Grothendieck construction for bicategories

Transcription

1 Grothendieck construction or bicategories Igor Baković Rudjer Bošković Institute Abstract In this article, we give the generalization o the Grothendieck construction or pseudo unctors given in [5], which provides a biequivalence between the 2-categories o pseudounctors and ibrations. We deine the second Grothendieck construction which to any pseudo 2-unctor associates a 2-ibration, deined by Hermida [6], and we prove the biequivalence between the 2-category o pseudo 2-unctors and the 2-category o 2-ibrations. 1 Introduction Throughout many areas o mathematics, we are repeatedly aced with two kinds o descriptions o mathematical objects o interest. First, such objects may be described by having certain property, and the ormalism underlying this approach is given by the comprehension scheme o set theory, which says that given any property P, there is a set S, consisting exactly o the elements having the property P. The most advanced orm o this ormalism is the categorical comprehension scheme, given by Gray in [3], in order to avoid a clearly strong dependence o such approach on the axiomatization o the set theory. The second kind o description, is the one in which those properties are encoded as the part o the structure o mathematical objects. The transition between this two approaches is given by the axiom o choice. One o the central themes in the work o Grothendieck, deals with this paradigm, where objects are characterized by certain universal properties, and ater the application o the axiom o choice, these properties give rise to coherence laws which become the part o the structure o derived objects. To illustrate all this by an example, we consider the Grothendieck construction, given in [5], which or any category C, provides the biequivalence : P sf (C) F ib(c) between the 2-category P sf (C) whose objects are pseudounctors P : C op Cat, and the 2-category o ibrations (or ibered categories) F ib(c) over C, also introduced in [5]. The ibered category F : E C is the unctor, which is characterized by the property which associates to any pair (, E), where : X Y is a morphism in C and E is an object in E 1

2 such that F (E) = X, a cartesian morphism : F E, having certain universal property, such that F ( ) =. Ater we choose a cartesian liting or each such pair, their universal property gives rise to the coherence law or the natural transormations which are part o the data or the obtained pseudounctor. Moreover, or any pseudounctor P : C op Cat, this coherence laws are precisely responsible or the associativity and unit laws o the composition in the category C P obtained rom the Grothendieck construction. In this article, we describe the generalization o the Grothendieck construction, or bicategories or weak 2-categories (in the language o Baez), which we thereore call the second Grothendieck construction. More precisely, or any category bicategory B, we provide the triequivalence : P s2f (B) 2F ib(b) between the 3-category P s2f (B) whose objects are pseudo 2-unctors P : B coop 2 Cat (or equivalently homomorphisms o 3-categories given in [2]), and the 3-category o 2- ibrations (or ibered 2-categories) 2F ib(b) over B. The objects o the 3-category 2F ib(b) are (weak) 2-unctors F : E B, again characterized by the existence o cartesian litings, now or both 1-morphisms and 2-morphisms. In the strict case, when both E and B are strict 2-categories, their deinition is given by Hermida in [6]. Although he doesn t deine explicitly 2-ibrations or bicategories, in his other article [7], he proposes their deinition by using the coherence or the birelection o bicategories and their homomorphisms (or weak 2-unctors) into (strict) 2-categories and (strict) 2-unctors. Here we give their explicit deinition, using the properties o certain biadjunctions given by Gray, in his monumental work [4]. We prove that such 2-ibrations precisely arise by applying the second Grothendieck construction to a general pseudo 2-unctor, and that strict 2-ibrations correspond to strict 2-unctors 2-unctors P : B coop 2 Cat. One o the main motivations or this work, was to describe the second Grothendieck construction as the interpretation theorem or the third nonabelian cohomology H 3 (B, K) with the coeicients in the bundle o 2-groups K over the objects B 0 o the bigroupoid B. In the case o groupoids, this approach was taken in [1], where the second nonabelian cohomology o groupoids, is deined by H 2 (G, K) := [B, AUT (K)] in which [G, AUT (K)] is the set deined in terms o connected components o pseudounctors P : G op AUT (K) to the ull sub 2-groupoid o the 2-groupoid Gpd o groupoids, induced naturally be the bundle o groups K over the objects G 0, o the groupoid G. The Grothendieck construction here is an interpretation theorem or the classiication o the groupoid ibrations, seen as short exact sequences o groupoids 1 K G B 1 over the same underlying set o objects M, by means o the higher Schreier theory. 2

3 2 The 2-category o pseudo 2-unctors Deinition 2.1. A pseudo 2-unctor is a weak 3-unctor F : B coop 2 Cat rom the (strict) 2-category B coop to the (strict) 3-category 2-Cat. It consists o the ollowing data: a 2-category F (x), or every object x in B, which we also write F x, a (weak) 2-unctor F (): F (y) F (x), or any 1-morphism : x y in B, which we abbreviate by : F y F x, a pseudo natural transormation F (α): F (g) F (), or any 2-morphism α: g in B, which we also write as α : g, whose components are described by the square g α E (E) (E) g (e) α E (e) g (F ) α F g (F ) or any 1-morphism e: E F in F y, whose coherence is given by the diagram g (F ) g g (e ) (e) g (E) id g (G) α e α F g (e e) α e α E (F ) α G (e ) (e) id (E) (G) (e e) or any composable pair o 1-morphisms E e F e G in the 2-category F y, 3

4 pseudo natural transormation B 2 F 2 2C 2 H χ B 1 F 1 2C 1 where B 2 denotes the disjoint union o products o categories B(y, z) B(x, y), indexed by all triples o objects (x, y, z) in B, and the 2-unctor F 2 : B 2 2C 2 sends each product category B(y, z) B(x, y) to the product o 2-categories [F (y), F (z)] [F (x), F (y)], o 2-unctors, pseudo natural transormations and modiications. We denoted by : 2C 2 2C 1 the composition o 2-unctors, which restricts to the 2- unctor : [F (y), F (z)] [F (x), F (y)] [F (x), F (z)], or any triple (x, y, z) o objects in B. Thus, or any object in B 2, which is just composable pair x g y z o 1-morphisms in B, we have a component χ g, : g (g) o the pseudo natural transormation χ, which is a 1-morphism in 2C 1, that is again a pseudo natural transormation. Also, or any morphism in B 2, which is a just a horizontally composable pair x α y g β z k l o 2-morphisms in B, we have a component χ β,α : g (g) o the pseudo natural transormation χ, which is a 2-morphism in 2C 1, that is a modiication. Thus, the data or the pseudo natural transormation χ, are given by the square k l χ l,k (lk) α β χ β,α (β α) g χ g, (g) 4

5 The coherence or the pseudo natural transormation χ is given by the diagram k l α γ δ χ l,k β g id m n (δβ) (γα) χ β,α χ δ,γ χ g, (l k) (β α) (δ γ) id (g ) (n m) ((δβ) (γα)) χ n,m and the coherence or the modiication is given by the commutative cube k l (χ l,k ) G (G) (lk) (G) (α β ) G (β α) (χ β,α ) G G (lk) (u) k l (u) g (G) (g) (G) (χ g,k ) G (α β ) (χ u l,k ) u k g (u) (χ g, ) u k l (H) (lk) (gk) (u) (H) (χ l,k ) H (α β (β α) ) H (χ β,α ) H (β α) G H g (H) (g) (H) (χ g,k ) H 5

6 or any vertically composable pair o 2-morphisms g x α k γ y β l δ z m n in the 2-category B 2. It has components given by the square g (G) (χ g, ) G (g) (G) g (p) (χ g, ) p (g) (p) g (H) (χ g, ) H (g) (H) or any 1-morphism p: G H in the 2-category F z, whose coherence is given by the 6

7 diagram g (H) g (r) (χ g, ) H g (p) g (G) id g (K) g (rp) (χ g, ) p (χ g, ) r (g) (H) (g) (g) (r) (p) id (g) (G) (g) (K) (χ g, ) G (g) (rp) (χ g, ) K or any composable pair o 1-morphisms H p G r K in the 2-category F z, which means that the pasting o upper two squares is equal to the bottom square given by the component (χ g, ) rp : (χ g, ) K g (rp) (χ g, ) G (g) (rp) (which we omitted rom the diagram too avoid too many labels), since the two triangles are given just by identity 2-morphisms corresponding to two dierent strict 2-unctors, or any composable triple x y g z h w in B, a modiication g h χ h,g (hg) χ g, h ω h,g, χ hg, (g) h χ h,g (hg) 7

8 whose coherence is given by the commutative cube g h ( χ h,g ) M (M) (hg) (M) (χ g, h ) M (ω h,g, ) M (χ hg, ) M g h (m) (g) h (M) (hg) (M) (χ h,g ) M g h (N) (χ g, h ) m ( χ h,g ) m (hg) (m) ( χ h,g ) N (hg) (N) (χ h,g ) m (hg) (m) (χ g, h ) N (g) h (m) (g) h (N) (χ hg, ) m (χ hg, ) N (ω h,g, ) N (χ h,g ) N (hg) (N) or any 1-morphism m: M N in the 2-category F w, or any 1-morphism : x y in B, two modiications i x µ,ix η x ρ i y η y µ i y, λ such that ollowing axioms are satisied: 8

9 the 3-cocycle condition given by a commutative cube g h k χ h,g k (hg) k χ g, h k ω h,g, k χ hg, k g χ k,h (g) h k (hg) k χ h,g k ω k,h,g χ k,hg ω k,h,g g (kh) χ kh,g (khg) χ k,hg χ g, (kh) (g) χ k,h ω χ khg, k,hg, ω kh,g, (g) (kh) χ kh,g (khg) commutative pyramid 9

10 g λ g η y g µ g, ρ i yg µ g,i y g µ i y, g ω h,i y, µ g, g µ g, (g) 10

11 3 2-ibrations The ollowing deinition o biadjunction between two weak 2-unctors is taken rom [8]. Deinition 3.1. A weak 2-unctor F : B C is called a let biadjoint to a weak 2-unctor G: C B i there is an equivalence in a 2-category Hom[B op, Hom[C, Cat]]. C(F (?), ) B(?, G( )) In the above deinition, C(F (?), ): B op Hom(C, Cat) is a weak 2-unctor sending each object x in B to the representable 2-unctor C(F (x), ): C Cat, and similary B(?, G( )): B op Hom(C, Cat) is a weak 2-unctor sending each object x in B to the weak 2-unctor B(x, G( )): C Cat. Let E be a 2-category. The 2-category E o 1-morphisms, associated to E has 1- morphisms o E or objects, thus E0 = E 1. A 1-morphism rom : x y to g : z w is a triple (a, φ, b) consisting o 1-morphisms a: x z, b: y w and a 2-morphism φ: g a b as in the diagram x a z φ g y b w and a 2-morphism rom (a, φ, b) to (c, ψ, d) is a pair (ϱ, ϑ) o 2-morphisms ϱ: a c and ϑ: b d such that the diagram a x ϱ z c y ψ b ϑ w g (in which we have omitted φ in order to avoid too many labels) commutes. d 11

12 Deinition 3.2. A weak 2-unctor F : E B is called a 2-ibration i there exist a 2- unctor L: (B, F ) E which is right biadjoint right inverse to the weak 2-unctor S := (F [1], E δ 1 ): E (B, F ). The 2-unctor S : E (B, F ) is then deined on any object a: p q o E, by S(a) = (F (p), F (a), q). The existence o its right biadjoint right inverse L: (B, F ) E means that or any object (x,, q) in (B, F ), and any object g in E, we have an equivalence Hom (B,F ) (S(g), (x,, q)) Hom E (g, L(x,, q)) o categories, which is pseudo natural in both variables. Thus, or any object (x,, q) in (B, F ) where : x F (q) is a 1-morphism in B, we have L(x,, q) =, where : (q) q is a biuniversal 1-morphism in E, Deinition 3.3. Let F : E B be a weak 2-unctor between bicategories. A 2-cleavage consists o the ollowing data: or each 1-morphism : x y in B, a 2-unctor : E y E x between the ibers, and a pseudo natural transormation θ : J x J y, where J x : E x E is an inclusion 2-unctor o the iber 2-category E x, or each 2-morphism φ: g in B, a modiication Ω φ : θ g θ φ such that ollowing axioms hold: F ((θ ) E ) = or each component o the pseudo natural transormation, indexed by the object E in E y, F ((Ω φ ) E ) = φ or each component o the modiication, indexed by the object E in E y. Remark 3.1. The data or the 2-cleavage correspond precisely to the universal lax 3-cocone g E y φ E x Ω φ J y J x θ E which represent the bicategory E as the lax 3-colimit o the lax 3-unctor corresponding to the 2-cleavage (where we omitted the pseudo natural transormation θ g rom the back ace). 12

13 Deinition 3.4. Let F : E B be a 2-unctor. A 1-morphism : x 1 x 2 in E is 1-cartesian i it is cartesian or the underlying unctor o F. This means that or any 1-morphism h: x 0 x 2 in E, such that there exists a 1-morphism b: F (x 0 ) F (x 1 ) in B, or which F (h) = F () g, pictured by x 0 g h x 1 x 2 F F (x 0 ) F (h) b F (x 1 ) F (x 2 ) F () then there exists a unique 1-morphism g : x 0 x 1, such that h = g and F (g) = b. A 1-morphism : x 1 x 2 is 2-cartesian i it is 1-cartesian and i or any 2-morphism φ: h h in E, and any 2-morphism β : b b in B, such that F (φ) = F () β, x 0 F (x 0 ) h φ h F b β F (h) b F F (h (φ) ) x 1 x 2 F (x 1 ) F () F (x 2 ) there exists a unique 2-morphism γ : g g, such that F (φ) = γ and F (γ) = β. Remark 3.2. The universal property or 2-cartesian 1-morphisms does not imply the universal property or 1-cartesian 1-morphisms. To see this, we take 2-cartesian : x 1 x 2 1-morphism as above, and we consider a 1-morphism h: x 0 x 2 as the identity 2- morphism. By the universal property or 2-morphisms, any 2-morphism β : b b or which F (h) = F () β gives a unique 2-morphism γ : g g, such that h = γ and F (γ) = β. But this gives two actorizations h = g and h = g on the level o 1-morphisms. 13

14 Deinition 3.5. Let F : E B be a 2-unctor. A 2-morphism σ : in E is 1-cartesian i or any 2-morphism α: F (h) g in E, such that there exists a 2-morphism γ : F (h) F () b in B, or which F (h) F (x 0 ) γ b F (x 1 ) F (x 2 ) F () F (σ) = F ( ) F (h) F (x 0 ) F (x 2 ) F (α) b F ( ) F (x 1 ) then there exists a unique 2-morphism φ: h g, such that α = (σ g)φ, as in h x 0 x 2 φ σ g and F (φ) = γ. x 1 = h x 0 x 2 α g x 1 A 2-morphism σ : is 2-cartesian i it is 1-cartesian and i its 1-target : x 1 x 2 is 2-cartesian 1-morphism. The ollowing deinition is given by Hermida in [6]. Deinition 3.6. A (strict) 2-unctor F : E B is a 2-ibration i it satisies ollowing conditions: or any pair (, E), where : x y is a 1-morphism in B, and E is an object in E y, there exists a 2-cartesian 1-morphism : (E) E, such that F ( ) =, or any pair (φ, E), where φ: g is a 2-morphism in B, and E is an object in E y, there exists a 2-cartesian 2-morphism φ: g, such that F ( φ) = φ. Ater we describe the universal properties o cartesian 1-morphisms and 2-morphisms corresponding to 2-ibrations o bicategories, we will prove the ollowing important characterization. Proposition 3.1. Let F : E B be a (weak) 2-unctor between bicategories. Then F is a 2-ibration i and only i it has a 2-cleavage. 14

15 4 The second Grothendieck construction Given a pseudo 2-unctor F : B coop 2 Cat as in the previous section, we will construct a 2-category E, as a lax 2-colimit o F. It will be a total 2-category o the corresponding 2-ibration P : E B. The objects o E are pairs (x, E) where x is an object o B, and E is an object o the 2-category F (x). A 1-morphism between two objects (x, E) and (y, F ) is a pair (, a): (x, E) (y, F ) such that : x y is a 1-morphisms in B, and m: E (F ) is a 1-morphism in F (x). A 2-morphisms between two 1-morphisms (, a) and (g, b) is a pair (α, φ): (, a) (g, b) such that α: g is a 2-morphisms in B, and φ: αf b a is a 2-morphism in F (x) g (F ) b α F φ E (F ) a where α F : g (F ) (F ) is the component indexed by the object F in F (y), o the pseudo natural transormation F (α): F (g) F (). For any composable pair (x, E) (,a) (y, F ) (g,b) (z, G) o 1-morphisms in E, we deine the composition (g, ba) = (g, b) (, a) where g : x y is the composition o 1-morphisms in B, and ba: E (g) (G) is deined by the composition E a (F ) (b) g (G) (χ g, ) G (g) (G) and in the general case this composition will not be strictly associative (like in the oridnary Grothendieck construction), unless the pseudo natural transormation χ is strict natural transormation. Thus, this composition will be coherently associative, and we deine associativity coherence or any three horizontally composable 2-morphisms in B, g h x α y β z γ w k l m 15

16 as ollowing. For any composable triple (x, E) (,a) (y, F ) (g,b) (z, G) (h,c) (w, H) o 1- morphisms in E, we have the composition (h(g), c(ba)) = (h, c) ((g, b) (, a)) where h(g): x y is the composition o 1-morphisms in B, and c(ba): E (h(g)) (H) is deined by the composition E a (F ) (b) g (G) (χ g, ) G (g) (G) (g) (c) (g) h (H) (χ h,g ) H (h(g)) (H) o 1-morphisms in the 2-category F x. Since (hg, cb) = (h, c) (g, b) where cb: F (hg) (H) is given by the composition F b g (G) g (c) g h (H) (χ h,g ) H (hg) (H) we have that ((hg), (cb)a) = ((h, c) (g, b)) (, a) = (hg, cb) (, a) is given by the composition E a (F ) which is equal to the composition (cb) (hg) (H) (χ hg, ) H ((hg)) (H) E a (F ) (b) g (G) g (c) g h (H) ((χ h,g ) H ) (hg) (H) (χ hg, ) H ((hg)) (H) o 1-morphisms in the 2-category F x. The resulting two compositions ((hg), (cb)a) and (h(g), c(ba)), will generally not be equal, i two natural 2-morphisms in the diagram E a (F ) (b) g (G) g (c) g h (H) ((χ h,g ) H ) (hg) (H) (χ g, ) G (χ g, ) c (χ g, ) h (H) (ω h,g, ) H (χ hg, ) H (g) (G) (g) (c) (g) h (H) (χh,g ) H (hg) (H) are not identities, that is i χ is not strict natural transormation and i the modiication ω is not identity. The reason is that ((hg), (cb)a) is obtained by the composition o E a (F ) (b) g (G) with the top and right edges o the diagram, while (h(g), c(ba)) is obtained by the composition o the same 1-morphism with the let and bottom edges o the diagram. We deine the 2-morphism α c,b,a : ((hg), (cb)a) (h(g), c(ba)) by the pasting composite o the 1-morphism E a (F ) (b) g (G) with the component 2-morphisms in the interiors o the above diagram, or more explicitly α c,b,a := ((χ h,g ) H (χ g, ) c (b) a)((ω h,g, ) H g (c) (b) a) 16

17 For vertical composition o any two 2-morphisms α g α h in B x α g α y x α α y h h the vertical composition o two 2-morphisms (, a) (α,φ) (g, b) (α,φ ) (h, c) in E is given h (F ) c α F φ E g (F ) b φ h (F ) c E φ φ (α α) F by φ φ := φ (α F φ). a α F (F ) a (F ) 17

18 For any horizontally composable pair o 2-morphisms in B, x α y g β z k l and or any two horizontally composable pair o 2-morphisms in E which covers them k (F ) u α F φ E (F ) a l (G) v βg ψ F g (G) b their horizontal composition ψ φ: vu ba is represented by the 2-morphism in the triangle (lk) (G) vu ψ φ E (g) (G) ba (β α) G 18

19 This horizontal composition is deined by the pasting composition o the diagram k l (G) k (v) k (ψ) k (F ) k g (G) k (b) u α F (α ) b φ E (F ) g (G) a (b) (k β ) G (α g ) G which is equal to the pasting composition o the diagram (χ l,k ) G (lk) (G) (χ β,k ) G (β k) G (gk) (G) (χ g,k ) G (χ g,α) G (g α) G (χ g, ) G (g) (G) E u k (F ) k (v) k l (G) (χ l,k ) G (lk) (G) φ a α F (α ) v (α l ) G (χ l,α ) G (l α) G (F ) (v) (ψ) l (G) (χ l, ) G (l) (G) (b) ( β ) G (χ β, ) G (β ) G g (G) (χ g, ) G (g) (G) This two pasting composites are equal since the irst is obtained by the pasting composite 19

20 o the top and ront aces o the diagram k l (χ l,k ) G (G) (lk) (G) k (k β ) G (β k) (χ β,k ) G G (l α) (v) k (ψ) G k (F ) k g (G) (gk) (G) (χ g,k ) G α F (F ) k (b) (α l ) G (α ) (χ v l,α ) G (α ) b (α g ) G l (G) (χ l, ) G ( β ) (v) G (χ β, ) G (ψ) (b) g (G) (l) (G) (χ g,α ) G (β ) G (χ g, ) G (g) (G) (g α) (G) and the second one is obtained by pasting back and bottom aces, consisting o squares whose edges are given by broken arrows. This diagram commutes since the prism is a coherence, whose square which shares with the cube is an identity 2-morphism o the 1- morphism (α β ) G : k l (G) g (G), which is a component o the pseudo natural transormation α β : k l g, indexed by the object G in the 2-category F z. The cube is just the coherence or the modiication χ which is a component indexed by the horizontal composition β α o the pseudo natural transormation χ, so the vertical edge o the above horizontal composition is equal to the component (β α) G : (lk) (G) (g) (G) indexed by the same horizontal composite. In order to prove that E is really a bicategory, we need to show that the Godement interchange law holds, and that the above horizontal composition is associative up to the coherent associativity isomorphism given by α c,b,a : ((hg), (cb)a) (h(g), c(ba)), or any three composable 1-morphisms. For any three horizontally composable 2-morphisms in B, g h x α y β z γ w k l m 20

21 and or any three horizontally composable 2-morphisms in E which covers them k (F ) u α F φ E (F ) a l (G) v βg ψ F g (G) b m (H) w γh ρ G h (H) c we need to show that α w,v,u ((ρ ψ) φ) = (ρ (ψ φ)) α c,b,a. The 2-morphism ρ (ψ φ) on right hand side is given by the pasting k l (G) k (v) k (ψ) k (F ) k g (G) k (b) u α F α c φ E (F ) g (G) a (b) (βk) G (α g ) G (lk) m (χ m,lk ) H (H) (mlk) (H) (lk) (w) ((lk) γ) ) H (χ γ,lk ) H (lk) (ρ) (χ l,k ) G (lk) (G) (lk) h (H) (hlk) (H) (lk) (c) (χ h,lk ) H (χ β,k ) G (χ g,α ) G (βk) ) G (βk) c ((βk) h ) H (χ h,βk ) H (γlk) H (hβk) H (gk) (G) (gk) h (H) (hgk) (H) χ g,k ) G (gk) (c) (χ h,gk ) H (gα) G (gα) c ((gα) h ) H (χ h,gα ) H (hgα) H (g) (G) (g) h (H) (hg) (H) χ g, ) G (g) (c) (χh,g ) H o the above diagram in which horizontal compositions are denoted by the concatenation. 21

22 By the deinition, the horizontal composition ρ ψ is given by the pasting o the diagram l m (χ m,l ) H (H) (ml) (H) l (w) (l γ ) H (χ γ,l ) H l (ρ) l (G) l h (H) (hl) (H) l (c) (χ h,l ) H v β G (β (β ) h ) G c (χ h,β ) H ψ F g (G) g h (H) (hg) (H) (χh,g ) H b g (c) (γ l) G (h β) H and the horizontal composition (ρ ψ) φ on the let hand side o the equation is given by k l m k ((χ m,l ) H ) (H) k (ml) (χ ml,k ) H (H) (mlk) (H) k l (w) (k l γ ) H (k (γl) ) H, k l (ρ) k ((χ γ,l ) H ) (χ γl,k ) H k l (G) k l h (H) k (hl) (H) (hlk) (H) k l (c) k ((χ h,l ) H ) (χ hl,k ) H k (v) k (βg ) (k β h ) G (k (hβ) ) H k ((β ) c ) k ((χ h,β ) H ) (χ hβ,k ) H k (ψ) k (F ) k g (G) k g h (H) k (hg) (H) (hgk) (H) k (b) k g (c) k ((χ h,g ) H ) (χ hg,k ) H u α F (α g ) G (α g h ) H (α (hg) ) H α c (α g ) u α ((χ hg,l ) H ) (χ hg,α ) H φ E a (F ) g (G) g h (H) (hg) (H) (hg) (H) (b) g (c) ((χ h,g ) H ) (χhg, ) H the pasting o the above diagram, in which all compositions are denoted by concatenation. By the above construction, we have the ollowing main theorem. (γlk) H (hβk) H (hgα) H 22

23 Theorem 4.1. For any pseudo 2-unctor F : B coop 2 Cat, the second Grothendieck construction gives a 2-ibration F : E B. Proo. We deine the 2-unctor F : E B on any object (x, E) o E by F(x, E) := x. Also or any 1-morphism (, a): (x, E) (y, F ) we deine F(, a) :=, and or any 2-morphism (α, φ): (, a) (g, b) we deine F(α, φ) := α. It is easy to see that such 2-unctor is really a 2-ibration. 23

24 Reerences [1] Blanco, V.; Bullejos, M.; Faro, E. Categorical non-abelian cohomology and the Schreier theory o groupoids. Math. Z. 251 (2005), no. 1, 41 59, [2] R. Gordon, A. J. Power, and R. Street, Coherence or tricategories, Memoirs Amer. Math. Soc. 117 (1995) Number 558, [3] Gray, John W. The categorical comprehension scheme Category Theory, Homology Theory and their Applications, III (Battelle Institute Conerence, Seattle, Wash., 1968, Vol. Three) pp Springer, Berlin, [4] Gray, John W. Formal category theory: adjointness or 2-categories, Lecture Notes in Mathematics, Vol. 391, Springer-Verlag, Berlin-New York, xii+282 pp. [5] A. Grothendieck, Catégories ibrées et descente, Exposé VI in Rêvetements étales et groupe ondamental, editor A. Grothendieck, Seminaire de Geometrie Algebrique I, LNM 224, Springer-Verlag, 1971, [6] C. Hermida, Some properties o F ib as a ibred 2-category. J. Pure Appl. Algebra 134 (1999), no. 1, [7] C. Hermida, Descent on 2-ibrations and strongly 2-regular 2-categories. Appl. Categ. Structures 12 (2004), no. 5-6, , [8] G.M. Kelly, Elementary observations on 2-categorical limits, Bull. Austral. Math. Soc., Vol. 39 (1989), , 24