Fibrations of bicategories

Transcription

1 Fibrations o bicategories Igor Baković Faculty o Natural Sciences and Mathematics University o Split Teslina 12/III, Split, Croatia Abstract The main purpose o this work is to describe a generalization o the Grothendieck construction or pseudounctors which provides a biequivalence between the 2-categories o pseudounctors and ibrations. For any bicategory B, we describe a tricategory o B- indexeded bicategories which are the weakest possible generalization o pseudounctors to the immediate next level. Then we introduce a notion o a ibration o bicategories, which generalizes 2-ibrations o strict 2-categories deined by Hermida. Finally we give an explicit construction o an analog o the Grothendieck construction which provides a triequivalence between the 3-category o B-indexeded bicategories and the 3-category o 2-ibrations over B. 1

2 1 Introduction The main purpose o this paper is to introduce the notion o a ibration o bicategories as a natural generalization o many existing amiliar notions existing in the category theory. This theory has its historical origins in two separate lines o mathematical development: ibration o categories or ibred categories, introduced by Grothendieck [29] or purely geometric reasons, and bicategories, introduced by Bénabou [7], as the algebraic structures which are the weakest possible generalization o categories, to the immediate next level. The irst coming-together o the these two lines o development can be traced in mostly unpublished work by Bénabou, in his investigations o logical aspects o ibred categories, and his attempt to give oundations o naive category theory [8] by using ibred categories. It was already pointed by Street that ibrations o bicategories are quite dierent rom ibrations in bicategories, in the paper [53] bearing that name. The simple observation that unctors which are equivalences o categories are not necessarily ibred categories, led him to introduce a bicategorical notion o two-sided ibration, which generalized his earlier notion o two-sided ibrations in 2-categories [52]. These ibrations are certain spans in a bicategory, and by using this construction, it was possible to express ibred and coibred categories [25] as such spans in a 2-category Cat o small categories. The main distinction between ibrations o bicategories and ibrations in bicategories rom the previous paragraph is that ibrations o bicategories should be seen as ibrations in the tricategory Bicat o bicategories, their homomorphisms, pseudonatural transormations and modiications. Tricategories were introduced by Gordon, Power and Street [24] as the weakest possible generalization o bicategories, on the 3-dimensional level. Their motivation to introduce tricategories came rom several dierent sources. One o these sources was within category theory and Walters s work on categories enriched in a bicategory [54], [55] which required monoidal bicategories which are precisely one object tricategories. Another example o such a structure is the bicategory o relations in a regular category explored by Carboni and Walters [17]. There were also another sources o motivation to introduce tricategories rom outside category theory. Namely, homotopy 3-types o Joyal and Tierney [41] are special types o tricategories. Monoidal bicategories bear the same relation to the algebraic aspects o Knizhnik-Zamolodchikov equations [35], [36] analogous to the relation between monoidal categories and Yang-Baxter equation. Gordon, Power and Street deined trihomomorphisms between tricategories as the weakest possible generalization o homomorphisms o bicategories, on the 3-dimensional level. Special types o trihomomorphisms, whose domain is a ixed bicategory and codomain the tricategory Bicat, are generalization o pseudounctors, extensively used by Grothendieck, who showed the biequivalence between them and ibred categories. Such special trihomomorphisms will be called indexed bicategories, as they are natural generalization o indexed categories used by Janelidze, Schumacher and Street [39] in the context o categorical Galois theory. Indexed bicategories will play a undamental role in this paper since they are objects o a tricategory which is triequivalent to the 3-category o ibrations o bicategories. 2

3 This triequivalence is a generalization o the amous Grothendieck construction [29], which or any category C, provides a biequivalence : Cat C op F ib(c) (1.1) between the 2-category Cat Cop whose objects are indexed bicategories or pseudounctors P : C op Cat, and the 2-category F ib(c) o ibered categories over C, introduced in [29]. 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 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 introduce the generalization o the Grothendieck construction, which or any bicategory B, provides a triequivalence : Bicat B coop 2F ib(b) (1.2) between a tricategory Bicat Bcoop whose objects are indexed bicategories P : B coop Bicat, and the 3-category 2F ib(b) o 2-ibrations o bicategories (or ibered bicategories) over B. The objects o the 3-category 2F ib(b) are strict homomorphism P : 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, the deinition o such ibrations was given by Hermida in [33]. Although he did not deine explicitly 2-ibrations or bicategories, in his subsequent paper [34], he proposed their deinition by using the birelection o the category Bicat o bicategories and their homomorphisms into the category 2-Cat o (strict) 2-categories and (strict) 2-unctors. Here we give their explicit deinition, using properties o certain biadjunctions introduced by Gray in his monumental work [27]. There are two major reasons why we restricted our attention to strict homomorphisms o bicategories. The irst one is that these are the morphisms o a category Bicat s, which is essentially algebraic in the sense o Freyd [21], and as such Bicat s is both complete and cocomplete, and in act locally initely presentable [1]. More precisely, Bicat s is a category o Eilenberg-Moore algebras or some monad K 2 on the category Cat-Gph o Cat-graphs. For any symmetrical monoidal category V satising certain (co)completeness conditions, Wol showed in [56] that the orgetul unctor U V : V-Cat V-Gph is monadic, with a let adjoint F V : V-Gph V-Cat. This was the starting point o Batanin s work [6] who gave an explicit description o a monad K n on n-gph, deined by U n F n : n-gph n-gph in the special case when V is a symmetrical monoidal category n-gph o n-graphs. In this way, Batanin realized weak n-categories as objects in a category o Eilenberg-Moore algebras or such monads, whose morphisms are strict, preserving all their structure precisely. 3

4 The practical importance o the act that the category Bicat s o bicategories and their strict homomorphisms is both complete and cocomplete, unlike the category Bicat, is that it allows us to deal with strict ibers o any strict homomorphism P : E B o bicategories. Strict ibers o such homomorphisms are deined as their (strict) pullbacks in Bicat s by a strict homomorphism rom the ree bicategory on one object. I we would have been working instead with general homomorphisms o bicategories, as morphisms o the category Bicat, we would be orced to deal with homotopy ibers which would inevitably cause many (unnecessary) technical diiculties in our constructions. This leads us to the second, even more important reason, why we restricted our attention to strict homomorphisms o bicategories. Even when we apply the Grothendieck construction (6.2) to the most general indexed bicategory P : B coop Bicat, whose codomain is the tricategory Bicat o homomorphisms o bicategories, pseudonatural transormations and modiications, we still obtain a ibration o bicategories which is strict homomorphism o bicategories! On the other side, the same result is obtained when we apply the Grothendieck construction (6.2) to an indexed 2-category P : B coop 2-Cat whose codomain is the 3- category 2-Cat o 2-categories and 2-unctors, 2-transormations and modiications. Although it may seem that strict homomorphisms o bicategories are rare in practice in category theory or its applications, they occur quite naturally in the homotopy theory, were Lack used them [44] in order to introduce a closed model structure on the category 2-Cat o 2-categories and 2-unctors, which he extended to the category Bicat s o bicategories and their strict homomorphisms, in his subsequent paper [45]. These model structures are closely related to the model structure o Moerdijk and Svensson in [51], and our ibrations o bicategories naturally it in these examples. Strict homomorphisms were also used by Hardie, Kamps and Kieboom who deined ibrations o bigroupoids in [32], generalizing the notion o ibration o 2-groupoids by Moerdijk [50]. They used Brown s construction [11] in order to derive an exact nine term sequence rom such ibrations, and they applied their theory to the construction o a homotopy bigroupoid o a topological space [31]. One o the irst author s motivations or this work, was to use the Grothendieck construction (6.2) as an interpretation theorem or the third nonabelian cohomology H 3 (B, K) o a bigroupoid B with coeicients in a bundle o 2-groups K. In the case o groupoids, this approach was taken in [9], where the second nonabelian cohomology o a groupoid G H 2 (G, K) := [G, AUT (K)] is the set [G, AUT (K)] o connected components o pseudounctors P : G op AUT (K) to the ull sub 2-groupoid AUT (K) o the 2-groupoid Gpd o groupoids, induced naturally be the bundle o groups K over the objects G 0 o the groupoid G. Then the Grothendieck construction (6.1) was used as an interpretation theorem or the classiication o ibrations o groupoids, 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. 4

5 2 Bicategories and their homomorphisms Bicategories were originally introduced by Bénabou [7]. In this section we will not ollow Bénabou s original presentation o bicategories, but rather we will use a more recent approach based on simplicial techniques. Deinition 2.1. Skeletal simplicial category consists o the ollowing data: objects are inite nonempty ordinals [n] = {0 < 1 <... < n}, morphisms are monotone maps : [n] [m], which or all i, j [n] such that i j, satisy (i) (j). We also call the topologist s simplicial category, and this is a ull subcategory o the algebraist s simplicial category, which has an additional object [ 1] =, given by a zero ordinal, that is an empty set. Skeletal simplicial category may be presented by means o generators as in a diagram [0] [1] [2] [3] 0 and relations given by the maps i : [n 1] [n] or 0 i n, called coace maps, which are injective maps that omit i in the image, and the maps σ i : [n] [n 1] or 0 i n 1, called codegeneracy maps, which are surjective maps which repeat i in the image. These maps satisy ollowing cosimplicial identities: j i = i j 1 (i < j) σ j σ i = σ i σ j+1 (i j) σ j i = i σ j 1 (i < j) σ j i = id (i = j, i = j + 1) σ j i = i σ j+1 (i > j + 1). We will use the ollowing actorization o monotone maps by coaces and codegeneracies. Lemma 2.2. Any monotone map : [m] [n] has a unique actorization given by = n i 1 n 1 i 2... i n s+1 s σ m t j t...σ m 2 j (2.1) σ m 1 j 1 (2.2) where 0 i s < i s 1 <... < i 1 n, 0 j t < j t 1 <... < j 1 m and n = m t + s. Proo. The proo is a direct consequence o the injective-surjective actorization in Set and cosimplicial identities. Deinition 2.3. An internal simplicial object S in a category C is a preshea S : op C on the skeletal simplicial category with values in C. 5

6 We can regard each ordinal [n] as a category whose objects are elements o a set {0, 1,..., n}, and such that there exists a unique morphisms rom i to j i and only i i j. In this way, we obtain a ully aithul embedding rom the skeletal category i: Cat (2.3) into the category Cat o small categories and unctors between them, since a monotone maps between any two ordinals [k] and [m] are precisely unctors between corresponding categories i([k]) and i([m]). We shall write [C, D] or D C or a category whose objects are unctors rom C to D, and whose morphisms are natural transormations between them. We will denote by an image o the zeroth ordinal [0] by a ully aithul embedding (2.3), but more oten we will abbreviate notation and use the term [k] to denote both the k-th ordinal and the category i([k]). We shall always identiy categories C and C with C, and use two natural embeddings 0 C : C [1] C 1 C : C [1] C For any category C, we also have two unctors rom the category C [1] o morphisms o C C 0 : C [1] C C 1 C [1] C (2.4) (2.5) whose objects are morphisms o C, and whose morphisms are their commutative squares. We call the two unctors (2.5) the codomain and the domain unctor, respectively. More generally, or any ordinal [n] we have a category C [n] whose objects are strings o n composable morphisms o C, and whose morphisms are appropriate commutative diagrams. These categories C [n] together with degeneracy unctors C σ i : C [n 1] C [n] or all 0 i n 1, and ace unctors C i : C [n] C [n 1] or all 0 i n, orm an internal simplicial object in Cat. The category C [2] o composable pairs o morphisms o C is a ollowing pullback in Cat C [2] C 2 C [1] C 0 C 0 (2.6) C [1] C 1 C and we adopt the ollowing convention: or any unctor P : D C, the irst o the symbols E C D and D C E will denotes the pullback o P and D 0, and the second one is the pullback o D 1 and P. 6

7 Deinition 2.4. A Cat-graph B consists o a discrete category B 0 o objects, and a category B 1 o morphisms, together with two unctors B 1 D 1 D 0 B 0 (2.7) where D 1 : B 1 B 0 is called a domain unctor and D 0 : B 1 B 0 a codomain unctor. There exists a category Cat-Gph whose objects are Cat-graphs, and morphisms F : B G o Cat-graphs consists o two unctors F 0 : B 0 G 0 and F 1 : B 1 G 1 such that diagrams B 1 F 1 G 1 B 1 F 1 G 1 D 0 D 0 D 1 D 1 (2.8) B 0 F 0 G 0 B 0 F 0 G 0 commute. I or any two objects x and y o a Cat-graph B, a category B(x, y) is deined as a pullback B(x, y) J x,y B 1 (2.9) T (D 0,D 1 ) (x,y) B 0 B 0 then and morphism F : B G o Cat-graphs may be seen as a unction F 0 : B 0 G 0 together with a amily o unctors F x,y : B(x, y) G(F 0 (x), F 0 (y)) or any two objects x, y in B 0. Let the two categories B 2 and B 3 be deined as the ollowing two pullbacks in Cat B 2 D 0 B 1 B 3 D 0 B 1 D 2 D 1 D 3 D 2 (2.10) B 1 D 0 B 0 B 1 D 0 B 0 o horizontally composable pairs and triples, respectively, o 1- and 2-morphisms in B. 7

8 Deinition 2.5. A bicategory B consists o the ollowing data: two categories, a discrete category B 0 o objects, and a category B 1 o morphisms, whose objects, g, h,... are called 1-morphisms o the bicategory B and whose morphisms φ, ψ, θ,... are called 2-morphisms o the bicategory B, and their composition in the category B 1, which we call the vertical composition o the bicategory B, is denoted by the concatenation ψφ, whenever the composition o φ and ψ is deined two unctors D 0, D 1 : B 1 B 0, called target and source unctors, respectively, a unit unctor I : B 0 B 1, any a horizontal composition unctor H : B 2 B 1, whose value H(g, ) on object (g, ) in B 2 is denoted by g. For any two objects x and y in B 0, we denote by B(x, y) a hom-category whose objects are 1-morphisms : x y in B such that D 0 () = y and D 1 () = x. a natural isomorphism B 3 H Id B1 B 2 Id B1 H α H (2.11) B 2 H B 1 natural isomorphisms B 2 S 1 S 0 H λ ρ B 1 B 1 B 1 (2.12) where the unctor S 0 : B 1 B 2 is deined by the composition B 1 (D 0,Id B1 ) B 1 B0 B 0 I Id B1 B 1 B0 B 1, and the unctor S 1 : B 1 B 2 is deined by the composition B 1 (Id B1,D 1 ) B 0 B0 B 1 Id B1 I B 1 B0 B 1, so that the values o these two unctors on any 1-morphism : x y in B are S 0 () = (, i x ) and S 1 () = (i y, ), respectively, and similarly or 2-morphisms in B. 8

9 This data are required to satisy the ollowing axioms: or any object (k, h, g, ) in B 4 we have the ollowing commutative pentagon ((k h) g) α k,h,g α k h,g, (k (h g)) (k h) (g ) α k,h g, α k,h,g k ((h g) )) k (h (g )) k αh,g, (2.13) or any object (g, ) in B 2 we have the ollowing commutative triangle α g,iy, (g i y ) g (i y ) ρ g g λ g (2.14) Remark 2.6. Note that a deinition o the horizontal composition unctor H : B 2 B 1 implies that or any composable pair o morphisms in the category B 2, as in a diagram x 1 φ 1 y φ 2 2 ψ 1 z ψ 2 we have a ollowing Godement interchange law h 1 h 2 (ψ 2 ψ 1 )(φ 2 φ 1 ) = (ψ 2 ψ 1 ) (φ 2 φ 1 ). (2.15) 9

10 Example 2.7. (Strict 2-categories) A bicategory in which associativity and let and right identity natural isomorphisms are identities is called a (strict) 2-category. A basic example is the 2-category Cat o small categories, unctors and natural transormations. Example 2.8. (Monoidal categories) Any monoidal category M may be seen as a bicategory or which the discrete category o objects is a terminal category with only one object and one morphism. Consequently, any strict monoidal category M may be seen as a strict 2-category with a terminal category o objects. Then it is clear that or any object x in the bicategory B, a hom-category B(x, x) is a (weak) monoidal category, and similarly or any object x in the strict 2-category C, a hom-category C(x, x) is a (strict) monoidal category. Example 2.9. (Bicategory o spans) Let C be a cartesian category, that is a category with pullbacks. First we make a choice o the pullback u y v q v p h u g z or any such diagram x z g y in a category C. Then there exists a bicategory Span(C) o spans in the category C whose objects are the same as objects o C. For any two objects x, y in Span(C), a 1-morphism u: x y is a span u g x y and a 2-morphism a: u w between any two such spans is given by a commutative diagram u g x a y l m w 10

11 The vertical composition o 2-morphisms is given by the composition in C. The horizontal composition o 1-morphisms is given by the pullback u v g h k x y z u y v p q u v g h k x y z and the horizontal identity i x : x x or this composition is a span x id x id x x x. It is obvious that the horizontal composition is not strictly associative, but it ollows rom the universal property o pullbacks that it is associative up to canonical isomorphisms which deine components o the associativity coherence or given choices o pullbacks. The ollowing two examples o bicategories are important and they can be both applied to the special case o bicategory Span(C) o spans in a cartesian category C rom the previous example. Example (Bimodules) Let Bim denote the bicategory whose objects are rings with identity. For any two rings A and B, Bim(A, B) will be a category o A-B-bimodules and their homomorphisms. Horizontal composition is given by the tensor product, and associativity and identity constraints are the usual ones or the tensor product. Example (Bimodules internal in a bicategory) Let B be a bicategory, whose homcategories have coequalizers, which are preserved by a (pre)composition with 1-morphisms in B. Then a construction rom a previous example can be extended to such bicategory B in the ollowing way. There exists a bicategory Bim(B) whose objects are pairs (x, M), 11

12 where x is an object in the bicategory B, and M is a monoid in a monoidal hom-category B(x, x). This means that the 1-morphism M : x x is supplied with two 2-morphisms µ M : M M M and ɛ M : i x M in B(x, x), such that the ollowing diagrams (M M) M µ M ι M M M α M,M,M µ M M (M M) ιm µ M M M µ M M ɛ M ι M ι i x M M ɛ M M M M i x µ M λ M ρ M M commute. We will usually reer to a monoid M in B(x, x) without mentioning explicitly the these two 2-morphisms. Then or any monoid M in B(x, x), and any monoid N in B(y, y), a 1-morphism L: x y is called a N-M-bimodule i there are 2-morphisms φ: N L L and ψ : L M L such that the diagrams (N N) L µ N ι L N L (L M) M ψ L M α N,N,L φ α L,M,M ψ N (N L) ιn N L φ φ L L (M M) ιl µ M L M φ L (N L) M φ ι M L M α N,L,M ψ N (L M) ι M ψ M L φ M 12

13 ɛ i y L N ι L N L φ λ L L ι L ɛ M L M L i x ψ ρ L L commute. Then a 1-morphism rom (x, M) to (y, N) in Bim(B) is an N-M-bimodule L. A 2-morphism between N-M-bimodules K and L in Bim(B) is a 2-morphism ν : K L in B such that diagrams N K κ K K M θ K ι N ν ν ν ι M ν N L φ L L M ψ L commute. For any N-M-bimodule L: x y, which we denote by N L M, and or any P -N-bimodule T : y z, denoted by P T N, their horizontal composition is a tensor product P T N N L M which is a P -M-bimodule T N L: x z obtained as a coequalizer in the category B(x, z) T N L ψ L T φ T L e T N L. The composition o 2-morphisms is induced by the universal property o the coequalizer, and the associativity coherence ollows since these coequalizers are preserved by a composition with 1-morphisms in B. Remark I the monoidal category Ab o abelian groups, is seen as a bicategory ΣAb with one object, with the usual tensor product o abelian groups as a horizontal composition, then the construction o a bicategory Bim(ΣAb) rom the previous example yields a usual bicategory Bim o bimodules rom Example 1.9, so that we have an identity o bicategories Bim(ΣAb) = Bim 13

14 We can combine the two previous examples and construct a bicategory Bim(Span(C)) o bimodules in a bicategory Span(E) o spans in a cartesian category E. For any object B 0 in the category E, the span B 1 d 0 d 1 B 0 B 0 (2.16) is a monoid i there exists a morphism m: B 1 B0 B 1 B 1 which is a morphism o spans B 1 B0 B 1 d 0 d 0 d 1 d 2 B 0 m B 0 d 0 d1 and a morphism i: B 0 B 1 which is a part o the ollowing morphism o spans B 1 B 0 id B0 id B0 i d 0 d1 B 1 B 0 B 0 These two morphisms o spans are required to satisy the associativity and identity constraints, which means that we have ollowing commutative diagrams B 1 B0 B 1 B0 B 1 d 0 d 1 d 2 m id B1 d 1 d 2 d 3 id B1 m B 1 B0 B 1 d 0 d 2 B 0 m B 0 d 1 d 2 d 0 d 1 B 1 B0 B 1 d 0 d 1 m B 1 14

15 B 1 d 0 d 1 i id B1 idb1 i B 1 B0 B 1 d 0 d 0 m id B1 d 1 d 2 d 1 d 2 d 0 d 1 B 1 B0 B 1 d 0 d 1 m B 1 B 0 B 0 This means that the span (2.16) is an underlying graph o an internal category B in E. Thereore, the objects o the bicategory Bim(Span(E)) are internal categories in E. For any two internal categories B and C, a 1-morphism rom B to C is a span K 0 µ 0 µ 1 (2.17) B 0 C 0 together with a morphism l : B 1 B0 K 0 K 0 which is a part o a morphism o spans B 1 B0 K 0 d 0 pr 1 µ 1 pr 2 B 0 l C 0 µ 0 µ1 K 0 and a morphism r : K 0 C0 C 1 K 0 which is a part o the ollowing morphism o spans K 0 C0 C 1 µ 0 pr 1 d 1 pr 2 B 0 r C 0 µ 0 µ1 K 0 Obviously, these two morphisms are let and right action o internal categories B and C respectively, and by constructing the category o elements K or these two actions, one 15

16 obtains the span o internal categories which is a discrete ibration [52], [53] rom B to C D 0 K D 1 B C (2.18) Finally, 2-morphisms between such discrete ibrations are just morphisms between their underlying spans. Thereore, the bicategory Bim(Span(E) is isomorphic to the bicategory P ro o prounctors or distributors. 16

17 The ollowing result is a typical instance o the coherence theorem or bicategories, which roughly says that any diagram in the bicategory which is made o associativity, let and right identity coherence isomorphisms must commute [49]. Theorem Let B be a bicategory. Then the diagrams α iz,g, (i z g) i z (g ) λ g i z λ g g α g,,ix (g ) i x g ( i x ) ρ g g ρ g (2.19) commute or any pair o 1-morphisms x y g z in B. Proo. For any triple o 1-morphisms x y g z h w, we consider the diagram ((h i z ) g) α h,i z,g α h i z,g, (ρ h g) (h (i z g)) (h g) (h i z ) (g ) (h λ g) ρ h (g ) α h,g, α h,i z g, h (g ) α h,i z,g h (λ g ) h λ g h ((i z g) )) h (i z (g )) h α i z,g, in which two triangles (beside the bottom one) commute because o the triangle coherence or identities, and two deormed squares commute by the naturality o associativity coherence. Since all the terms are 2-isomorphisms, then the bottom triangle also commutes. By taking h = i z, we obtain the identity i z (λ g α iz,g, ) = i z (λ g ) 17

18 rom which it ollows that the back ace o the cube i z ((i z g) ) λ (i z g) λ g (i z g) i z (λ g ) i z (g ) λ g g α i z,g, i z α iz,g, i z (i z (g )) λ i z (g ) λ g i z (g ) i z λ g i z (g ) λ g g commutes. The top, bottom and right aces commute rom the naturality o the let identity coherence, and the right ace commutes trivially. Since all edges are 2-isomorphisms we conclude that the ront ace also commutes, which proves that the irst triangle in lemma commutes. Similarly, we prove the commutativity o the other triangle. The statement preceding the theorem, can be made more precise as an instance o a supercoherence introduced by Jardine in [40]. For a bicategory B, and any ordinal [n], we can consider the category B n o horizontally composable strings o 1-morphisms in B, in an analogy with (2.10). A string rom [n 0 ] to [n k ] is a actorization o a map : [n 0 ] [n k ] n 0 θ 1 n 1 θ 2... θ k n k (2.20) in the category, such that each θ i is either coace or codegeneracy, but (2.20) does not necessarily have a canonical orm (2.2). Then i we deine ace unctors D i : B n B n 1 by ( n, n 1,..., 3, 2 ) i = 0 D i ( n, n 1,..., 2, 1 ) = ( n,..., i+1 i,..., 1 ) 0 < i < n (2.21) ( n 1, n 2,..., 2, 1 ) i = n by composing out i th object, and i we deine the degeneracy unctors S i : B n B n+1 by ( n, n 1,..., 1, i x0 ) i = 0 S i ( n, n 1,..., 2, 1 ) = ( n,..., i+1, id xi, i,..., 1 ) 0 < i < n (2.22) (i xn, n,..., 2, 1 ) i = n expanding the i th object by its identity morphism, the collection o categories B n and ace and degeneracy unctors orm a supercoherent structure, which we will make precise by irst introducing a notion o a homomorphism o bicategories. 18

19 Deinition A homomorphism F : B B o bicategories consists o the ollowing data: a (discrete) unctor F 0 : B 0 B 0, and a unctor F 1 : B 1 B 1, two natural transormations B 2 F 2 B 2 B 0 F 0 B 0 H µ H I η I (2.23) B 1 F 1 B 1 B 1 F 1 B 1 given by components µ g, : F (g) F () F (g ) and η x : i F (x) F (i x), respectively. These data are required to satisy the ollowing axioms: or any object (h, g, ) in B 3 there is a commutative diagram (F (h) F (g)) F () µ h,g F () F (h g) F () µ h g, F ((h g) ) a F (h),f (g),f () F (h) (F (g) F ()) F (a h,g, ) F (h) µ g, F (h) F (g ) µ h,g F (h (g )) (2.24) or any object in B 1 there is a commutative diagram F () i F (x) F () η x F () F (i x ) µ,ix F ( i x ) ρ F () F (ρ ) (2.25) F () F () 19

20 or any object in B 1 there is a commutative diagram i F (y) F () η y F () F (i y ) F () µ iy, F (i y ) λ F () F (λ ) (2.26) F () F () I the components µ g, : F (g) F () F (g ) and η x : i F (x) F (i x) o two natural transormations (2.23) are the identity 2-morphisms, which means that the two diagrams (2.23) commute strictly, then we say that the homomorphism F : B B is strict. Remark I both B and B are strict 2-categories, the the coherence (2.24) becomes F (h) F (g) F () µ h,g F () F (h g) F () F (h) µ g, µ h g, F (h) F (g ) µ h,g F (h g ) and the coherence or the right (2.25) and let identity (2.26) become the two diagrams F () F (i x ) F () η x µ,i x F () F () F (i y ) F () η y F () µ i y, F () F () The proo o the ollowing important result can be ound in Bénabou s paper [7]. Theorem There exist categories Bicat and Bicat s whose objects are bicategories, and morphisms are homomorphisms and strict homomorphisms o bicategories, respectively. In the rest o the paper, we will be mostly concerned with strict homomorphisms o bicategories, since all the examples o ibrations o bicategories in Section 4 will be such strict homomorphisms o bicategories. Now, we will give two basic examples o (nonstrict) homomorphism o bicategories. 20

21 Example (Weak monoidal unctors) Since any two monoidal categories M and M may be seen as bicategories ΣM and ΣM with only one object, any weak monoidal unctor F : M M may be seen as a homomorphism ΣF : ΣM ΣM o bicategories ΣM and ΣM. Example (Pseudounctors) For any category C, any pseudounctor F : C op Cat may be seen as a homomorphism rom a locally discrete bicategory C op. The collection o categories B n, ace unctors (2.21) and degeneracy unctors (2.22) D 1 B 0 B 1 D 0 D 2 D 0 B 2 D 3 B 3... (2.27) may be presented as in the above diagram in Cat, in which we denoted only extremal ace unctors D 0, D n : B n B n 1, while we omitted all degeneracy unctors S i : B n B n+1. These unctors do not satisy simplicial identities, since there exist natural isomorphisms D 0 α: D i D j D j 1 D i (i < j) α: S i S j S j+1 S i (i j) α: D i S j S j 1 D i (i j) α: D i S j Id (i = j, i = j + 1) α: D i S j S j 1 D i (i > j + 1) (2.28) all o which are denoted by α. The three unctors D 0, D 1, D 2 : B 2 B 1 satisy identities D 0 D 1 = D 0 D 0 D 1 D 2 = D 1 D 1 D 0 D 2 = D 1 D 0 (2.29) where the irst and the second identity correspond to the compatibility o the horizontal composition with the target and source unctors, respectively, and the third identity correspond to the second pullback in a diagram (2.10). The irst nontrivial simplicial natural isomorphisms in (2.28) arise rom the associativity coherence (2.11) in the bicategory B α: D 1 D 2 D 1 D 1 (2.30) which is the only nontrivial simplicial identity between the ace unctors rom B 3 to B 1. The let and right identity coherence (2.12) provide the ollowing two natural isomorphisms λ: D 1 S 1 Id B1 ρ: D 1 S 0 Id B1 (2.31) and these natural isomorphisms, together with (2.30), satisy three commutative diagrams corresponding to the associativity, let and right identity coherence, which are the only nontrivial coherence conditions out o seventeen commutative diagrams described in [40], so that collection o categories and unctors (2.27), together with natural isomorphisms (2.28) constitute a supercoherent structure, which we call a supercoherent nerve N B o B. 21

22 For any such supercoherent structure N B, we consider the set Str([n], [m]) o strings (2.20) rom [n] to [m], and to any string (θ k,..., θ 2, θ 1 ) we associate the composite unctor θ k... θ 1 : B n B m deined with respect to the supercoherent structure N B. A coherence isomorphism (θ k,..., θ 1 ) (γ l,..., γ 1 ) (2.32) is deined to be a composite o natural isomorphisms o the orm ζ k... ζ i+1 α ɛ ζ i... ζ 1, where ɛ = ±1 and each ζ i is either a ace or a degeneracy o N B. It ollows that θ k... θ 1 = γ l... γ 1 i there exist a coherence isomorphism (2.32), so that we can consider a category Str(θ) whose objects are all string (θ k,..., θ 2, θ 1 ) such that θ = θ k... θ 1, and whose morphisms are coherence isomorphisms. The ollowing supercoherence theorem is proved in [40]. Theorem Let N B be a supercoherent nerve o a bicategory B. Then the category Str(θ) is a trivial groupoid. Remark Since a trivial groupoid is a category with exactly one morphism between any two objects, the supercoherence theorem asserts that any two strings (2.20) representing the same monotone map can be connected by a path o coherence isomorphism (2.32), and moreover any two such paths have equal composites, which means that all diagrams o coherence isomorphisms necessarily commute. We will justiy the term a supercoherent nerve, by the construction which associates to any supercoherent structure a pseudosimplicial category. Deinition A pseudosimplicial category B is a pseudounctor B : op Cat rom the skeletal simplicial category to the 2-category Cat o small 2-categories. Now, suppose that N B is a supercoherent nerve o the bicategory B. For any monotone map θ : [n] [m], we choose its canonical orm (2.2), and we deine the induced unctor θ = S j1... S jt D is... D i1 : B n B m (2.33) with respect to the supercoherent structure N B. I τ : [m] [k] is another monotone map, the the two unctors τ θ and (τθ) come rom strings which have the same composition, and by the supercoherence theorem there exists a unique coherence natural isomorphism µ τ,θ : τ θ (τθ). (2.34) Then we immediately obtain the ollowing result whose proo can be also ound in [40]. Theorem Let N B be a supercoherent nerve o a bicategory B. Then the assignment (2.33), together with natural isomorphisms (2.34) deines a pseudosimplicial category. Proo. The proo is essentially the consequence o the supercoherence theorem since it involves diagrams o coherence natural isomorphism (2.34) which necessarily commute. 22

23 For any strict homomorphism o bicategories P : E B, and every object x in B, a iber bicategory E x is a (strict) pullback E x J x E T P (2.35) J x B where J is the ree bicategory on one object. Let us recall that a ree bicategory is a unctor F : Cat Gph Bicat s (2.36) rom the category Cat-Gph o Cat-graphs to the category Bicat s o bicategories and their strict homomorphisms. Objects o the category Cat-Gph are graphs G consisting o the set G 0 o objects together with a category G(x, y) or any two objects x, y in G 0. A morphism F : G H o Cat-graphs consists o the unction F 0 : G 0 H 0 and unctors F x,y : G(x, y) H(F 0 (x), F 0 (y)) or any two objects in G 0. The ree bicategory unctor (2.36) can be described on the level o objects in the ollowing way. For any Cat-graph G, the ree bicategory FG has an underlying 2-truncated globular set whose objects FG 0 is the set G 0. The set FG 1 o 1-morphisms is deined inductively to include identity 1-morphism i x as new object o G(x, x) or any object x in G 0, then all objects o the category G(x, y) or any two objects x, y in G 0, and or any objects in G(x, y) and g in G(y, z) their horizontal composition g. The set FG 2 o 2-morphisms is deined inductively by taking new 2-isomorphisms α h,g, : (h g) h (g ) or any object in G(x, y), g in G(y, z) and h in G(z, w), with λ : i x and ρ : i x or any element in G(x, y), and their horizontal compositions with all morphisms in G(x, y) or any two objects x, y in G 0. Then we orm vertically composable strings o the horizontal compositions o these 2-morphisms, and we quotient out by the equivalence relation generated by naturality o the associativity, let and right identity coherence 2-morphisms, the Godement interchange law, and the compatibility o the horizontal composition with the vertical composition in categories G(x, y) or any two objects x, y in G 0. Then the ree bicategory F( ) on the Cat-graph whose G 0 = { } and G(, ) = is actually a bigroupoid J with a unique object, 1-morphisms are horizontal compositions i, i i, (i i ) i, i (i i ), ((i i ) i ) i, (i i ) (i i ),... o an identity 1-morphism i : and whose 2-morphisms are horizontal compositions o coherence 2-isomorphisms α i,i,i : (i i ) i i (i i ) and λ i = ρ i : i i i. 23

24 The ree bicategory unctor (2.36) has a right adjoint, which is the orgetul unctor U : Bicat s Cat Gph (2.37) taking a bicategory B to its underlying graph UB with UB 0 = B 0 and UB(x, y) = B(x, y). This means that the ree bicategory unctor (2.36) has the ollowing universal property: or any bicategory B and any morphism G: G UB rom a graph G to an underlying graph UB o B, there exists a unique strict homomorphism G: FG B o bicategories G η G UFG G U G (2.38) such that diagram commutes, where η G : G UFG is component o the unit o adjunction. A strict homomorphism x: J B takes the unique object o J to the object x in B and the identity 1-morphism i : in J to an identity 1-morphism i x : x x in B. Actually, x: J B is uniquely determined by (2.38) by the image o the unique object since J is the ree bicategory on the object. Thereore, E x is a bicategory whose objects are all those objects in E mapped to the object x in B and whose 1-morphisms are all those 1-morphisms j : E F whose image by P are parenthesized strings o 1-morphisms i x, i x i x, (i x i x ) i x, i x (i x i x ), ((i x i x ) i x ) i x, (i x i x ) (i x i x ),... (2.39) and whose 2-morphisms φ: j k are mapped by P to such strings o identity 2-morphisms. Thereore, any 1-morphism j : F E in E x is mapped to a 1-morphism o the type (2.39) and we deine its order ɛ(j) as the number o occurrences o identity morphisms i x in P (j). We can use a supercoherent nerve N B o the bicategory B with aces D j : B n B n 1, or any 1 j n, and degeneracies S j : B n 1 B n, or any 1 j n 1, as unctors between categories B n o horizontally composable strings o n morphisms, which satisy pseudosimplicial identities. Then any parenthesized string o the type (2.39) may be expressed as D i1... D in S n 0 (i x) where S n 0 (i x) is a sequence o n + 1 identity morphisms UB (i x, i x,..., i x, i x ) }{{} n+1 times and D j : B n B n 1 actors out j th object in such strings or any 1 j n. The indices are such that 1 = i 1 i 2... i n n and i n 1 < i n when i n = n. Thereore, we introduce a notation θ(j) = (i 1, i 2,..., i ɛ(j) 1 ), and we usually write P (j) = i θ(j) x when P (j) = D i1... D iɛ(j) 1 S ɛ(j) 1 0 (i x ) such that 1 i 1 i 2... i ɛ(j) 1 ɛ(j) and i ɛ(j) 2 < i ɛ(j) 1 when i ɛ(j) 1 = ɛ(j). 24

25 Deinition A pseudonatural transormation σ : F G is deined by the data: a natural transormation σ 0 : F 0 G 0 between (discrete) unctors, a natural isomorphism B 1 G 1 B 1 F 1 σ 1 σ 0 B 1 σ 0 B 1 whose component indexed by an object : x y in B 1 is given by a 2-morphism σ : G() σ x σ y F () as in a diagram F (x) σ x G(x) F () σ G() F (y) σ y G(y) These data are required to satisy the ollowing axioms: or any object (g, ) in B 2, there is a commutative diagram (G(g) G()) σ x α G(g),G(),σx G(g) (G() σ x ) G(g) σ G(g) (σ y F ()) µ G g, σ x G(g ) σ x α 1 G(g),σy,F () (G(g) σ y ) F () (2.40) σ g σ g F () σ z F (g ) (σ σ z (F (g) F ()) z F (g)) F () σ z µ F g, α σz,f (g),f () 25

26 or any object x in B there is a commutative diagram i G(x) σ x λ σx σ x ρ 1 σx σ x i F (x) η G x σ x σ x η F x (2.41) G(i x ) σ x σ i x σ x F (i x ) Remark I both B and B are strict 2-categories, then the coherence (2.40) becomes a commutative diagram G(g) G() σ x G(g) σ G(g) σ y F () µ G g, σx G(g ) σ x σ g F () σ g σ z F (g ) σ z µ F g, σ z F (g) F () The second coherence (2.41) becomes the commutative diagram σ x ηx G σ x σ x ηx F G(i x ) σ x σ x F (i x ) σ i x 26

27 Deinition For any two pseudonatural transormations, a modiication Γ: σ σ B σ F Γ G σ consists o the ollowing data: or each object x in B, a 2-morphism Γ x : σ x σ x in the bicategory B such that the ollowing diagram B σ x F (x) Γ x G(x) F () σ x σ G() σ y F (y) Γ y G(y) commutes, which means that a diagram o 2-morphisms in the bicategory B σ y G() σ x G() Γ x G() σ x σ σ σ y F () Γ y F () σ y F () commutes. 27

28 Theorem Homomorphisms rom a bicategory B to E are objects o a bicategory Hom(B, E) (2.42) where 1-morphisms are pseudonatural transormations and 2-morphisms are modiications. Proo. The horizontal composition ξ σ o two pseudonatural transormations σ : P R and ξ : R S has a component indexed by an object B in B, deined by an identity (ξ σ) B := ξ B σ B (2.43) and or any 1-morphism : A B in the bicategory B, the pasting composite o a diagram P (A) σ A R(A) ξ A S(A) P () σ R() ξ S() P (B) σ B R(B) ξ B S(B) deines the component (ξ σ) o the pseudonatural transormation ξ σ by an identity (ξ σ) := (ξ B σ )(ξ σ A ). (2.44) The associativity coherence or this composition is given by an invertible modiication α ω,ξ,σ : (ω ξ) σ ω (ξ σ) (2.45) whose components or any three pseudonatural transormations P σ R ξ S ω T [(ω ξ) σ] A P (A) α ωa,ξ A,σ AT (A) [ω (ξ σ)] A P () [ω (ξ σ)] T () [(ω ξ) σ] B P (B) α ωb,ξ B,σ BT (B) [ω (ξ σ)] B 28

29 are given by associativity coherence 2-morphisms α ωa,ξ A,σ A : (ω A ξ A ) σ A ω A (ξ A σ A ). For a homomorphism R: B E o bicategories, an identity pseudonatural transormation ι R : R R (2.46) has an identity 1-morphism i R(B) : R(B) R(B) as a component (ι R ) B indexed by an object B in B, and its component (ι R ) : R() i R(B) i R(A) R() indexed by any 1-morphism : A B in B, is as in a diagram R(A) i R(A) R(A) R() R(B) λ R() R() ρ 1 R() i R(B) R() R(B) and is equal to ρ 1 R() λ R(). Then or any pseudonatural transormations σ : P R and ξ : R S, a let and right identity coherence modiications λ σ : ι R σ σ (2.47) ρ ξ : ξ ι R ξ (2.48) have components or any object B in B given by the let and right identity coherence λ σb : i R(B) σ B σ B (2.49) ρ ξb : ξ B i R(B) ξ B (2.50) respectively. The horizontal composition o any two modiications in Hom(B, E) P σ Γ R ξ Ω S τ ζ is deined by the horizontal composition o its components indexed by objects B in B (Ω Γ) B := Ω B Γ B (2.51) 29

30 Associativity and identity coherence or the horizontal composition ollows immediately rom associativity and identity coherence o the horizontal composition in the bicategory E. Vertical composition o modiications σ Γ τ Π π in Hom(B, E) is deined by (ΠΓ) B := Π B Γ B (2.52) and the two compositions (2.51) and (2.52) satisy the Godement interchange law. 30

31 3 Indexed bicategories and tricategories All deinitions in this section are taken rom [24], and Bicat denote a tricategory o bicategories, their homomorphisms, pseudonatural transormations and modiications. Deinition 3.1. A tricategory T consists o the ollowing data: a set T 0 whose elements are called objects o T or any two objects x and y in T, a bicategory T (x, y) which we call a hom-bicategory, whose objects, g, h,... are called 1-morphisms o T, whose 1-morphisms φ, ψ, θ,... are called 2-morphisms o T, and whose 2-morphisms Π, Θ,... are called 3-morphisms o T. The horizontal composition o 1-morphisms φ and ψ in T (x, y) is denoted by ψ φ, and the vertical composition o 2-morphisms Π and Θ in T (x, y) is denoted by the concatenation ΘΠ, whenever these compositions are deined or any three objects x, y and z in T, a homomorphism o bicategories : T (y, z) T (x, y) T (x, z) (3.1) called a composition, whose coherence conditions are not named or any object x in T, a homomorphism o bicategories where I is the ree 2-category on one object, I x : I T (x, x) (3.2) or any three objects x, y and z in T, a pseudonatural equivalence T (z, w) T (y, z) T (x, y) Id T (x,y) T (y, w) T (x, y) Id T (z,w) α (3.3) T (z, w) T (x, z) T (x, w) or any two objects x and y in T, pseudonatural equivalences T (x, y) I y Id T (x,y) Id T (x,y) I x λ ρ T (y, y) T (x, y) T (x, y) T (x, y) T (x, x) (3.4) 31

32 or any ive objects x, y, z, w, t in T, an isomodiication π inside a cube 1 T α T 3 1 α T 3 1 T 2 α T 3 1 T 2 1 α 1 T 2 α T 1 (3.5) or any three objects x, y and z in T, an isomodiication µ inside a pyramid T 2 Id T1 I Id T1 λ Id T 1 T T 2 Id T1 ρ α T 2 T 1 µ (3.6) 32

33 or any three objects x, y and z in T, two isomodiications λ and ρ inside diagrams T 3 I 1 1 1, λ 1 T 2 T 2 1 α T I 1 T 2 λ T T I 1 1 ρ T 2 T 2 1 α T 1 I T 2 ρ T (3.7) Example 3.2. (Tricategory Bicat) Bicategories are objects o the tricategory Bicat, and or any two bicategories B and E a hom-bicategory Bicat(B, E) is the one (2.42) given in Theorem A composition homomorphism : Bicat(E, P) Bicat(B, E) Bicat(B, P) (3.8) is deined on the level o objects or any two homomorphisms F : B E and G: E P by G F = GF. (3.9) The composition o homomorphisms o bicategories is strictly associative, by Theorem 2.16, so that the tricategory Bicat has an underlying category which we denote by the same name Bicat. For any two pseudonatural transormations B F φ E G ψ P L M the composition homomorphism (3.8) is deined in the ollowing way. First, we deine a pseudonatural transormation ψ F : G F M F whose component indexed by an object 33

34 B in B is (ψ F ) B = ψ F (B), and whose component indexed by a 1-morphism : B C in B is (ψ F ) = ψ F () as in a diagram GF (B) ψ F (B) MF (B) GF () ψ F () MF () GF (C) ψ F (C) MF (C) Second, we deine a pseudonatural transormation M φ: M F M G whose component indexed by an object B in B is (M φ) B = M(φ B ), and whose component (M φ) MF (B) M(φ B ) MG(B) MF () (M φ) MG() MF (C) M(φ C ) MG(C) indexed by a 1-morphism : B C in B, is deined by a composition o 2-morphisms MG() M(φ B ) µ G(),φ B M(G() φ B ) M(φ ) M(φ C F ()) µ 1 φ C,F () M(φ C ) MF (). It is easy to see that both ψ F and M φ are pseudonatural transormations, and these are 1-morphisms in the hom-bicategory Bicat(B, P) which will also be denoted by ψf and M φ, respectively. Then we use horizontal composition (2.43) and (2.44) in the bicategory Bicat(B, P) to deine the composition (3.8) o pseudonatural transormations ψ and φ by ψ φ = (ψ F ) (M φ) (3.10) which we will also write in abbreviated orm ψ φ = ψf Mφ. Example 3.3. (Tricategory o spans in Cat) There is tricategory Span(Cat) o spans in the 2-category Cat, whose objects are small categories. For any two small categories C and D, a 1-morphism (F, G, G): C D is a span C F G 34 G D

35 and a 2-morphism (φ, A, ψ): (F, G, G) (L, L, M) in Span(Cat) is a diagram G F G C φ A ψ D L M L consisting o a unctor A: G L together with natural transormations φ: L A F and ψ : M A G. A 3-morphism β : (φ, A, ψ) (µ, B, ν) in Span(Cat) is given by a diagram G F G µ ν C β A B D L M L in which β : A B is a natural transormation, such that µ(l β) = φ and ν(m β) = ψ. It is obvious that or any two small categories C and D, a hom-bicategory Span(Cat)(C, D) is actually a strict 2-category. The composition homomorphism : Span(Cat)(D, E) Span(Cat)(C, D) Span(Cat)(C, E) (3.11) is deined on the level o spans o categories G H F G H K C D E by the composite let and right legs o a diagram where a square is a pullback in Cat P 1 P P 2 G H F G H K C D E 35

36 It is obvious that an identity I C : C C or the horizontal composition (3.11) is a span C Id C Id C C C. and that the horizontal composition (3.11) is nor strictly associative or unital, but it ollows rom a universal property o pullbacks that it is associative and unital up to canonical isomorphisms o categories. From the previous example, it ollows that or any small category B, the ollowing span D 1 B I D 0 B B (3.12) is a monoid in a monoidal 2-category Span(Cat)(B, B), where B I is a category o morphisms. By identiying any unctor P : E B with a span (I E, E, P ), we have a composition (B, P ) P 1 P 2 B I D 1 E (3.13) D 0 P I E B B E which means that or a slice 2-category (Cat, B) o unctors over B, we have a 2-unctor (B, ): (Cat, B) (Cat, B) (3.14) which is a actually a 2-monad on (Cat, B) by the commutativity o the ollowing diagram (B 3, P ) F D 1 P 2 (B D 2 2, P ) B D 1 P 1 B E P 2 D 1 P 1 B (B 2, P ) P P 2 I E (B, P ) 36

37 Then a pseudoalgebra structure or the 2-monad (3.14) is given by a unctor P : E B and a unctor M : (B, P ) E which is a part o a morphism o spans (B, P ) Q 1 P B M ϑ E P I E E such that the ollowing diagram in Span(B, E) (B 2, P ) F D 1 P 2 (B, P ) D 2 B D 1 P 1 β B E P 2 D 1 P 1 B (B, P ) ϑ P P 2 I E E commutes. The let leg o a span (4.17) is a unctor rom the comma category (B, P ) D 1 P 1 : (B, P ) B. (3.15) and this unctor is a actually a ibration which we prove by the ollowing theorem. Theorem 3.4. For any unctor P : E B, the unctor(3.15) is a ibration o categories, called a canonical ibration associated to P. 37

38 Deinition 3.5. Let B be a small bicategory. A B-indexed bicategory is a trihomomorphism F : B coop Bicat o tricategories rom B coop to the tricategory Bicat which consists o: or every object x in B, a bicategory F(x), which we will also denote by F x, or every two objects x and y in B, a homomorphism o bicategories F x,y : B(x, y) op Hom(F y, F x ) (3.16) rom the opposite B(x, y) op o the category B(x, y) o 1-morphisms and 2-morphisms with x and y as a 0-source and 0-target, respectively, into the bicategory Hom(F y, F x ). For any 1-morphism : x y in B, its image F x,y () by (3.16) is a homomorphism x y F x,y F x F y (3.17) on the right side, and or any 2-morphism β : g in B, its image F x,y (β) by (3.16) x β y F x,y F x β F y (3.18) g g is a pseudonatural transormation β : g on the right side o (3.18). or any composable pair β g γ h o 2-morphisms in B, an isomodiication µ γ,β : β γ (γβ) (3.19) which satisy coherence with respect to the vertical composition o 2-morphisms in B (β γ ) δ µ γ,β δ (γβ) δ µ δ,γβ [δ(γβ)] α β,γ,δ id (3.20) β (γ δ ) β (δγ) β µ δ,γ µ δγ,β [(δγ)β] and such that the ollowing normalization conditions or modiications are satisied µ ιg,β = λ β (3.21) 38

39 µ β,ι = ρ β (3.22) where modiications on the right side are the components o the let and right identity coherence modiications (2.47) and (2.48) in the bicategory Hom(F x, F x ), respectively. or any 1-morphisms : x y and g : y z in B, two pseudonatural equivalences χ g, : g (g ) (3.23) χ g, : (g ) g (3.24) in the bicategory Hom(F x, F x ), together with isomodiications Θ g, : χ g, χ g, ι g (3.25) Λ g, : χ g, χ g, ι (g ) (3.26) such that (3.23) and (3.24) are adjoint pseudonatural equivalences. or any horizontally composable pair o 2-morphisms in B g x β y γ z k l an isomodiication χ γ,β : (γ β) χ l,k χ g, (β γ ), as in a ollowing diagram k l χ l,k (l k) β γ χ γ,β (γ β) (3.27) g χ g, (g ) 39

40 or any three x y g w h z 1-morphisms in B, an isomodiication g h χ g, h (g ) h χ h,g [(h (g )] χ h,g ω h,g, α h,g, (3.28) (h g) χ h g, [(h g) ] This data are required to satisy the ollowing coherence conditions: commutative cube g h k χ h,g k (h g) k g χ k,h χ g, h k θ x ω h,g, χ k,h g [k (h g)] α h,g, g (k h) χ k h,g (g ) h k [(k h) g] ω h,g, k θ h g, k (g ) χ k,h χ g, (k h) χ h,g k ω k h,g, (g ) (k h) [h (g )] k α h,g, k [(h g) ] k ω k h,g, χ k,h (g ) χ (k h) g, χ k (h g), χ k,(h g) ω k,h,g χ k h,g [k (h (g ))] (k α h,g, ) [k ((h g) )] α k,h,g ω h,g, θ z α k,h g, [(k (h g)) ] (α k,h,g ) [(k h) (g )] [((k h) g) ] α k h,g, 40