Universal Classification of Topological Categories

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1 University of Miami Scholarly Repository Open ccess Dissertations Electronic Theses and Dissertations Universal Classification of Topological Categories Marta lpar University of Miami, Follow this and additional works at: Recommended Citation lpar, Marta, "Universal Classification of Topological Categories" (2012). Open ccess Dissertations This Open access is brought to you for free and open access by the Electronic Theses and Dissertations at Scholarly Repository. It has been accepted for inclusion in Open ccess Dissertations by an authorized administrator of Scholarly Repository. For more information, please contact

2 UNIVERSITY OF MIMI UNIVERSL CLSSIFICTION OF TOPOLOGICL CTEGORIES By Marta lpar DISSERTTION Submitted to the Faculty of the University of Miami in partial fulfillment of the requirements for the degree of Doctor of Philosophy Coral Gables, Florida December 2012

3 2012 Marta lpar ll Rights Reserved

4 UNIVERSITY OF MIMI dissertation submitted in partial fulfillment of the requirements for the degree of Doctor of Philosophy UNIVERSL CLSSIFICTION OF TOPOLOGICL CTEGORIES Marta lpar pproved: Marvin V. Mielke, Ph.D. Professor of Mathematics M. Brian Blake, Ph.D Dean of the Graduate School Shulim Kaliman, Ph.D. Professor of Mathematics Victor C. Pestien, Ph.D. ssociate Professor of Mathematics Shihab S. sfour, Ph.D. Professor of Engineering

5 LPR, MRT (Ph.D., Mathematics) Universal Classification of Topological Categories (December 2012) bstract of a dissertation at the University of Miami. Dissertation supervised by Professor Marvin V. Mielke. No. of pages in text. (94) The main purpose of this dissertation is to construct, for various well known families of topological categories and some of their generalizations, a member of the family that is universal in the sense that every member of the family is isomorphic to the pullback, along its so called classifying functor, of the said universal family member. This is carried out by first constructing a topological category that is universal for the family of all topological categories and then by defining various family universal categories by describing their classifying functors. further refinement is made by placing restrictions on the classifying functors themselves, thus obtaining various restricted families of topological categories along with their corresponding restricted universal categories. These constructions and results are first described in the more general setting of horizontal structures. We will show that all horizontal structures can be obtained by pulling back the universal horizontal structure along an appropriate classifying functor and as a consequence, by restriction, every topological category can be realized as the pullback, along its classifying functor, of the universal topological category.

6 CKNOWLEDGEMENTS I would like to thank my advisor Dr. Marvin Mielke for introducing me to the subject of Categorical Topology, for all his help and guidance throughout my thesis work and for his seemingly infinite patience. iii

7 CONTENTS 1. Introduction 1 2. Foundations 4 Horizontal Structures Universal Horizontal Structure Topological Structures Universal Topological Category Classification of General Topological Structures 57 Categories of Topological Posets Categories of Fuzzy Topological Posets Level topologies Classification of General Convergence Structures 76 Poset Properties Stack and Filter Convergence Structures Generalized Stack Convergence Structures Generalized Filter Convergence Structures References 93 iv

8 CHPTER 1 INTRODUCTION Topological functors were introduced by Herrlich (in terms of the existence of certain initial lifts in [10]) and by Wyler (in terms of contravariant functors to the category of complete lattices in [21]) among others in the early 1970s as a result of axiomatizing the properties that many categories share: topological and pretopological spaces [16], filter and stack convergence spaces [18], limit spaces [17], bornological spaces [1] etc. The most important feature of these categories is the existence of final (and initial) structures, in particular, for topological spaces the formation of induced (and coinduced) topologies, which can be viewed as lifting properties of the underlying set functor U : Top Sets. In general, a functor E B between categories E and B is said to be a topological functor and E a topological category over B, if it satisfies certain horizontal lifting conditions. Several aspects of topological spaces can be extended to topological categories, which can then serve as realms in which to formulate and test various topological notions. Many of the familiar examples of topological categories, nearness and uniform spaces [9], grill spaces [18], along with the examples listed above result from generalizing some particular aspect of the notion of topological space, while others, such as: the categories of pairs, relations, preorders and families have other origins. 1

9 2 In the second chapter the 2-category of horizontal structures over a category (denoted by H ) is defined as a supercategory of topological categories over. The so called horizontal morphisms of these horizontal structures behave much like those morphisms (continuous functions) f : (X, τ) (Y, σ) in Top where σ is the topology coinduced on Y by f, in the sense that given any other topology ρ on Y such that f : (X, τ) (Y, ρ) is continuous (f = f in Sets), f factors through f in Top. Using the Grothendieck construction, H will be shown to be naturally equivalent to the functor category CT. Horizontal structures, just as topological structures, are pullback stable; this fact will lead to the definition of a contravariant 2-functor H : CT 2 Cat which assigns H to a category, and for a functor F : B, H(F ) : H B H acts via the canonical pullback construction. The natural (strong-lax) isomorphism based on the Grothendieck construction shows that H is a representable 2-functor. The representing object must be a suitable collection of categories large enough to contain the large categories used later (in particular the category of posets), but it must be a category itself and as such an object in CT; its objects must form a proper class (the objects of CT form a conglomerate). This universe of categories will be denoted by CT. With CT the representing object for H, the universal object is the image of the identity functor 1 : CT CT under the functor Yoneda Lemma. CT by the enriched version of the 1 will be referred to as the universal horizontal structure CT and denoted by CT ; the universality of CT structure is isomorphic to a pullback of CT means that every horizontal P CT along some functor. In particular, considering functors in CoP CT, the resulting universal object is the universal topological category CoP ; (CoP is the category of cocomplete posets with cocontinuous functors).

10 3 In the third chapter we will describe the classifying functors for some familiar topological categories and for their more general versions. The categories discussed all resemble Top, the category of topological spaces and continuous functions. The generalization of Top is achieved in two steps. First a topology will be defined on a suitable poset rather than as a subset τ of the powerset P (X) of some set X; then the topology will be viewed in terms of its characteristic function χ τ : P (X) 2 which will then be replaced by a function T : M with appropriate properties. The elements of the poset M give the degree of membership of the elements of the poset in the topology defined by T. The definition of T will include as special cases all of the different versions of fuzzy topologies that appear in the literature ([4],[8],[19],[24]). The universal member of this family of topological categories will be identified through its classifying functor. The fourth chapter deals with categories whose objects are (po)sets with some kind of convergence structure defined on them. Stack convergence spaces, filter and local filter convergence spaces, limit spaces and pretopological spaces will be described through their respective classifying functors. Each convergence structure is defined via a convergence function q : S(X) P (X) or q : F (X) P (X) where S(X) and F (X) denote the set of all stacks and filters on X, respectively and q associates to a stack (filter) the set of points of X to which it converges. To generalize the categories listed above, P (X) will again be replaced by an appropriate poset; as a consequence stacks and filters will become down segments and ideals of the poset (due to the reverse ordering). The classifying functors will be subfunctors of one another, since they are obtained by putting restrictions on the convergence function. There will thus be obtained a restricted family universal category for each of the convergence types discussed.

11 CHPTER 2 FOUNDTIONS Horizontal Structures Definition 2.1 cf. Definition in [3] 2-category C is defined by the following data: (1) a class of objects called 0-cells, (2) for each pair a, b of 0-cells a category C(a, b) (often required to be small); (the objects of C(a, b) are called 1-cells and its arrows are called 2-cells), (3) for each triple a, b, c of 0-cells, a bifunctor c abc : C(a, b) C(b, c) C(a, c), (4) for each 0-cell a, a functor u a : 1 C(a, a). These data are required to satisfy the usual associativity and unit axioms. Given 1-cells f, g, h in C(a, b) and 2-cells α : f g and β : g h, the composition of α and β in the category C(a, b) will be denoted by β α, and given 2-cells α : f g in C(a, b) and φ : k l in C(b, c) a f α b g k φ l c 4

12 c abc (α, φ) : kf lg will be denoted by φ α. The composition map c abc being a functor implies that the interchange law (ψ β) (φ α) = (β α) (ψ φ) holds 5 for the 2-cells as pictured: a α b β φ ψ c. Depending on possible size restrictions imposed on the collection of objects and morphisms of categories, we ll adopt the following notation: CT will denote the quasicategory of all categories (as in Definitions 3.49 and 3.50 in[1]), Cat the category of all small categories and CT will denote a category with categories (not necessarily small) as objects such that CT is an object in CT. CT, Cat and CT are 2-categories. If is a 2-category, so are op ( 1-cells reversed) and co (2-cells reversed). Given a category, the functor category CT is a 2-category with functors F : CT the 0-cells, natural transformations α : F G the 1-cells and for α, β : F G, the 2-cells (modifications) m : α β are defined as follows: each 1- cell α : F G is a collection of functors (arrows in CT) {α a : F (a) G(a)} a and a 2-cell m : α β is then a collection of natural transformations {m a : α a β a } a such that for f : a b in, 1 Gf m a = m b 1 F f (see diagram below) where 1 F f and 1 Gf are the identity natural transformations on the functors F f and Gf respectively, and denotes the standard horizontal composition of natural transformations (cf. page 43 in [14]).

13 6 a f F (a) F f α a m a G(a) β a Gf b F (b) α b m b G(b) β b Each m a again consists of components (arrows in the category G(a)) corresponding to objects x of F (a): (m a ) x : α a (x) β a (x). (To avoid the clutter of multiple pairs of parentheses, (m a ) x will sometimes be written simply as m ax : for example, the image of the arrow (m a ) x under a functor φ a will be written as φ a (m ax ) rather than φ a ((m a ) x ) ). Given functors F, G : CT and natural transformations α, β, γ : F G, composition of arrows (2-cells) m : α β and n : β γ in CT (F, G) is defined for a by (n m) a = n a m a where n a m a is the standard vertical composition of natural transformations (page 42 in [14]) with the component for x F (a) given by the composition of arrows in the category G(a) : (n m) ax = n ax m ax : α a (x) γ a (x). This definition implies that composition of arrows in CT (F, G) is associative. The identity 2-cell 1 α : α α on α : F G consists of the collection of identity arrows {1 αa (x) a and x F (a)} in G(a). Given a triple of objects F, G, M in CT, the composition functor c F GM : CT (F, G) CT (G, M) CT (F, M) is defined for α : F G and φ : G M by the composition of functors; for an object a in, c F GM (α, φ) a = φ a α a. (Note: c F GM (α, φ) will be denoted by φα; thus (φα) a = φ a α a.)

14 The horizontal composition c F GM (m, n) = n m of 2-cells m and n as pictured 7 below α φ F m G n M β ψ is given for an object a in F (a) α a m a G(a) β a φ a n a M(a) ψ a by (n m) a = n a m a : φ a α a ψ a β a where n a m a is again the horizontal composition of natural transformations: for x F (a), (n a m a ) x is given by the diagonal arrow of the commutative square below. (φ a α a )(x) φ a (m ax ) (φ a β a )(x) (n a ) α a(x) (ψ a α a )(x) ψ a (m ax ) (n a ) β a(x) (ψ a β a )(x) For c F GM to be a functor, it must preserve identities and composition: Given an identity two cell (1 α, 1 φ ) in CT (F, G) CT (G, M), the component of c F GM (1 α, 1 φ ) corresponding to objects a in and x in F (a) is 1 φa (α a (x)), the identity arrow on the object φ a (α a (x)) in the category M(a), which is the same as the corresponding component of 1 cf GM (α,φ) = 1 φα. c F GM preserving the composition of 2-cells translates for m 1, m 2, n 1 and n 2 as pictured below m 1 n 1 F G M m 2 n 2 to (m 2 m 1 ) (n 2 n 1 ) = (n 1 m 1 ) (n 2 m 2 ), which for an object a in becomes the standard interchange law for the natural transformations (m 1 ) a, (m 2 ) a, (n 1 ) a

15 and (n 2 ) a. Given an object F in CT, the unit 1-cell in CT (F, F ) is the 8 identity natural transformation 1 F : F F consisting of the identity functors 1 F (a) : F (a) F (a) for an object a in and the unit 2-cell on 1 F consists of the identity arrows {1 x x F (a)}. The associativity and unit axioms hold as a direct consequence of the definitions involved. Definition 2.2 Let P : E B be a functor. Given an object b B, the categorical fiber of P over b is the subcategory P 1 (b) of E defined to have as objects e E such that P (e) = b and as morphisms f : e 1 e 2 in E such that P (f) = 1 b. The morphisms in the categorical fiber over any object are called vertical morphisms. V (E) will denote the subcategory of E that has the same objects as E, and its morphisms are the vertical morphisms of E. (V (E) is a subcategory of E, since composition of vertical morphisms gives vertical morphisms and the identity morphisms are vertical.) Definition 2.3 Given a category and subcategories R and L, (R, L) is called a splitting of, if for each f : a b in, f has a unique decomposition as f = lr with l L and r R. Definition 2.4 Given P : E B and a splitting (H, V ) of E, (E, V, H) will be called a horizontal structure over B if it satisfies the following properties: (i) V is a subcategory of V (E), and (ii) each morphism f : a b in B lifts uniquely to any given domain in the fiber over a in E, with the lift f in H. The morphisms of H will be called horizontal morphisms. Then given a horizontal structure over

16 B, every morphism t : e 1 e 2 in E factors uniquely as t = vh with v V and h H, and h is the unique horizontal lift of P (t) to domain e 1. 9 Remarks 2.5 (i) H and V, being subcategories, contain all the identity morphisms of E; condition (ii) in the definition above then implies that H V = {1 e } e E. (ii) For morphisms f : a b and g : b c in B, if f is the horizontal lift of f to domain e in E and ḡ is the horizontal lift of g to cod( f), then ḡ f = gf: H being a subcategory means that ḡ f is in H, therefore both ḡ f and gf are horizontal lifts of gf to the same domain; by the uniqueness of horizontal lifts then, ḡ f = gf. (iii) The horizontal lift of an isomorphism is an isomorphism: Suppose f : a b is an isomorphism in B, f is its horizontal lift to domain e in E and f 1 is the horizontal lift of f 1 to domain cod( f) def = ē ; by (ii) above, f 1 f = f 1 f = 1 a which means that both f 1 f and 1 e are horizontal lifts of 1 a to domain e; then by the uniqueness of horizontal lifts, we have that f 1 f = 1 e and by a similar argument f f 1 = 1ē. Hence f 1 is the inverse of f in E. Definition 2.6 Define the 2-category H B of horizontal structures over B as follows: The 0-cells are horizontal structures (E, V, H) P B as in Definition 2.4. The 1-cells are functors Φ : E 1 E 2 such that for the 0-cells (E 1, V 1, H 1 ) P 1 B and (E 2, V 2, H 2 ) P 2 B, P 1 = ΦP 2 and that Φ preserves both the horizontal and vertical morphisms, i.e., h H 1 Φ(h) H 2 and v V 1 Φ(v) V 2. (Such functors also preserve the horizontal-vertical decomposition of arrows.) The 2-cells of H B are natural transformations α : Φ Ψ such that for each object e in E 1,

17 α e : Φ(e) Ψ(e) is a vertical morphism in V 2. For each pair (E 1, V 1, H 1 ) P 1 B and (E 2, V 2, H 2 ) P 2 B of 0-cells, the 1-cells Φ : E 1 E 2 and 2-cells α : Φ Ψ 10 must form a category (will be denoted by H B (E 1, E 2 )). Composition of arrows in the category H B (E 1, E 2 ) is defined componentwise: given objects Φ, Ψ, and Υ and arrows α : Φ Ψ and β : Ψ Υ in H B (E 1, E 2 ), E 1 Ψ Φ α E 2, the β Υ composition β α : Φ Υ is defined for each object e in E 1 by (β α) e = β e α e : Φ(e) Υ(e). The associativity of composition of 2-cells then follows from the associativity of composition of morphisms in E 2. Since both α e : Φ(e) Ψ(e) and β e : Ψ(e) Υ(e) are in V 2 and V 2 is closed under composition, β e α e is in V 2 as well, so β α is well defined. For each triple (E i, V i, H i ) P i B, i = 1, 2, 3 of 0-cells, we must have a bifunctor c : H B (E 1, E 2 ) H B (E 2, E 3 ) H B (E 1, E 3 ). For 1-cells, c is the usual composition of functors. For 2-cells α : Φ 1 Ψ 1 and β : Φ 2 Ψ 2, E 1 Φ 1 α E 2 ψ 1 Φ 2 β E 3 Ψ 2 c(α, β) = β α : Φ 2 Φ 1 Ψ 2 Ψ 1 is defined by the usual horizontal composition of natural transformations, i.e., for each object e of E 1, as the diagonal of the commutative square below: Φ 2 Φ 1 (e) Φ 2 (α e ) β Φ1 (e) Ψ 2 Φ 1 (e) Ψ 2 (α e ) Φ 2 Ψ 1 (e) β Ψ1 (e) Ψ 2 Ψ 1 (e) The functoriality of c follows from the standard interchange laws for the vertical and horizontal composition of natural transformations: (δ β) (γ α) = (β α) (δ γ).

18 11 E 1 α E 2 β γ δ E 3 s the unit 1-cells are the identity functors and the unit 2-cell on a 1-cell Φ : E E is the identity natural transformation given by the identity arrows 1 e : e e for any object e of E, the required unit axioms and associativity axioms hold. Definition 2.7 (cf. Definition in [3]) Given two 2-categories and B, a functor F : B on the underlying categories and B is a 2-functor if for each pair of objects a and b in, F induces a functor F ab : (a, b) B(F a, F b) such that F ab is compatible with composition and units, i.e., such that the following diagrams commute: (a, b) (b, c) F ab F bc c abc (a, c) F ac B(F a, F b) B(F b, F c) c F af bf c B(F a, F c) u a 1 (a, a) F aa u F a B(F a, F a) Definition 2.8 (cf. Definition in [3]) Given two 2-categories, B and two 2-functors F, G : B, a natural transformation α : F G is a 2- natural transformation, if for each pair of objects a, b in, the following diagram of categories commutes:

19 12 (a, b) G ab F ab B(F a, F b) R α B(Ga, Gb) S α B(F a, Gb) Both functors R α and S α above act via composition: for f : F a F b, R α (f) = α b f and for a 2-cell φ in B(F a, F b), R α (φ) = 1 αb φ; for g : Ga Gb, S α (g) = g α a and for a 2-cell γ in B(Ga, Gb), S α (γ) = γ 1 αa. The purpose of the following ( ) is to define a 2-functor K : CT op 2 Cat, where 2 Cat is the category of 2-categories, 2-functors and 2-natural transformations. Construction 2.9 Given categories and B, a functor F : B induces a 2-functor F : CT B CT as follows: For a functor T : B CT, F (T ) = T F ; given a pair of objects T 1, T 2 in CT B, F must define a functor F 1,2 : CT B (T 1, T 2 ) CT (T 1 F, T 2 F ). Given a natural transformation α : T 1 T 2 with components {α b : T 1 (b) T 2 (b) b B}, F 1,2(α) : T 1 F T 2 F is defined to be the natural transformation with (F 1,2(α)) a = α F a as its component corresponding to an object a in. F 1,2(α) will be denoted by α F. Given 1-cells α, β and a 2-cell m : α β in CT B (T 1, T 2 ) α T 1 m T 2 β with components the arrows {m bx : α b (x) β b (x)} in T 2 (b) for b in B and x T 1 (b), the corresponding 2-cell F 1,2(m) (denoted below for short by m F )

20 13 T 1 F α F m F T 2 F β F has components {(m F a ) x : α F a (x) β F a (x)} for a in and x in T 1 (F (a)). For an identity 2-cell 1 α on α : T 1 T 2, F 1,2(1 α ) has components the identity arrows α F a (x) 1 α F a (x) in T 2 F (a); F 1,2 then preserves identities. Since the vertical composition of 2-cells is defined componentwise, by composition of arrows in a category, F 1,2 preserves composition as well. The compatibility of F 1,2 with horizontal composition is equivalent to the commutativity of the diagram below (the subscripts for the composition functor c were omitted). CT B (T 1, T 2 ) CT B (T 2, T 3 ) F 1,2 F 2,3 CT (T 1 F, T 2 F ) CT (T 2 F, T 3 F ) c CT B (T 1, T 3 ) F 1,3 c CT (T 1 F, T 3 F ) For a 1-cell (α, φ) in CT B (T 1, T 2 ) CT B (T 2, T 3 ), c(α, φ) = φα is defined via composition of functors; for an object b in B, (φα) b = φ b α b : T 1 (b) T 3 (b) and then F 1,3(φα) is the natural transformation with component for a in the functor (φα) F a = φ F a α F a : T 1 F (a) T 3 F (a). Going the other way around the diagram gives first the natural transformation (F 1,2 F 2,3)(α, φ) = (α F, φ F ), whose image under the composition functor c is again φ F α F with components φ F a α F a for an object a in. The diagram then commutes for 1-cells. For appropriate 2-cells m and n, the commutativity of the diagram translates to (n m) F = n F m F. The components of both 2-cells corresponding to an object

21 14 a in are natural transformations φ F a α F a ψ F a β F a (where m : α β and n : φ ψ). By the definition of n m, (n m) F a = n F a m F a. The diagram then commutes for 2-cells as well. The unit axiom is equivalent to the commutativity of the diagram below: u T 1 CT B (T, T ) F u T T T F CT (T F, T F ) Since the unit 1-cells have as components the identity functors and the unit 2-cells have identity arrows as components both of which are preserved by F, the diagram above does commute. F : CT B CT is then a 2-functor. Lemma 2.10 natural transformation α : F G for the functors F, G : B induces a 2-natural transformation α : F G where F, G : CT B CT are the 2-functors defined in the previous construction. Proof. The component α T of α corresponding to an object T in CT B must be shown to be a natural transformation (a 1-cell in CT ) α T : T F T G. The component (α T ) a of α T corresponding to an object a is the functor defined by the image of the arrow α a : F (a) G(a) in B under the functor T ; (α T ) a = T (α a ) : T F (a) T G(a). Given a pair of objects T 1, T 2 in CT B, the following diagram must commute for α to be a 2-natural transformation:

22 15 CT B (T 1, T 2 ) G 1,2 F 1,2 CT (T 1 F, T 2 F ) R α CT (T 1 G, T 2 G) S α CT (T 1 F, T 2 G) where R α and S α are as in Definition 2.8. For a 1-cell φ : T 1 T 2 and an object a in, the commutativity of the diagram means T 2 (α a ) φ F a = φ Ga T 1 (α a ); this equality follows from the naturality of φ: T 1 F (a) T 1 (α a ) φ F a T2 F (a) T 2 (α a ) T 1 G(a) φ Ga T2 G(a) For a 2-cell m in CT B (T 1, T 2 ) φ T 1 m T 2 ψ the commutativity of the square above translates to the commutativity of the diagram below: α T 1 =T 1 (α) T 1 F T 1 G φ F m F T 1 F ψ F φ G m G T 2 G ψ G α T 2 =T 2 (α) Since 1 T2 (α) m F = m G 1 T1 (α) follows from the definition of 2-cells in CT, α is a 2-natural transformation.

23 16 Proposition 2.11 The previous two constructions define a 2-functor K : CT op 2 Cat with K acting on 0, 1 and 2-cells as follows: K() = CT, K(F ) = F and K(α) = α. Proof. Given categories and B, K must induce a functor K B : CT(, B) 2 Cat( CT B, CT ) such that the following diagram commutes: CT(, B) CT(B, C) K B K BC c CT(, C) K,C 2-Cat(CT B, CT ) 2-Cat(CT C, CT B ) c 2-Cat(CT C, CT ) For 1-cells F : B and G : B C, (GF ) = F G holds since for T in CT C, (GF ) (T ) = T (GF ) and F (G (T )) = F (T G) = (T G)F ; for φ : T 1 T 2 in CT C, both (GF ) (φ) and F G (φ) result in the natural transformation φ GF : T 1 GF T 2 GF ( as defined in Construction 2.9). Given 2-cells φ, γ (and then φ and γ ) as pictured, CT F 1 φ F 2 F 1 φ F 2 B K CT B G 1 γ G 2 G 1 γ G 2 C CT C the commutativity of the diagram translates to the equation (γ φ) = φ γ. For T in CT C, by Lemma 2.10, the component of (γ φ) corresponding to T is the

24 17 natural transformation (γ φ) T : T G 1F 1 T G 2 F 2 that has as the component for a in the functor ((γ φ) T ) a : T G 1 F 1 (a) T G 2 F 2 (a) which is the image of the arrow (γ φ) a : G 1 F 1 (a) G 2 F 2 (a) in C under the functor T : ((γ φ) T ) a = T ((γ φ) a ), where (γ φ) a is the diagonal of the commutative square (in C) below. G 1 F 1 (a) γ F 1a G 2 F 1 (a) G 1 (φ a ) (γ φ) a G 1 F 2 (a) γf2 a G 2 (φ a ) G 2 F 2 (a) Since (for i = 1, 2) F i G i (T ) = T G if i, by the definition of F i (Construction 2.9) F i (γ T ) = (γ T ) F i and on the other hand (by Lemma 2.10), (γ T ) F i = T (γ Fi ) and φ T G i = T G i (φ), applying the functor T to the diagram above gives the commutative square, whose diagonal, by the definition of the horizontal composition of natural transformations in 2-Cat is ((φ γ ) T ) a : T G 1 F 1 (a) = F 1 G 1(T )(a) T (γ F1 a )=F 1 (γ T )(a) T G 2 F 1 (a) = F 1 G 2(T )(a) T G 1 (φ a )=φ T G 1 ((γ φ ) T ) a T G 2 (φ a )=φ T G 2 T G 1 F 2 (a) = F 2 G 1(T )(a) T (γ F2 a )=F 2 (γ T )(a) T G 2 F 2 (a) = F 2 G 2(T )(a) Thus (γ φ) = γ φ since the components of the natural transformations are equal. K B must also honor the vertical composition of natural transformations:

25 18 α B. The component of K(β α) = (β α) corresponding to an object a β in and T in CT B is T (β a α a ), whereas the corresponding component of K(β) K(α) = β α is T (α a )T (β a ); since T is a functor, we have that T (β a α a ) = T (α a )T (β a ). s a direct consequence of the definitions involved, all unit axioms are satisfied as well, and therefore K is well defined. Definition 2.12 (cf. Definition in [3]) 2-functor F : B is called a lax 2-functor, if it preserves composition and identities up to coherent 2-cells, i.e., for every triple of objects a, b, c in, there is a natural transformation δ abc and for every object a in, there is a natural transformation ε a such that the following diagrams commute. (a, b) (b, c) (a, c) F ab F bc B(F a, F b) B(F b, F c) B(F a, F c) c δ abc c F ac u a 1 (a, a) ε a u F a F ac 1 B(F a, F a) The natural transformations δ and ε must satisfy the following coherence axioms. (The component of δ abc corresponding to the pair of arrows (f, g) will be denoted by δ f,g rather than (δ abc ) (f,g).)

26 Coherence with composition: for any triple of arrows a must have δ gf,h (1 F h δ f,g ) = δ f,hg (δ g,h 1 F f ). f b g c 19 h d in, we Coherence with units: δ 1a,f (1 F f ε a ) = 1 F f and δ f,1b (ε b 1 F f ) = 1 F f (see diagrams below). 1 F h δ f,g F h F g F f F h F (g f) δ gf,h δ f,g 1 F f δ f,hg F (h g) F f F (h g f) 1 F f ε a F f 1 F a F f F (1 a ) ε b 1 F f 1 F b F f F (1 b ) F f 1 F f δ 1 a,f 1 F f δ f,1b F f F (f 1 a ) 1 F f F f F (1 b f) 1 F f F will be called a strong-lax 2-functor when the natural transformations δ abc and ε a are natural isomorphisms for any objects a, b and c. (Such functors are also referred to as pseudo, or weak functors.) The purpose of the constructions outlined in ( ) is to define a strong-lax 2-functor H : CT op 2 Cat. Remark 2.13 Given a morphism Φ : [(E 1, V 1, H 1 ) P 1 B] [(E 2, V 2, H 2 ) P 2 B] of horizontal structures over B, if h : x y is the horizontal lift of h : a b to domain x in E 1, then Φ( h) : Φ(x) Φ(y) is the horizontal lift of h to domain Φ(x) in E 2 : since Φ is assumed to preserve horizontal morphisms, Φ( h) is horizontal, and P 1 = P 2 Φ implies that it covers h, so it is the horizontal lift of h to domain

27 20 Φ(x) by the uniqueness of horizontal lifts. dopting the notation h for a lift of an arrow h to a domain in E 1, and h for the lift of h to a domain in E2, we have that Φ( h) = h. Construction 2.14 Given a functor F : B and a horizontal structure (E, V, H) P B over B, the canonical pullback of E P B along F defines a horizontal structure (E F, V F, H F ) P F over. The objects of E F are pairs (a, x) where a and x are objects in and E respectively such that F (a) = P (x); the morphisms are pairs (f, g) with f a morphism in and g a morphism in E such that F (f) = P (g). Composition of morphisms is defined componentwise. The horizontal structure (E F, V F, H F ) is given by V F = {(a, x) (1 a,g) (a, y)} with g V and by H F = {(a, x) (f,h) (b, y)} where h is in H. In the pullback square below, the functors P F and G are defined in the obvious way: P F (a, x) = a, P F (f, g) = f and similarly G(a, x) = x, G(f, g) = g. It follows from the definitions that both V F and H F are subcategories of E F, and clearly V F V (E F ). (E F, V F, H F ) P F G (E, V, H) P F B The unique horizontal lift of f : a b in to domain (a, x) in E F is (f, F f) : (a, x) (b, x) where F f is the unique horizontal lift of F f : F (a) F (b) to domain x in E; thus x = Cod(F f). The unique factorization of a morphism (a, x) (f,g) (b, y) in E F is (f, g) = (1 b, v)(f, h) where vh is the unique factorization of g in E.

28 Remark 2.15 The canonical pullback of E P along the identity functor 1 : results in an isomorphic copy E 1 of E with objects (a, x) such that P (x) = a and morphisms (f, g) such that P (g) = f; thus the objects and morphisms 21 of E 1 are of the form (P (x), x) and (P (g), g), respectively. Lemma 2.16 Given a morphism Φ : E 1 E 2 of horizontal structures over B and a functor F : B, Φ induces a morphism Φ F : E F 1 E F 2 of the pullback horizontal structures. Proof. On objects Φ F (a, x) = (a, Φ(x)) and on morphisms Φ F (f, g) = (f, Φ(g)). Since Φ F preserves vertical morphisms and P 1 Φ F = P F, Φ F is a morphism of horizontal structures, i.e., it is a one cell in H. Construction 2.17 Given morphisms Φ, Ψ : E 1 E 2 of horizontal structures over B and a natural transformation α : Φ Ψ, pulling α back along a functor F : B gives a natural transformation α F : Φ F Ψ F. For an object (a, x) in E F 1, α F (a,x) = (1 a, α x ) : Φ F (a, x) = (a, Φ(x)) Ψ F (a, x) = (a, Ψ(x)), where α x : Φ(x) Ψ(x) is the component of the natural transformation α for the object x in E 1 ; (1 a, α x ) is clearly a vertical morphism, and for (a, x) (f,g) (b, y) in E F 1, the following diagram commutes (a, Φ(x)) (f,φ(g) (b, Φ(y)) (1 a,α x ) (a, Ψ(x)) (f,ψ(g)) (1 b,α y ) (b, Ψ(y)) since by the naturality of α, Ψ(g)α x = α y Φ(g).

29 22 Construction 2.18 Given a functor F : B, F gives rise to a 2-functor H F : H B H as follows: (E, V, H) in Construction For a pair of objects (E i, V i, H i ) P B is sent to (E F, V F, H F ) P F as P i B, i = 1, 2 in HB, H F must define a functor H FE1,E 2 : H B (E 1, E 2 ) H (E F 1, E F 2 ). (We ll write H F1,2 instead of H FE1,E 2.) For a morphism Φ : E 1 E 2 of horizontal structures (an object in H B (E 1, E 2 ) as defined in 2.6), H F (Φ) = Φ F as in Lemma For a natural transformation α : Φ Ψ (an arrow in H B (E 1, E 2 )), H F (α) = α F : Φ F Ψ F as defined in Construction Given arrows Φ α Ψ β Υ in H B (E 1, E 2 ), (β α) F = (β F α F ), since for an object (a, x) in E1 F, (β α) F (a,x) = (1 a, (β α) x ) = (1 a, (β x α x )) = (1 a, β x ) (1 a, α x ) = β(a,x) F αf (a,x) = (βf α F ) (a,x). For the identity 2-cell 1 : Φ Φ in H B (E 1, E 2 ), 1 F : Φ F Φ F is clearly the identity 2-cell in H (E1 F, E2 F ). H(F ) must also satisfy the following compatibility conditions to be a 2-functor. Compatibility with composition translates into the commutativity of the following diagram for objects E 1, E 2 and E 3 in H B. H B (E 1, E 2 ) H B (E 2, E 3 ) c H B (E 1, E 3 ) H F 1,2 H F 2,3 H F 1,3 H (E F 1, E F 2 ) H (E F 2, E F 3 ) c H (E F 1, E F 3 ) For the diagram above to commute on the object level, we must have (ΨΦ) F = Ψ F Φ F for functors Φ : E 1 E 2 and Ψ : E 2 E 3. Given an object (a, x) in E1 F, (ΨΦ) F (a, x) = (a, ΨΦ(x)) and (Ψ F Φ F )(a, x) = Ψ(a, Φ(x)) = (a, ΨΦ(x)); similarly for an arrow (f, g) : (a, x) (b, y) in E1 F, since the functors Φ and Ψ act on the second coordinate, (ΨΦ) F (f, g) = Ψ F (Φ F (f, g)) = (f, ΨΦ(g)). For the diagram to commute for arrows α : Φ 1 Ψ 1 and β : Φ 2 Ψ 2 we must

30 23 have β F α F = (β α) F. For an object (a, x) in E1 F, the components of the natural transformations in question are (by Construction 2.17 and Definition 2.6) as follows: (β α) F (a,x) = (1 a, (β α) x ) = (1 a, Ψ 2 (α x ) β Φ1 (x)) and (β F α F ) (a,x) = Ψ F 2 (α(a,x) F ) βf Φ F 1 (a,x) = ΨF 2 (1 a, α x ) β(a,φ F 1 (x)) = (1 a, Ψ 2 (α x )) (1 a, β Φ1 (x)) = (1 a, Ψ 2 (α x ) β Φ1 (x)). Compatibility with units means that for an object E in H B, with 1 E : E E the identity functor, (1 E ) F = 1 E F and that for a functor Φ in H B, with 1 Φ the identity 2-cell on Φ we have (1 Φ ) F = 1 Φ F. By Lemma 2.16, (1 E ) F (a, x) = (a, 1 E (x)) = (a, x) for an object (a, x) in E F and similarly for a morphism (f, g) in E F, (1 E ) F (f, g) = (f, 1 E (g)) = (f, g), so (1 E ) F = 1 E F. By Construction 2.17, (1 Φ ) F (a,x) = (1 a, (1 Φ ) x ) = (1 a, 1 Φ(x) ) and (1 Φ F ) (a,x) = (1 a, 1 Φ(x) ), we have that (1 Φ ) F = 1 Φ F, and the 2-functor H F : H B H is well defined. Remark 2.19 For the identity functor 1 : the corresponding 2-functor H 1 : H H is an isomorphism of categories, since pulling back an object P E 1 1 in H along the identity functor results in an isomorphic copy E 1 1 of E 1 (see Remark 2.15), and for horizontal structures E 1 and E 2 over, H 1 : H (E 1, E 2 ) H (E 1 1, E 1 2 ) gives a bijection on both the classes of 1-cells and 2-cells. Lemma 2.20 natural transformation θ : F G of functors F, G : B defines a 2-natural transformation H θ : H F H G where H F, H G : H B H are the 2-functors defined in Construction Proof. For an object (E, V, H) P B of H B, the corresponding arrow H θe : E F E G is the functor defined as follows: For an object (a, x) in E F, H θe (a, x) =

31 24 (a, x) where x is the codomain of the horizontal lift θ a : x x of θ a : F (a) G(a) to domain x in E. For a morphism (f, g) : (a, x) (b, y), H θe (f, g) : (a, x) (b, y) is constructed as follows: let h be the horizontal lift of h = Gf θ a = θ b F f : F (a) G(b) to domain x in E and Gf be the horizontal lift of Gf to domain x. We then have both x θ a x Gf z and h covering h. Since H is a subcategory and both θ a and Gf are in H, so is Gf θ a and then by the uniqueness of horizontal lifts we have Gf θ a = h : x z. We also have x g y θ b y in E; let vh be the horizontal-vertical decomposition of θ b g. Since P (h ) = P (vh ) = P (θ b g) = θ b P (g) = θ b F f = h, h is also a horizontal lift of h, so h = h. Then z = cod(h ) = dom(v), and we have x Gf z v y. For morphisms then, H θe is defined by H θe (f, g) = (f, vgf) : (a, x) (b, y). We must still show that H θe preserves composition and identities. Given morphisms (a, x) (f,g) (b, y) (f,g ) (c, t) in E F, we have that H θ (f, g ) H θ (f, g) = (f f, v Gf vgf) where v is the vertical component of the horizontal-vertical decomposition of θ c g (see figure below) and H θ (f f, g g) = (f f, v G(f f)) where v is the vertical component of the horizontal-vertical decomposition of θ c g g. (θ c g g = v (G(f f) θ a ).) Thus it is left to show that v Gf vgf = v G(f f). Let v h be the horizontal-vertical decomposition of Gf vgf. Then Gf vgf = v h means that h covers Gf Gf = Gf f and it is a horizontal morphism with domain x; the same holds for Gf f, so by the uniqueness of horizontal lifts, h = Gf f. Then we have that on one hand θ c g g = v Gf θ b g = v Gf vgf θ a = v v h θ a = v v G(f f) θ a ; on the other hand, it follows from the definition of v, that θ c g g = v G(f f)θ a. Then by the uniqueness of horizontal-vertical decompositions, we have that v v = v, which then shows that v G(f f) = v v G(f f) = v v h = v Gf vgf.

32 25 E F (f,g ) (f,g) (b,y) (a,x) (c,t) H θe > f > f a b c E G (f,v Gf ) (f,vgf) (c, t) (b,ȳ) (a, x) F G θ x g F (a) F f θ a y g θ a F (b) F f x Gf z θ b t G(a) θ b F (c) ȳ θ c v G(b) θ c G(f f) v v t G(c) v P E B Given an object (a, x) in E F, 1 (a,x) = (1 a, 1 x ); we must show that H θe (1 a, 1 x ) = (1 a, 1 x ). By the definition of H θe, H θe (1 a, 1 x ) = (1 a, v1 G(a) ) where v is the vertical component of the horizontal vertical decomposition of θ a 1 x = θ a ; since θ a : x x can be written as θ a = 1 x θ a, v = 1 x. Moreover, 1 G(a) being a horizontal lift is in H, and it is in V as well, since it covers the identity on G(a) in B; therefore 1 G(a) = 1 x which shows that v1 G(a) = 1 x. For H θ to define a 2-natural transformation, the following diagram of categories must commute: H B (E 1, E 2 ) H G(E1,E 2 ) H (E G 1, E G 2 ) H F(E1,E 2 ) H (E1 F, E2 F ) H (1,H θe2 ) H (H θe1,1) H (E F 1, E G 2 ) Following the object (functor) Φ : E 1 E 2 of H B (E 1, E 2 ) first horizontally and then vertically gives the functor H θe2 Φ F : E F 1 E G 2. Given an object (a, x) in

33 26 E1 F, Φ F (a, x) = (a, Φ(x)) and H θe2 (a, Φ(x)) = (a, Φ(x)) where Φ(x) is the codomain of the horizontal lift of θ a : F (a) G(a) to domain Φ(x) in E 2 ; following Φ in the other direction results in the functor Φ G H θe1. Now H θe1 (a, x) = (a, x) where x is the codomain of the horizontal lift of θ a to domain x in E 1, and Φ G (a, x) = (a, Φ( x)). Since by Remark 2.13 Φ(x) = Φ( x), we have that for objects, H θe2 Φ F = Φ G H θe1. For a morphism (f, g) : (a, x) (b, y) in E1 F, on one hand H θe2 Φ F (f, g) = H θe2 (f, Φ(g)) = (f, v Gf) where v is the vertical component of the horizontalvertical decomposition of θ b Φ(g) and Gf is the horizontal lift of Gf : G(a) G(b) to domain Φ( x) in E 2 ; on the other hand Φ G H θe1 (f, g) = Φ G (f, vgf) = (f, Φ(vGf)) where Gf is the horizontal lift of Gf to domain x in E 1 and v is the vertical component of the horizontal-vertical decomposition of θ b g in E 1. Since Φ preserves the horizontal-vertical decomposition of morphisms, the horizontalvertical decomposition of θ b Φ(g) = Φ( θ b g) is Φ(vG(f) θ a ) = Φ(v)G(f) θa. Thus Φ(v) = v and Φ(vG(f)) = v G(f); hence H θe2 Φ F = Φ G H θe1 for morphisms as well. The commutativity of the diagram for 2-cells translates to 1 Hθ2 α F = α G 1 Hθ2 : E F 1 Φ F α F E2 F Ψ F H θe2 1 E2 G and E1 F H θe2 H θe1 1 E1 G H θe2 Φ G α G E2 G. Ψ G Given an object (a, x) of E F 1, the corresponding component of both composite natural transformations (by Construction 2.17) is (1 a, α x ) : (a, Φ( x) (a, Ψ( x)). Corollary 2.21 If the natural transformation θ : F G in Lemma 2.20 is a natural isomorphism, then the components of H θ are isomorphisms as well.

34 27 Proof. Suppose the components θ a : F (a) G(a) of θ are invertible arrows in B; then given a horizontal structure E P B, the corresponding component H θe : E F E G is a functor with its inverse H 1 θ E : E G E F defined as follows. For an object (a, y) of E G, H 1 θ E (a, y) = (a, ȳ) with ȳ = cod(θa 1 ). Then by part (iii) of Remark 2.5, on the object level H θe H 1 θ E = 1 E G and H 1 θ E H θe = 1 E F. Given a morphism (f, g) : (a, x) (b, y) in E G H 1 θ E (f, g) def = (f, vf f) where v is the vertical component of the horizontal-vertical factorization of θ 1 b g with θ 1 b the lift of θ 1 b : G(b) F (b) to domain y in E. To show that H 1 θ E H θe = 1 E F on the morphism level, suppose (f, g) : (a, x) (b, y) is a morphism in E F ; H θe (f, g) = (f, vgf) where v is the vertical component of the horizontal-vertical factorization of θ b g (θ b is lifted to domain y to get θ b ) and H 1 θ E (f, vgf) = (f, v F f) where v is the vertical component of the horizontal-vertical decomposition of θ 1 b vgf. Since θ b g = vgf θ a by the horizontal-vertical factorization of θ b g, and θ 1 b vgf = v F fθa 1 we have that v F f = θ 1 b vgf θ a = θ 1 b θ b g = g. Thus H 1 θ E H θe = 1 E F and by a similar argument, H θe H 1 θ E (again using the horizontal-vertical factorization of θ 1 b vgf), = 1 E G. Hence pulling back a horizontal structure along isomorphic functors results in isomorphic horizontal structures: F = G E F = E G. Proposition 2.22 The preceding constructions define a strong-lax 2-functor H : CT op 2-Cat. Proof. For zero cells, H() = H, the 2-category of horizontal structures over (Def. 2.6). For each pair of objects, B in CT, we have a functor H,B : CT(, B) 2 Cat(H B, H ) defined for F : B by H,B (F ) = H F :

35 28 H B H as in Construction 2.18, and for θ : F G (F, G : B) by H,B (θ) = H θ as in Lemma Given F, G, K : B, α : F G and β : G K in CT(, B), we must have H,B (βα) = H,B (β)h,b (α), or using the notation of Lemma 2.20, H βα = H β H α. For an object (a, x) in E F, H α (a, x) = (a, x) E G where x is the codomain of the horizontal lift α a of α a : F (a) G(a) to domain x in E, and H β (a, x) = (a, x) E K where x is the codomain of the horizontal lift β a of β a : G(a) K(a) to domain x in E. H βα (a, x) = (a, ˆx) E K where ˆx is the codomain of the horizontal lift (βα) a of (βα) a : F (a) K(a) to domain x in E. Since H is assumed to be a subcategory of E, the composition of horizontal morphisms is horizontal; the uniqueness of horizontal lifts to a given domain then implies that β a α a = (βα) a and hence x = ˆx. Given a morphism (f, g) : (a, x) (b, y) in E F, on one hand H α (f, g) = (f, vgf) : (a, x) (b, ȳ) where v is the vertical component of the horizontal-vertical decomposition of α b g and H β (f, vgf) = (f, v Kf) : (a, x) (b, ȳ) where v is the vertical component of the horizontal-vertical decomposition of β b v Gf. On the other hand, H βα (f, g) = (f, ˆvKf) : (a, x) (b, ȳ) where ˆv is the vertical component of the horizontalvertical decomposition of (βα) b g = β b α b g (see diagram below). We must show then that v = ˆv: we have that β b α b g = ˆv(Kf (βα) a ) = ˆvKf β a α a, but also β b α b g = (β b v Gf)α a = (v Kf β a )α a, so v = ˆv. For the identity 2-cell 1 F : F F (F : B), we need to show that H 1F = 1 HF : E F E F. Clearly H 1F (a, x) = (a, x) since 1 x : x x is the horizontal lift of 1 : a a to domain x in E, and similarly for f : a b and g : x y, H 1F (f, g) = (f, vf f) where v is the vertical component of the horizontal-vertical decomposition of 1 y g = g = vf f.

36 29 (βα) a E E F E G E K α β a a x x x Kf Gf g y α b ȳ v β b ȳ v H β ; ; H α f > a b G F K α β β a F (a) α a G(a) K(a) F f F (b) Gf β b Kf α b G(b) K(b) B For every triple of objects, B and C in CT and functors F B G C, we have a natural isomorphism δ F,G : H F H G H GF ; the component of δ F,G corresponding to an object E P C of H C is given by the isomorphism of the categories (E G ) F = E (GF ) : The objects of E G P G B are pairs (b, x) such that G(b) = P (x) and then the objects of (E G ) F P F are pairs (a, (b, x)) with F (a) = P G (b, x) = b, so each object (a, (b, x)) (E G ) F is of the form (a, (F (a), x)) and will be identified with (a, x) E (GF ) ; since F (a) = b, GF (a) = G(b) = P (x), so (a, x) is an object in (E G ) F. The isomorphism works similarly for morphisms. For every object in CT, we have the natural isomorphism ε : 1 H H 1 that for an object E in H, identifies x in E with (P (x), x) in E 1 and similarly g : x y in E with (P (g), g) : (P (x), x) (P (y), y) in E 1 (cf. Remark 2.15).

37 30 For one cells F : B, H(F ) is a 2-functor defined as in Construction For two cells θ : F G, H θ : H F H G is the 2-natural transformation with components H θe : H F (E) = E F H G (E) = E G defined as in Lemma Definition 2.23 (Definition 1.1 in [20]) Given a functor F : B CT, the Grothendieck construction on F, denoted by BF, is the category with objects the pairs (b, x) with b an object of B and x an object of F (b). morphism f : (b, x) (c, y) of BF is a quadruple f = (x, f, α, y) such that f : b c is a morphism in B and α : F f(x) y is a morphism in F (c). Composition in BF is defined by (y, g, β, z) (x, f, α, y) = (x, gf, β F g(α), z) for morphisms (y, g, β, z) and (x, f, α, y) in F, such that gf is defined in B. For an object (b, x) in, 1 (b,x) = (x, 1 b, 1 x, x). B BF The functor U F : B associated to F : B CT is defined as U BF F (b, x) = b for objects and as U F (x, f, α, y) = f for morphisms. For U F : BF B, the categorical fiber (U F ) 1 (b) over an object b is isomorphic to F (b). Remarks 2.24 (i) Given a functor F : B CT, let F 0 = U 0 F : B Sets where U 0 : CT Sets is the functor which sends a category C to its set of objects. Then with F 0 : B Sets CT (where each set is viewed as a discrete category), for U F : F 0 B each fiber is a discrete category with the identities as the only morphisms and F 0 is the same category as the category of elements of F 0 in the proof of Proposition 1 in [15]. (ii) If F factors through POS (the category of partially ordered sets and order preserving functions), i.e., F : B POS CT, then a morphism f : b c in B lifts to a morphism f : (b, x) (c, y) in BF iff F f(x) y. This so called

38 lifting condition is the same when F : B CoP (where CoP is the category of cocomplete posets; see Definition 2.39). For f = (x, f, α, y), the vertical morphism α : F f(x) y is unique, thus f will be written as a triple (x, f, y) and the composition rule in Definition 2.21 simplifies to (y, g, z) (x, f, y) = (x, gf, z). 31 Lemma 2.25 For a functor F : B CT, (H, V ) defined as follows gives a splitting of the category F. H consists of all the objects of and all the B BF morphisms of the form (x, f, 1 F f(x), F f(x)) : (b, x) (c, F f(x)). V is the vertical subcategory of BF ; its morphisms then are of the form (x, 1 b, α, y) : (b, x) (b, y). Proof. morphism (x, f, α, y) : (b, x) (c, y) in BF factors as (x, f, α, y) = (F f(x), 1 c, α, y) (x, f, 1 F f(x), F f(x)); this factorization of (x, f, α, y) as a horizontal morphism followed by a vertical, is clearly unique. Lemma 2.26 The splitting of BF given in Lemma 2.25 above defines a horizontal structure over B. Proof. The unique horizontal lift of f : b c in B to domain (b, x) in BF is f = (x, f, 1 F f(x), F f(x)) : (b, x) (c, F f(x)). Lemma 2.27 natural transformation α : F G (where F, G : CT) induces a morphism α : F G of horizontal structures (Definition 2.6). Proof. α is defined for (a, x) in F, ( α) (a, x) = (a, α a (x)) and for (x, f, ϱ, y) : (a, x) (b, y) by ( α) (x, f, ϱ, y) = (α a (x), f, α b (ϱ), α b (y)). preserves identities, since for each object a in, α a : F (a) G(a) is a α

39 32 functor, so we have that for an identity arrow (x, 1 a, 1 x, x) : (a, x) (a, x), ( α) (x, 1 a, 1 x, x) = (α a (x), 1 a, α a (1 a ), α a (x)) = (α a (x), 1 a, 1 αa, α a (x)). To show that α preserves composition, the following equality must hold for morphisms (x, f, r, y) : (a, x) (b, y) and (y, g, t, z) : (b, y) (c, z): ( α) [(y, g, t, z) (x, f, r, y)] = ( α) (y, g, t, z) ( α) (x, f, r, y). By the definition of composition in F we have that ( α) [(y, g, t, z) (x, f, r, y)] = ( α) (x, gf, t F g(r), z) = (α a (x), gf, α c (t F g(r)), α c (z)); composing in G gives ( α) (y, g, t, z) ( α) (x, f, r, y) = (α b (y), g, α c (t), α c (z)) (α a (x), f, α b (r), α b (y)) = (α a (x), gf, α c (t) Gg(α b (r)), α c (z). The two morphisms then are the same if α c (t F g(r)) = α c (t) Gg(α b (r)); this equality follows from the naturality of α, which implies that α c F g = Gg α b (see diagram below). F (b) α b G(b) F g F (c) α c Gg G(c) s a direct consequence of the definitions involved U F = U G and α preserves vertical morphisms; it also preserves horizontal morphisms, since ( α) (x, f, 1 F fx, F fx) = (α a (x), f, α b (1 F fx ), α b (y)) = (α a (x), f, 1 αb F fx), α b (y)). α : F G is then a 1-cell in the 2-category H. Construction 2.28 category induces a 2-functor : CT H as follows: The image of a 0-cell T : CT under is T U T, the Grothendieck construction on T as given in Definition 2.23, and the image of a 1-cell α : T S is the functor α defined in the previous lemma. For any pair of objects T, S in CT, must give a functor CT (T, S) H ( T, S)

40 which will also be denoted by to avoid multiple subscripts. The image of a 2-cell m : α β, under the functor will be denoted by m (see diagram below) and its component 33 T α m β CT β T m α S S corresponding to an object (a, x) in T is the arrow in S given as follows: ( m) (a,x) = (α a(x), 1 a, (m a ) x, β a (x)) : (a, α a (x)) (a, β a (x)), which is a vertical morphism as required by Definition 2.5. For : CT (T, S) H ( T, S) to be a functor, it must preserve the vertical composition of 2-cells: given m : α β and n : β γ (with α, β, γ : T S) we have on one hand for objects a and x in and T (a) respectively that (n m) (a,x) = (n ax m ax ) = (α a (x), 1 a, n ax m ax, γ a (x)) and ont the other hand ( n) (a,x) ( m) (a,x) = (β a (x), 1 a, n ax, γ ( x)) (α a (x), 1 a, m ax, β ( x)) = (α a (x), 1 a, n ax G(1 a )(m ax ), γ ( x)). Since S(1 a ) = 1 Ga is the identity functor on S(a), S(1 a )(m ax ) = m ax and we have that (n m) = n m holds componentwise. also preserves identity ( ) 2-cells: 1 α = (α a (x), 1 a, (1 α ) ax, α a (x)) (Def 2.21) where (1 α ) ax = 1 αa (x). (a,x) The compatibility of the following diagram: with composition translates into the commutativity of CT (T, S) CT (S, R) CT (T, R) H ( T, S) H ( S, R) c c H ( T, R)

41 For a 1-cell (α, β) in CT (T, S) CT (S, R) both functors β α and βα take an object (a, x) in to (a, β a α a (x)) and an arrow (x, f, ϱ, y) to T (β a α a (x), f, β b α b (ϱ), β b α b (y)). For a 2-cell (m, n) in CT (T, S) CT (S, R) we must have that n m = m). This equality follows directly form applying the relevant definitions; for objects a in and x in T (a), the (n component of both 2-cells is the following arrow in R: (β a α a (x), 1 a, (n m) ax, η a γ a (x)) : (a, β a α a (x)) (a, η a γ a (x)) where α, β, γ and η are as pictured below. 34 α β T m S n R γ η CT α β T m S n R γ η The unit axiom also holds as a consequence of the definitions and hence : CT H as defined above is a 2-functor. functor T : B is an equivalence of 2-categories when there is a 2-functor S : B and 2-natural isomorphisms T S = 1 B and ST = 1. Lemma 2.29 The 2-functor of Construction 2.28 is an equivalence of the 2-categories CT and H for any category. Proof. 2-functor 1 isomorphisms θ and λ, such that 1 E : H CT will be defined along with natural θ 1 H and 1 λ 1 CT. Given P in H, the corresponding functor T E : CT is defined as follows.