (B A ) A (B (B A )) A ((B B ) A ) 1 A (γ1 A ) A (1 A ) (A 1 ) A α A,1 1 A A A α 1 and (B A ) (A B) B (A (A B)) B ((A A) B) 1 A (β1 B ) B (1 B) (B 1 ) B α B,1 1 B B B δ 1, where unmarked arrows are obtained from the Lemma 4.4 and α A,1, α B,1 are the arrows defined in the proof of last Lemma. Therefore A B Aut 0 (, ), and Aut() is closed under the multiplication. Lemma 5.17 (Inverse). Every u has an inverse. Proof. This follows from the definition of Aut 0 (). Combing these five lemmas and noting that 1 Aut 0 (), we finish the proof of that Aut() is a group. Moreover, it is direct to see that the multiplication and inverse maps are continuous, hence Aut() is a topological Lie group. Acknowledgements. The authors thank Lili Shen and Xiang Tang for helpful discussions. The first author was supported by NSFC (No ). The second author was supported by NSFC (No ). The third author was supported by IBS project (#IBS-R003-D1). REFERENCES [ALR] A. Adem, J. Leida, Y. Ruan, Orbifolds and stringy topology. Cambridge University Press, [CR1] W. Chen, Y. Ruan, Orbifold romov Witten theory, Cont. Math. 310 (2002), [CR2] W. Chen, Y. Ruan, A new cohomology theory for orbifold, Commun. Math. Phys. 248 (2004), [LT] E. Lerman, S. Tolman, Hamiltonian torus actions on symplectic orbifolds and toric varieties, Trans. Amer. Math. Soc., 349 (1997), No. 10, [L] T. Leinster, Basic bicategories, arxiv:math/ , [M] K. Mackenzie, Lie groupoids and Lie algebroids in differential geometry, Cambridge University Press, Cambridge, [MM] I. Moerdijk, J. Mrčun, Introduction to foliations and Lie groupoids, Cambridge University Press, Cambridge, [MZ] R. Meyer, C. Zhu, roupoids in categories with pretopology, Theory Appl. Categ. 30 (2015), [P] D. A. Pronk, Étendues and stacks as bicategories of fractions, Compos. Math. 102 (1996), [S] L. Shen, Adjunctions in quantaloid-enriched categories, PhD thesis, Sichuan University, [T1] M. Tommasini, Some insights on bicategories of fractions: representations and compositions of 2- morphisms, Theory Appl. Categ. 31 (2016), [T2] M. Tommasini, A bicategory of reduced orbifolds from the point of view of differential geometry, J. eom. Phys. 108 (2016),