Categories and Modules

Transcription

1 Categories and odules Takahiro Kato arch 2, 205

2 BSTRCT odules (also known as profunctors or distributors) and morphisms among them subsume categories and functors and provide more general and abstract framework to explore the theory of structures In this book we generalize and redevelop the basic notions and results of category theory using this framework of modules Topics Chapter introduces modules and cells among them module from a category to is a functor of the form op Set, assigning a set of arrows to each pair of objects x and a cell from a module to N sends each arrow of to an arrow of N The hom-functor of a category C forms an endomodule C C C called the hom of C, and the arrow function of a functor F C D forms a cell F C D from the hom of C to the hom of D odules and cells thus subsume categories and functors in this way and set up a more general and conceptual framework to explore the structure of mathematics Presheaves and copresheaves are called right and left modules respectively in this book and studied as special instances of modules Chapter 2 discusses the action of a module on its domain and codomain, the operation that yields the Yoneda embedding functor in the case of a hom endomodule The chapter introduces an important class of modules called representable, which each functor produces by the composition with the hom of its codomain Chapter 3 presents two variants of modules, namely collages and commas, which are special sorts of cospans and spans between two categories We will establish an isomorphism between the category of modules and the category of collages, and, later in Chapter 9, construct an adjoint equivalence between the category of commas and the category of collages Two forgetful functors from OD (the category of modules and cells) to CT are dened through construction of collages and commas, and it is shown that they form left and right adjoints of the embedding CT OD given by the hom operation C C Chapter 4 introduces the notion of frames of a module cylindrical frame of an endomodule abstracts a natural transformation between two functors, and a conical frame of a right (resp left) module abstracts a cone between an object and a functor Inner and outer cylinders, which are, so to speak, natural and extranatural transformations spanning a module, are dened using cylindrical frames Likewise, cones are dened along a module using conical frames Chapter 5 succeeds Chapter 2 and discusses the actions of the domain and codomain of a module on its arrows and frames It is shown, as a generalization of the Yoneda embedding, that these actions embed a module in (the hom of) the category of right modules (ie presheaves) over and the category of left module (ie copresheaves) over The Yoneda lemma is presented in a general form to state that the morphisms from the representable module of a functor F to an arbitrary module correspond one-to-one with the cylinders dened between F and Using the lemma, we show a variety of bijective correspondences between frames and cells Chapters 6, 7, 8, and 0 explore the fundamental concepts of category theory in the framework of modules Universals are dened along a module and we study if they are preserved by a cell The embedding of a module in the category of presheaves (resp copresheaves) makes the denition of a universal simple: an arrow of a module is universal if its image under the embedding is an isomorphism Lifts, of which Kan lifts are special instances, are dened as universal cylinders, and extensions, of which Kan extensions are special instances, are dened as universal cells We introduce the notion of pointwise lifts and show that the 2

3 notion subsumes that of pointwise extensions and vice versa The concept of an adjoint is extended to a cell We show that a cell preserves universals if it has adjoints, and deduce RPL (right adjoints preserve limits) as a corollary of this general mechanism The concept of density is also generalized for modules In fact, density is most naturally dened with modules The fact every presheaf is a colimit of representables is proved using the concept of density to show how the concept works in the framework of modules dvice on reading The exposition is carried out within the framework of set-enriched -dimensional categories Chapter 0 presents the notations and some elementary facts of category theory used in the sequel Size issues are not treated rigorously The specication of a universe is almost always implicit; a universe U is chosen so that given categories and modules become locally U-small, ie all hom-sets become U-small We just say small (resp large) instead of U-small (resp U-large) for U chosen implicitly category or a module constructed is called large if it is not, in general, locally small The choice of a universe is xed in each context This book is made up of sequences of Denition - Proposition - Theorem - Corollary, with each optionally preceded by Note and followed by Remark ach Proposition is tightly associated with the preceding Denition and states some straightforward consequences of it Two statements dual to each other are indicated by the symbols and Proofs are given for the assertions indicated by Parentheses and brackets serve mainly as punctuation to enhance readability; parentheses are used to delimit objects and arrows, whereas square (resp angle) brackets are used to delimit categories and functors (resp modules and cells) 3

4 Contents 0 Preliminaries 6 odules and Cells 0 odules 0 2 Cells 2 3 Cell morphisms 32 2 Slicing and ction 34 2 Slicing of modules ction of modules Yoneda functors and representations 4 3 Collages and Commas 46 3 Collages Commas 52 4 Frames 65 4 Cylindrical frames Conical frames Inner cylinders Outer cylinders Weighted cylinders 9 46 Cones Bicylinders and wedges Cones and wedges in a category 7 49 Cones and wedges in Set 28 5 Yoneda Lemma 36 5 Yoneda modules Yoneda morphisms Yoneda morphisms for cylinders 5 54 Yoneda morphisms for cones Correspondences between frames and cells 73 6 Universals 87 6 Units of one-sided modules Universal arrows Units of two-sided modules Universal cylinders Kan lifts 206 4

5 Contents 7 Limits 20 7 Universal cones Universal wedges Limits in a category Limits of modules 22 8 djunctions djunctions for categories Transformations of adjoints djunctions for modules quivalence of categories quivalence of modules djoint functor theorem Collages and Commas (continued) 26 9 Cylinder modules quivalence CO CLG quivalence [ ] [ ] quivalences [ ] [ ] and [ ] [ ] 276 0xtensions coyoneda lemma Universal cells Kan extensions Density 3 05 nds 37 List of Symbols 320 5

6 0 Preliminaries Given a universe U, Set U, Cat U, and CT U denote the categories of U-small sets, U-small categories, and locally U-small categories respectively By Set we mean Set U for U chosen implicitly, and similarly for Cat and CT 2 category is dened in terms of hom-sets Pairwise disjointness of hom-sets is not required Given a category C, an element f of a hom-set hom C (a, b) or a triple (a, f, b) such that f hom C (a, b) is called an arrow of C (or a C-arrow) and written f a b, a f b, or just f if its domain and codomain are understood or unimportant The composite of arrows f a b and g b c is written as f ` g = g f 3 The set of objects of a category C is denoted by C and the object function of a functor F is denoted by F 4 Italic small letters a, b, c, vary over objects and arrows When we write c C for a category C, c stands for an object or arrow of C 5 The evaluation of a functor F C B at c C is written as c` F = F (c) = F c, and the component of a natural transformation τ F G C B at an object c C is written as c` τ = τ c = τ c 6 The opposite of a category C is denoted by C, and the opposite of a functor F C B is denoted by F C B, or often by F C B using the same name as its original counterpart 7 Given categories and D, the product, coproduct, and functor categories are denoted by D, + D and [, D] respectively The terminal category and its sole object are denoted by ; the interval category is denoted by 2 and its objects are denoted by 0 and 8 Italic capital letters F, G, vary over both functors and natural transformations When we write F [, D], F D, or F D for categories and D, F stands for an object or an arrow of the functor category [, D], ie a functor or a natural transformation 6

7 0 Preliminaries 9 Let, D, and C be categories The composite of F D and G D C is written as The composition forms a bifunctor F ` G = G F (F, G) F ` G [, D] [D, C] [, C] 0 The bifunctorial operation [, ] on CT CT is dened in the following way: a) for a pair of categories and, [, ] is the category of functors ; b) for a functor P Y and a category, [P, ] [Y, ] [, ] is the precomposition functor dened by F ` [P, ] = P ` F for F [Y, ]; c) for a functor Q B and a category, [, Q] [, ] [, B] is the postcomposition functor dened by F ` [, Q] = F ` Q for F [, ] The right and left exponential transposes of a bifunctor K D C are denoted by K D [, C] K [D, C], and the (simple) transpose of a functor K [D, C] is denoted by K D [, C] These transpositions form iso functors as indicated in the following commutative diagram: [D, [, C]] [ D, C] [, [D, C]] 2 By the obvious isomorphism [, D] D, a functor (resp natural transformation) D is identied with an object (resp arrow) of D Given categories and D and given an object d D, the product functor d as in d D D d yields the section d D of D at d under the identication 7

8 0 Preliminaries 3 Given categories and C, the evaluation (e, F ) e` F [, C] C is identied with the composition (e, F ) e ` F [, ] [, C] [, C] Given an object e, the precomposition functor [e, C] [, C] C, evaluation at e, takes each F [, C] and yields the composite e F C, ie the evaluation e` F Note that the diagram [D, [, C]] [, [D, C]] commutes; that is, the evaluation [D,[e,C]] [D, C] [e,[d,c]] e K [D, C] of K at e is given by the composition D K [, C] [e,c] C 4 Given categories, D, C, and an object d D, the precomposition functor [ d, C] [ D, C] [, C], partial evaluation at d, takes each K [ D, C] and yields the composite d D K C, the right slice of K at d Note that the diagram [ D, C] [ d,c] [, C] [D, [, C]] [d,[,c]] commutes; that is, the partial evaluation of K D C at d is the same thing as the evaluation d D K [, C] of the right exponential transpose of K at d 8

9 0 Preliminaries 5 Given a category D, the unique functor D is denoted by D or just by Given categories and D, the product functor D as in D D D D yields the projection under the identication D D 6 Given categories D,, and C, the D-ary diagonal functor of the functor category [, C] is the precomposition functor [ D, C] [, C] [ D, C] sending each functor F C to the composite bifunctor D D F C, F duplicated across D s a special case, the D-ary diagonal functor [ D, C] C [D, C] of a category C sends each object c C to the constant functor c D C given by the composition Note that the diagrams D D c C [, C] [ D,C] [ D,[,C]] [ D,C] [ D, C] [D, [, C]] [ D, C] commute [, C] [,[ D,C]] [, [D, C]] 9

10 odules and Cells odules Denition right module over a category, written, is a functor Set Given a pair of right modules, N, a morphism Φ from to N, written Φ N, is a natural transformation Φ N Set left module over a category, written, is a functor Set Given a pair of left modules, N, a morphism Φ from to N, written Φ N, is a natural transformation Φ N Set Remark 2 The notation [, Set] (see Preliminaries(0)) is abbreviated to [ ] : a) for a category, [ ] denotes the category of right modules over ; b) for a functor K, [K ] [ ] [ ] denotes the precomposition functor given by the assignment K ` If is a small category, then the category [ ] is locally small K [K ] is contravariant functorial The notation [, Set] (see Preliminaries(0)) is abbreviated to [ ] : a) for a category, [ ] denotes the category of left modules over ; The assignment b) for a functor K, [ K] [ ] [ ] denotes the precomposition functor given by the assignment K ` If is a small category, then the category [ ] is locally small K [ K] is contravariant functorial 2 By denition, 3 [ ] = [ ] The assignment ; a left module over is the same thing as a right module over the opposite category If is a right module, then the image of an object (resp arrow) x under the functor Set is written (x) or just x For an object x, the set x is called the hom-set at x n element m of a hom-set x or a pair (x, m) such that m x is called an arrow of (or an -arrow) and written m x, x m, or just m if its domain is understood or unimportant 0

11 odules and Cells If is a left module, then the image of an object (resp arrow) a under the functor Set is written (a) or just a For an object a, the set a is called the hom-set at a n element m of a hom-set a or a pair (m, a) such that m a is called an arrow of (or an -arrow) and written m a, m a, or just m if its codomain is understood or unimportant 4 If Φ N is a right module morphism, then the component of the natural transformation Φ N Set at x is written x Φ The image of an - arrow m x under the function x Φ x x N is written m` x Φ = x Φ m or just m` Φ = Φ m If Φ N is a left module morphism, then the component of the natural transformation Φ N Set at a is written Φ a The image of an - arrow m a under the function Φ a a N a is written m` Φ a = Φ a m or just m` Φ = Φ m Proposition 3 right module morphism Φ N is invertible in the category [ ] if and only if every component x Φ is a bijection left module morphism Φ N is invertible in the category [ ] if and only if every component Φ a is a bijection Proof Immediate from the denition on noting that a natural transformation is invertible in a functor category if and only if every its component is invertible Remark 4 right (resp left) module morphism is called iso if it satises the equivalent conditions in Proposition 3 Denition 5 [two-sided] module from a category to a category, written, is a bifunctor Set Given a pair of modules, N, a morphism Φ from to N, written Φ N, is a natural transformation Φ N Set Remark 6 The notation [, Set] (see Preliminaries(0)) is abbreviated to [ ] : a) for a pair of categories and, [ ] denotes the category of modules ; b) for a pair of functors S and T D, [S T] [ ] [ D] denotes the precomposition functor given by the assignment [S T] ` If and are small categories, then the category [ ] is locally small The assignment (S, T) [S T] is contravariant bifunctorial 2 By denition, [ ] = [ ] = [ ] ; a two-sided module is the same thing as a right module (resp left module )

12 odules and Cells 3 If is a module, then the image of an object (resp arrow) (x, a) under the functor Set is written (x) (a) or just x a For a pair of objects x and a, the set x a is called the hom-set from x to a n element m of a hom-set x a or a triple (x, m, a) such that m x a is called an arrow of (or an -arrow) and written m x a, x m a, or just m if its domain and codomain are understood or unimportant 4 If Φ N is a module morphism, then the component of the natural transformation Φ N Set at (x, a) is written x Φ a The image of an -arrow m x a under the function x Φ a x a x N a is written m` x Φ a = x Φ a m or just m` Φ = Φ m 5 The canonical isomorphism yields a canonical isomorphism [ ] [ ] natural in By this isomorphism, a right module over is identied with a two-sided module from to the terminal category The canonical isomorphism yields a canonical isomorphism [ ] [ ] natural in By this isomorphism, a left module over is identied with a two-sided module from the terminal category to 6 There are obvious isomorphisms Set [ ] [ ] [ ], by which a set is identied with a module over the terminal category 7 Given a universe U, a module is called a) locally U-small if and are locally U-small and all hom-sets of are U-small; that is, if all hom-sets of,, and are U-small; b) U-small if it is locally U-small and if and are U-small; c) U-large if it is not locally U-small We just say small (resp large) instead of U-small (resp U-large) for U chosen implicitly Proposition 7 module morphism Φ N is invertible in the category [ ] if and only if every component x Φ a is a bijection Proof See the proof of Proposition 3 Remark 8 module morphism is called iso if it satises the equivalent conditions in Proposition 7 Denition 9 2

13 odules and Cells Let be a right module The composite g ` m = m g of an -arrow g y x and an -arrow m x, as indicated in g y x g ` m m, is the -arrow y dened by g ` m = g m, the image of m under the function g x y Let be a left module The composite m ` f = f m of an -arrow m a and an -arrow f a b, as indicated in m ` f m a f b, is the -arrow b dened by m ` f = m` f, the image of m under the function f a b 2 Let be a module The composite g ` m = m g of an -arrow g y x and an -arrow m x a, as indicated in y, is the -arrow y a dened by g x g ` m m a g ` m = g a m, the image of m under the function g a x a y a The composite m ` f = f m of an -arrow m x a and an -arrow f a b, as indicated in m x a, is the -arrow x b dened by m ` f f b m ` f = m` x f, the image of m under the function x f x a x b 3

14 odules and Cells Remark 0 module thus induces a composition law among the arrows of,, and Conversely, a module may be dened by giving a set of -arrows and a composition of them with -arrows and -arrows satisfying the associativity axiom, ie by giving a collage (see Section 3) from to 2 When a two-sided module is regarded as a right module, an -arrow m x a is written as m (x, a) and the compositions g y x m g ` m a x m ` f m a b f are written as (y, a) (g,a) (x, a) (g,a) ` m m (x, b) (x,f) (x, a) m (x,f) ` m When a two-sided module is regarded as a left module, an -arrow m x a is written as m (x, a) and the compositions g y x m g ` m a x m ` f m a b f are written as m (x, a) m (x, a) m ` (g,a) (g,a) m ` (x,f) (x,f) (y, a) (x, b) Denition The hom of a category C is the endomodule C C C given by the assignment (x, a) hom C (x, a) for x, a C Remark 2 For a pair of objects a, b C, the hom-set hom C (a, b) of a category C is the same thing as the hom-set a C b of the endomodule C Hereafter a hom-set is written as a C b rather than hom C (a, b); likewise, for an object c C and a C-arrow f a b, the functions hom C (c, f) hom C (c, a) hom C (c, b) hom C (f, c) hom C (b, c) hom C (a, c) are written as c C f c C a c C b f C c b C c a C c 4

15 odules and Cells Notation 3 In a diagram, a module and a module morphism Φ N are depicted as 2 Italic capital letters, N, vary over both modules and module morphisms When we write [ ],, or, stands for an object or an arrow of the module category [ ], ie a module or a module morphism (cf Preliminaries(8)) Denition 4 Φ N Given a right module (or module morphism) and a functor (or natural transformation) S as in S, their composite, written [S] or just S, is the right module (or module morphism) dened by the composition S Set Given a left module (or module morphism) and a functor (or natural transformation) T as in T D, their composite, written [T ] or just T, is the left module (or module morphism) D dened by the composition D T Set Remark 5 The composition (S, ) S [, ] [ ] [ ] is functorial in each variable, contravariant in S and covariant in If S is a functor, then S = S ` = ` [S ] for any [ ] The composition (, T ) T [ ] [D, ] [ D] is functorial in each variable, covariant in both and T If T D is a functor, then T = T ` = ` [ T] for any [ ] 5

16 odules and Cells Denition 6 Given a module (or module morphism), a functor (or natural transformation) S, and a second functor (or natural transformation) T, all as in S T D, their composite, written [S] [T ] or just S T, is the module (or module morphism) D dened by the composition Remark 7 The composition D S T Set (S,, T ) S T [, ] [ ] [D, ] [ D] is functorial in each variable, contravariant in S and covariant in and T If S and T D are functors, then (see Remark 6()) for any [ ] 2 s a special case, S T = [S T] ` = ` [S T] given S, their composite S is dened by the composition S Set If is the terminal category, then the composite S coincides with that dened in Denition 4 under the identication [ ] [ ] and [ ] [ ] given T D, their composite T D is dened by the composition D T Set If is the terminal category, then the composite T D coincides with that dened in Denition 4 under the identication [ ] [ ] and [ D] [ D] xample 8 Given a module and functors as in S T D 6

17 odules and Cells, their composite is the module S T D dened by e S T d = (e` S) (T d) for e and d D n S T-arrow m e d is given by an -arrow m e` S T d For an -arrow h e e and an S T-arrow m e d, their composite h ` m e d is given by the -arrow (h` S) ` m e ` S T d as indicated in h e e ` S h` S e e` S m h ` m T d ; similarly, for a D-arrow h d d and an S T-arrow m e d, their composite m ` h e d is given by the -arrow m ` (T h) e` S T d as indicated in e` S m T d d m ` h T h T d d h 2 s a special case of above, given a pair of functors as in S C C C T D, their composite is the module S C T D dened by e S C T d = (e` S) C (T d) for e and d D n S C T-arrow f e d is given by an C-arrow f e` S T d For an -arrow h e e and an S C T-arrow f e d, their composite h ` f e d is given by the C-arrow (h` S) ` f e ` S T d as indicated in h e e e ` S h` S e` S h ` f f T d ; similarly, for a D-arrow h d d and an S C T-arrow f e d, their composite f ` h e d is given by the C-arrow f ` (T h) e` S T d as indicated in e` S f T d d f ` h T h T d d h 3 Given a module morphism and functors as in S Φ D N, their composite is the module morphism S Φ T S T S N T D dened by e S Φ T d = (e` S) Φ (T d) for e and d D T 7

18 odules and Cells 4 Given a natural transformation, a module, and a functor as in S σ D S T, their composite is the module morphism σ T S T S T D which maps each S T-arrow m e d to the S T-arrow σ e ` m e d as indicated in e` S σ e e` S m` σ T m T d Similarly, given S τ D, their composite is the module morphism S τ S T S T D which maps each S T-arrow m e d to the S T -arrow m ` τ d e d as indicated in T T e` S m T d m` S τ τ d T d 5 The evaluation x a of a module (or module morphism) at (x, a) is identied with the composition x a For a pair of objects x and a, the precomposition functor [x a] [ ] Set, evaluation at (x, a), sends each module to the hom-set x a and sends each module morphism Φ N to its component x Φ a x a x N a at (x, a) 6 The partial evaluation of a module Y B at (x, a) is given by the composition Y x Y Y B a B B, yielding the slice of at (x, a), ie the module [x Y] [a B] Y B such that y [x Y] [a B] b = (x, y) (a, b) for y Y and b B The precomposition functor [x Y a B] [ Y B] [Y B], partial evaluation at (x, a), sends each module Y B to its slice at (x, a) 8

19 odules and Cells 7 Given a right module and a category, the composition yields the two-sided module by duplicating across The -ary diagonal functor of [ ] is the precomposition functor [ ] [ ] [ ] sending each right module to the composite module s a special case, the -ary diagonal functor [ ] Set [ ] of Set sends each set S to the constant left module S given by the composition S Given a left module and a category, the composition yields the two-sided module by duplicating across The -ary diagonal functor of [ ] is the precomposition functor [ ] [ ] [ ] sending each left module to the composite module s a special case, the -ary diagonal functor [ ] Set [ ] of Set sends each set S to the constant right module S given by the composition 8 The join of categories and is the module given by the composition S, where is the hom of the terminal category For every pair of objects x and a, the hom-set x a consists of a single arrow, the identity s a special case, the join is given by the composition ; dually, the join is given by the composition 9

20 odules and Cells Proposition 9 If Φ in xample 8(3) is a module isomorphism, so is the composite S Φ T Proof Note that S Φ T is given by the image of Φ under the precomposition functor [S T] Since any functor preserves isomorphisms, S Φ T is an isomorphism Proposition 20 Given, the associative law holds S S T D D S S T T = [S ` S] [T T ] T 2 Given, the associative law holds Proof Indeed, S T D S T = S T = S T S S T T = [S T ] ` [[S T ] ` ] = [[S T ] ` [S T ]] ` = [[S ` S] [T ` T ]] ` = [S ` S] [T T ] 2 By what we have just seen, S T = S [ ] T = [S ` ] T = S T and S T = S [ ] T = S [ T ] = S T Denition 2 Given a module, the opposite module is dened by the composition Set, where denotes the simple transposition (a, x) (x, a); similarly, given a module morphisms Φ N, the opposite module morphism Φ N is dened by the composition Φ Set N 20

21 odules and Cells Remark 22 The iso functor [ ] [ ] is given by the assignment and Φ Φ 2 For any module and any module morphism Φ, = Φ = Φ 3 For any category C, C = C ; that is, the hom of the opposite category is the opposite module of the hom 4 For any composite module S T, S T = T S ; that is, the opposite of the composite S T D is given by the composite D T S 5 The opposite of the module morphisms Φ N is often denoted by Φ N using the same name as its original counterpart (cf Preliminaries(6)) 6 The opposite of a right module is the left left module ; in fact, and are dened by the same functor 2 Cells = Set Denition 2 Let and N Y B be modules Given a pair of functors P Y and Q B, a module cell (or just a cell) Φ P Q N, depicted as P Φ Y N B Q, is dened by a module morphism Φ P N Q Remark 22 2

22 odules and Cells cell Φ P Q N sends each -arrow m x a to the N -arrow m` Φ x` P Q a, the image of m under the function x a x Φ a x P N Q a = (x` P) N (Q a) 2 Cells and module morphisms are regarded as special instances of each other cell Φ P Q N is thought of as a module morphism from to the composite module P N Q Conversely, a module morphism Φ N is depicted as Φ N 3 The identity module morphism yields the identity cell 4 ny composition trivially yields a cell P Y N Q B P N Q P Y N B Q ; a cell expresses an identity = P N Q Denition 23 P Y N Let and N Y be right modules Given a functor P Y, a right module cell Φ P N, depicted as B P Φ Y N, is dened by a right module morphism Φ P N Q 22

23 odules and Cells Let and N B be left modules Given a functor Q B, a left module cell Φ Q N, depicted as Φ N B Q, is dened by a left module morphism Φ N Q Remark 24 right (resp left) module cell in Denition 23 is regarded as a special instance of a two-sided module cell in Denition 2 where and B (resp and Y) are the terminal category under the identication in Remark 6(5) Conversely, by Remark 6(2), a two-sided module cell in Denition 2 is the same thing as a right (resp left) module cell depicted below: P Q Y B Φ N Φ N P Q Y B Denition 25 Let be a right module and N Y B be a two-sided module Given a functor P Y and an object b B, a right conical cell Φ P b N, depicted as P Φ Y N B b, is dened by a right module morphism Φ P N b Let be a left module and N Y B be a two-sided module Given an object y Y and a functor Q B, a left conical cell Φ y Q N, depicted as y Φ Y N B Q, is dened by a left module morphism Φ y N Q Remark 26 conical cell in Denition 25 is regarded as a special instance of a two-sided module cell in Denition 2 where (resp ) is the terminal category under the identication in Remark 6(5) 2 right (resp left) module cell in Denition 23 is regarded as a special instance of a right (resp left) conical cell in Denition 25 where B (resp Y) is the terminal category 23

24 σ odules and Cells Conversely, a right (resp left) conical cell in Denition 25 is the same thing as a right (resp left) module cell depicted below P Φ Y N b Φ y N B Q Denition 27 Given a pair of modules J D and, the module of cells J, J, [, ] [D, ], is dened by (S) J, (T ) = (J ) D (S T ) for S [, ] and T [D, ], where D is the hom of the module category [ D] Remark 28 For a pair of functors S and T D, the set (S) J, (T) consists of all cells S T J ; 2 If Θ S T J is a cell and σ S S is a natural transformation as in J D S S Θ T, then their composite is the cell J D S σ ` Θ dened by the module morphism σ ` Θ J S T given by the composition T J Θ S T σ T S T By xample 8(4), the cell σ ` Θ sends each J -arrow j e d to the -arrow j` σ ` Θ = σ e ` (j` Θ) e` S T d as indicated in e` S σ e e` S j` σ ` Θ j` Θ T d 24

25 odules and Cells If Θ S T J is a cell and τ T T D is a natural transformation as in J D S Θ T τ T, then their composite is the cell J D S Θ ` τ T dened by the module morphism Θ ` τ J S T given by the composition J Θ S T S τ S T By xample 8(4), the cell Θ ` τ sends each J -arrow j e d to the -arrow j` Θ ` τ = (j` Θ) ` τ d e` S T d as indicated in e` S j` Θ T d j` Θ ` τ τ d T d 3 If J is small and is locally small, then the module J, is locally small Proposition 29 Given a module J and a composite module P N Q as in P Y J P N Q N D B Q, the identity [, ] J,P N Q [D, ], ie, holds [,P] [, Y] J,N [D,Q] [D, B] J, P N Q = [, P] J, N [D, Q] 25

26 odules and Cells Proof For any S [, ] and T [D, ], S J, P N Q T = (J ) D (S P N Q T ) = (J ) D ([S ` P] N [Q T ]) = (S ` P) J, N (Q T ) = (S` [, P]) J, N ([D, Q] T ) = (S) [, P] J, N [D, Q] (T ) Denition 20 If Θ S T J is a cell and Φ N is a module morphism as in J D S Θ T Φ N, then their composite Θ ` Φ = Φ Θ is the cell J D S Θ ` Φ N dened by the module morphism Θ ` Φ J S N T given by the composition T J Θ S T S Φ T S N T Remark 2 See xample 8(3) for the module morphism S Φ T The cell Θ` Φ sends each J -arrow j e d to the N -arrow, the image of j under the composite function j` Θ ` Φ = j` Θ` Φ e` S T d e J d e Θ d e S T d = (e` S) (T d) (e` S) Φ (T d) (e` S) N (T d) Denition 22 Given a module J D and a module morphism Φ N, the module morphism, postcomposition with Φ, is dened by J, Φ J, J, N [, ] [D, ] (S) J, Φ (T) = (J ) D (S Φ T) for each pair of functors S and T D 26

27 odules and Cells Remark 23 The module morphism J, Φ maps each cell Θ S T J to the cell Θ ` Φ S T J N dened in Denition 20 2 The assignment Φ J, Φ is functorial; indeed the functor is dened by for S [, ], T [D, ], and [ ] J, [ ] [[, ] [D, ]] S J, T = (J ) D (S T ) Note By Remark 22(2), the following denition is regarded as a special case of Denition 20 and vice versa Denition 24 Given a pair of cells as in S J Θ D T, their composite Θ ` Φ = Φ Θ is the cell P Φ Y N B Q J D S ` P Θ ` Φ Y N dened by the module morphism Θ ` Φ J [S ` P] N [Q T] given by the composition B Q T J Θ S T S Φ T S P N Q T = [S ` P] N [Q T] Remark 25 The cell Θ ` Φ sends each J -arrow j e d to the N -arrow, the image of j under the composite function j` Θ ` Φ = j` Θ` Φ e` S` P Q T d e J d e Θ d e S T d = (e` S) (T d) (e` S) Φ (T d) (e` S) P N Q (T d) = (e` S` P) N (Q T d) 27

28 odules and Cells Proposition 26 odules and cells among them form a category with the composition given in Denition 24 and the identities given in Remark 22(3) Proof The only non-trivial part is the verication of the associativity of the composition Consider cells as in P Φ Q P Q Φ P Φ Q The composites (Φ ` Φ ) ` Φ and Φ ` (Φ ` Φ ) are dened by the module morphisms Φ ` P Φ Q ` [P ` P ] Φ [Q Q] and Φ ` P Φ ` P Φ Q Q respectively But, by the functoriality (see Remark 7()) and associativity (see Proposition 20) of the composition, we have Remark 27 Φ ` P Φ Q ` [P ` P ] Φ [Q Q] = Φ ` P Φ Q ` P P Φ Q Q = Φ ` P Φ ` P Φ Q Q Given a universe U, OD U denotes the category consisting of all locally U-small modules and all cells among them; OD U is U-large By OD we mean OD U for U chosen implicitly 2 Given a pair of categories and, there is a canonical embedding [ ] OD, identical on objects, given by the obvious arrow function (see Remark 22(2)) The embedding is not, in general, full Proposition 28 cell Φ P Q N is invertible in the category OD if and only if the functors P and Q are iso and the module morphism Φ P N Q is iso Proof If Φ P Q N has an inverse Ψ S T N, then P (resp Q) and S (resp T) are inverse to each other and Φ P N Q and P Ψ Q P N Q are inverse to each other Conversely, if P, Q, and Φ P N Q are iso, the inverse of Φ P Q N is given by the cell Ψ S T N with S = P, T = Q, and the module morphism Ψ N S T dened by Ψ = [P ] Φ [Q ] Denition 29 cell Φ P Q N is called iso if it satises the equivalent conditions in Proposition 28; 2 fully faithful if the module morphism Φ P N Q is iso Remark 220 In Proposition 227 we will see the relation between the notion of fully faithfulness for cells and that for functors 28

29 odules and Cells Note Proposition 29 allows the following denition Denition 22 Given a module J D and a cell P Φ Y N B Q, the cell [, ] J, [D, ] [,P] [, Y] J,Φ J,N [D,Q] [D, B], postcomposition with Φ, is dened by the postcomposition module morphism J, J,Φ J, P N Q = [, P] J, N [D, Q] with Φ P N Q Remark 222 The cell J, Φ sends each cell Θ S T J to the cell Θ ` Φ S ` P Q T J N dened in Denition 24 Proposition 223 The assignment Φ J, Φ is functorial Proof Clearly, the assignment Φ J, Φ preserves the identities To verify that it preserves the composition, let Φ and Ψ be a composable pair of cells and consider the cells J, Φ, J, Ψ, and J, Φ ` Ψ depicted in [, ] J, [D, ] P Φ Q Y N B [,P] [, Y] J,Φ J,N [D,Q] [D, B] P Z Ψ L Q [,P ] C [, Z] J,Ψ J,L [D,Q ] [D, C] [, ] J, [D, ] P ` P Z Φ ` Ψ L Q Q [,P ` P ] C [, Z] J,Φ ` Ψ J,L [D, C] [D,Q Q] We need to verify that the composition of the cells J, Φ and J, Ψ yields the cell J, Φ ` Ψ First note that [, P ` P ] = [, P] ` [, P ] and [D, Q Q] = [D, Q ] [D, Q] by the functoriality of the operations [, ] and [D, ] The cell J, Φ ` J, Ψ is dened by the module morphism J, Φ ` [, P] J, Ψ [D, Q] and the cell J, Φ ` Ψ is dened by the module morphism J, Φ ` P Ψ Q But, by Remark 23(2) and Proposition 29, J, Φ ` P Ψ Q = J, Φ ` J, P Ψ Q = J, Φ ` [, P] J, Ψ [D, Q] 29

30 odules and Cells Remark 224 Given a small module J D, the functor J, OD OD is dened by the object function J, and the arrow function Φ J, Φ, extending the functor J, in Remark 23(2) as shown in [ ] OD J, J, [[, ] [D, ]] OD, where denotes the canonical embedding in Remark 27(2) Denition 225 The hom of a functor H C B is the cell C C C H H B B B H given by the arrow function of H; that is, for each pair of objects x, a C, the function a H b a C b (a` H) B (H b) is given by Remark 226 H a,b hom C (a, b) hom B (H a, H b), f H f The naturality of the module morphism H C H B H C C follows from the functoriality of H 2 Hereafter, given a functor H, each component of the arrow function of H is written as a H b Proposition 227 functor H C B is iso (resp fully faithful) if and only if the hom cell H H H C B is iso (resp fully faithful) (see Denition 29) Proof Immediate from the denitions Theorem 228 The hom operation dened in Denition and Denition 225 embeds CT in OD Specically, the assignment C C forms a faithful functor CT OD, injective on objects Proof The verication of the functoriality is straightforward The faithfulness and the injectiveness on objects are evident 30

31 odules and Cells Denition 229 Given a cell and commutative quadrangles of functors as in R F P Y S P Y Φ N Q B B G Q, their pasting composite R F P R Φ F Y S N G B Q is dened by module morphism given by the composition R Φ F R F R P N Q F = P S N G Q R Φ P N Q F Remark 230 pasting composition R F S P Y Φ N Q B G with both ends being identities, yields a cell R F R Φ F S N G, ie a module morphism R Φ F R F S N G Proposition 23 In Denition 229, if Φ is fully faithful, so is the cell R Φ F Proof Immediate from Proposition 9 Proposition 232 Cells and commutative quadrangles of functors as in R F P P Φ Y S Y N B Q G B Q K Z T K Z Ψ L L C C H L 3

32 odules and Cells yield the same cell R Φ ` Ψ F = R Φ F ` S Ψ G P ` K L Q R F T L H irrespective of the order of the horizontal and vertical compositions Proof The horizontal composition followed by the vertical composition yields the cell R Φ F` S Ψ G, and the vertical composition followed by the horizontal composition yields the cell R Φ ` Ψ F, as shown in R F P R Φ F Q Y S N G B R F B K Z S Ψ G T L H L P ` K C Z T P ` K Z Φ ` Ψ L L Q C C H L Q R F R F P ` K R Φ F ` S Ψ G Z T L H L Q P ` K R Φ ` Ψ F C Z T L H C L Q The cell R Φ F` S Ψ G is dened by the module morphism R Φ F` P S Ψ G Q and the cell R Φ ` Ψ F is dened by the module morphism R Φ ` P Ψ Q F But 3 Cell morphisms Denition 3 Given a pair of cells R Φ ` P Ψ Q F = R Φ F ` R P Ψ Q F = R Φ F ` [R ` P] Ψ [Q F] = R Φ F ` [P ` S] Ψ [G Q ] = R Φ F ` P S Ψ G Q P Φ Y N Q S Ψ B Y N B, a morphism from Φ to Ψ, written τ Φ Ψ N, is a pair (τ, τ 2 ) of natural transformations τ P S Y τ 2 Q T B such that the quadrangle x` P τ x commutes for every -arrow m x a m` Φ Q a τ 2 a x` S m` Ψ T a T 32

33 odules and Cells Remark 32 The identity morphism of a cell Φ P Q N is given by the pair of the identity natural transformations; that is, Φ = ( P, Q ) 2 The composition of two cell morphisms τ Φ Ψ N and σ Ψ Ω N is given componentwise; that is τ ` σ = (τ ` σ, τ 2 ` σ 2 ) Proposition 33 Given a pair of modules and N, all cells N and morphisms among them dene the cell category [ N ] with the identities and the composition dened in Remark 32 Proof Self explanatory Remark 34 If is small and N is locally small, then the category [ N ] is locally small Proposition 35 cell morphism τ Φ Υ N is invertible in the category [ N ] if and only if τ and τ 2 are natural isomorphisms Proof Immediate from the denitions Remark 36 cell morphism is called iso if it satises the equivalent conditions in Proposition 35 Denition 37 The hom of a natural transformation τ F G C B is the cell morphism τ F G C B, given by the pair (τ, τ) Theorem 38 Given a pair of categories C and B, the hom operation dened in Definition 225 and Denition 37 embeds the functor category [C, B] in the cell category [ C B ] Specically, the assignment H H forms a faithful functor [C, B] [ C B ], injective on objects Proof The verication of the functoriality is straightforward The faithfulness and the injectiveness on objects are evident 33

34 2 Slicing and ction 2 Slicing of modules Denition 2 Let and be categories The right exponential transposition [, Set] (see Preliminaries()) is denoted by [ ] [, [, Set]] [, [ ]] The right exponential transpose of a module is a covariant functor [ ] [ ] and the right exponential transpose of a module morphism Φ N is a natural transformation [Φ ] [ ] [N ] [ ] The evaluation of the functor at a, ie the partial evaluation of the module at a, is written (a) or just a Similarly, the evaluation of the natural transformation Φ at a, ie the partial evaluation of the module morphism Φ at a, is written Φ a The left exponential transposition [, Set] (see Preliminaries()) is denoted by [ ] [, [, Set]] [, [ ]] or [ ] [, [ ] ] The left exponential transpose of a module is a contravariant functor [ ] [ ] 34

35 2 Slicing and ction and the left exponential transpose of a module morphism Φ N is a natural transformation [ Φ] [ N ] [ ] [ ] The evaluation of the functor at x, ie the partial evaluation of the module at x, is written (x) or just x Similarly, the evaluation of the natural transformation Φ at x, ie the partial evaluation of the module morphism Φ at x, is written x Φ Remark 22 The partial evaluation of a module at a yields the right slice of at a, ie the right module a given by for x x a = x a The partial evaluation of a module at x yields the left slice of at x, ie the left module x given by for a x a = x a 2 The partial evaluation of at an -arrow f s t yields the right module morphism f s t which maps each -arrow m x s to the -arrow m ` f x t as indicated in x m s m` f The partial evaluation of at an -arrow f s t yields the left module morphism f t f t s which maps each -arrow m t a to the -arrow f ` m s a as indicated in f s t f m m a 3 35

36 2 Slicing and ction The partial evaluation of a module morphism Φ N at a yields the right slice of Φ at a, ie the right module morphism Φ a a N a given by for each x x Φ a = x Φ a The partial evaluation of a module morphism Φ N at x yields the left slice of Φ at x, ie the left module morphism x Φ x x N given by for each a x Φ a = x Φ a 4 The partial evaluation a of a module at a is identied with the composition a (cf xample 8(5)) The partial evaluation x of a module at x is identied with the composition x (cf xample 8(5)) 5 For any module, [ ] = [ ] ; that is, the opposite of the right exponential transpose of is the left exponential transpose of the opposite module xample 23 The diagonal functors in xample 8(7) and the exponential transpositions form the following commutative diagrams (cf Preliminaries(6)): [ ] [ ] [ ] [,[ ]] [ ] [ ] [, [ ]] [ ] [,[ ] ] [, [ ] ] [ ] = [, Set] [ ] = [, Set] [ ] [,[ ]] [ ] [,[ ]] [ ] [, [ ]] [ ] [, [ ]] 36

37 2 Slicing and ction Proposition 24 For any module morphism Φ N, the following conditions are equivalent: Φ is iso; 2 the right module morphism Φ a a N a is iso for every a ; 3 the left module morphism x Φ x x N is iso for every x Proof Since x Φ a = x Φ a = x Φ a for any x and a, the assertion is immediate on recalling Remark 4 and Remark 8 Proposition 25 Given a module (or module morphism) and a functor (or natural transformation) K as in K, the right exponential transpose of the composite K is given by the composition ; that is, [ ] K [ K ] = [ ] K Given a module (or module morphism) and a functor (or natural transformation) K as in K, the left exponential transpose of the composite K is given by the composition K [ ] ; that is, [ K ] = K ` [ ] Proof For any e, [ K ] e = K e = (K e) = [ ] (K e) = [[ ] K] e Proposition 26 Given a functor K and a module as in K, the right exponential transpose of the composite module K is given by the composition ; that is, [K ] [ ] [ ] [ K ] = [K ] [ ] 37

38 2 Slicing and ction Given a functor K and a module as in K, the left exponential transpose of the composite module K is given by the composition [ ] [ K] [ ] ; that is, Proof For any a, [ K ] = [ ] ` [ K] [ K ] a = K a = K a = [K ] a = [K ] ([ ] a) = [[K ] [ ]] a Denition 27 Given a cell P Φ Y N B Q the right slice of Φ at a is the right conical cell a P Φ a Y N dened by the right slice of the module morphism Φ P N Q at a, ie the right module Φ a a P N Q a B Q a the left slice of Φ at x is the left conical cell x x` P x Φ Y N dened by the left slice of the module morphism Φ P N Q at x, ie the left module x Φ x x P N Q Proposition 28 cell is fully faithful if and only if every its right (resp left) slice is fully faithful Proof Immediate from Proposition 24 B Q 38

39 22 ction of modules 2 Slicing and ction Denition 22 Let be a category and be a module The right action of on the functor category [, ] is the covariant functor [ ] [, ] [ ] dened by for K [, ] [ ] K = K the left action of on the functor category [, ] is the contravariant functor [ ] [, ] [ ] dened by for K [, ] K ` [ ] = K Remark 222 module acts on a functor K by the composition K and yields the module K such that x K e = x (K e) for x and e module acts on a functor K by the composition K and yields the module such that for e and a K e K a = (e` K) a 2 39

40 2 Slicing and ction module acts on a natural transformation τ S T by the composition and yields the module morphism τ S T τ S T which maps each S-arrow m x e to the T-arrow m ` τ e x e as indicated in x m S e m` τ module acts on a natural transformation τ S T by the composition and yields the module morphism S T τ e T e τ τ T S which maps each T -arrow m e a to the S -arrow τ e ` m e a as indicated in e` S 3 By Remark 22(4), τ e τ m e` T m a the right exponential transpose [ ] [ ] of a module is identied with the right action [ ] [, ] [ ] of on the functor category [, ] the left exponential transpose [ ] [ ] of a module is identied with the left action [ ] [, ] [ ] of on the functor category [, ] 40

41 2 Slicing and ction 4 For any category and any module, [ ] [ ] ; that is, the opposite of the right action of on [, ] is (identied with) the left action of the opposite module on [, ] Theorem 223 Given a category and a module, the diagrams commute [, ] [, ] [, ] [ ] [, [ ]] [ ] Proof Immediate from Proposition 25 [, ] [, [ ] ] Corollary 224 Given a category and a module, the quadrangles [,] [, ] [, ] [,] [ ] commute Proof The diagrams [ ] [ ] [ ] [ ] [ ] [,] [, ] [, ] [ ] [,[ ]] [, ] [, [ ]] [ ] [, ] [, ] [,] [, ] [ ] [, [ ] ] [,[ ] ] [ ] commute by Theorem 223, and yield the desired commutative quadrangles by xample Yoneda functors and representations Denition 23 The right Yoneda functor for a category is the functor [ ] [ ] given by the right exponential transpose of the hom ; in short: [ ] = [ ] 4

42 2 Slicing and ction The left Yoneda functor for a category is the functor [ ] [ ] given by the left exponential transpose of the hom ; in short: [ ] = [ ] Remark 232 The right Yoneda functor sends each object x to the right module x, called the representable right module of x, and sends each -arrow f s t to the right module morphism f s t which maps each -arrow h x s to the -arrow h ` f x t as indicated in x h s h` f f t (cf Remark 22(2)) The left Yoneda functor sends each object a to the left module a, called the representable left module of a, and sends each -arrow f s t to the left module morphism f t s which maps each -arrow h t a to the -arrow f ` h s a as indicated in f s t f h h a (cf Remark 22(2)) Denition 233 representation of a right module is a pair (r, Υ) consisting of an object r and a right module isomorphism Υ r representation of a left module is a pair (r, Υ) consisting of an object r and a left module isomorphism Υ r Remark

43 2 Slicing and ction right module is called representable if it has a representation; that is, if it is isomorphic to the representable right module r for some object r, which is said to represent left module is called representable if it has a representation; that is, if it is isomorphic to the representable left module r for some object r, which is said to represent Denition 235 Let and be categories The right general Yoneda functor for the functor category [, ] is the functor [ ] [, ] [ ] given by the right action of the hom on the functor category [, ]; in short: [ ] = [ ] The left general Yoneda functor for the functor category [, ] is the functor [ ] [, ] [ ] given by the left action of the hom on the functor category [, ]; in short: [ ] = [ ] Remark 236 The right general Yoneda functor sends each functor G to the module G, called the corepresentable module of G, and sends each natural transformation τ S T to the module morphism τ S T which maps each S-arrow h x a to the T-arrow h ` τ a x a as indicated in x h S a h` τ τ a T a (cf Remark 222(2)) The left general Yoneda functor sends each functor F to the module F, called the representable module of F, and sends each natural transformation τ S T to the module morphism τ T S 43

44 2 Slicing and ction which maps each T -arrow h x a to the S -arrow τ x ` h x a as indicated in (cf Remark 222(2)) x` S τ x τ h x` T h a 2 The right Yoneda functor [ ] [ ] for a category is identied with the right general Yoneda functor [ ] [, ] [ ] for the functor category [, ] The left Yoneda functor [ ] [ ] for a category is identied with the left general Yoneda functor for the functor category [, ] [ ] [, ] [ ] Proposition 237 Given categories,, and, the quadrangles [,] [, ] [, ] [,] [ ] [ ] [ ] [ ] [ ] [ ] commute Proof This is a special case of Corollary 224 where is given by the hom of (resp ) Denition 238 Let be a module corepresentation of is a pair (R, Υ) consisting of a functor R and a module isomorphism Υ R representation of is a pair (R, Υ) consisting of a functor R and a module isomorphism Υ R Remark 239 module is called corepresentable if it has a corepresentation; that is, if it is isomorphic to the corepresentable module R for some functor R, which is said to corepresent 44

45 2 Slicing and ction 2 3 representable if it has a representation; that is, if it is isomorphic to the representable module R for some functor R, which is said to represent representation of a right module is identied with a corepresentation of a two-sided module where is the terminal category representation of a left module is identied with a representation of a two-sided module where is the terminal category corepresentation of a module is expressed by a fully faithful cell Υ representation of a module is expressed by a fully faithful cell R R Υ 4 xample 230 shows that not all modules are representable xample 230 Let and be discrete categories correspondence R from to, ie a subset of, denes a module R by { } x R a = if (x, a) R otherwise R is representable (resp corepresentable) if and only if R is a function (resp ) Proposition 23 Let be a module functor R and a module morphism Υ R form a corepresentation of if and only if for every a the object R a and the right module morphism Υ a a (R a) form a representation of the right module a functor R and a module morphism Υ R form a representation of if and only if for every x the object x` R and the left module morphism x Υ x (x` R) form a representation of the left module x Proof By Proposition 24, Υ R is iso i is iso for every a Υ a a R a = (R a) 45

46 3 Collages and Commas 3 Collages Denition 3 collage from a category to a category, written, is dened by a category, collage category, satisfying the following conditions: a) the coproduct category + is a full subcategory of and = + ; b) a x = if x and a Remark 32 The inclusion of the coproduct category + into the collage category is denoted by + or by 2 collage from a category to the terminal category is called a right collage over ; the inclusion of into the collage category is denoted by collage from the terminal category to a category is called a left collage over ; the inclusion of into the collage category is denoted by 3 collage is called locally small if the collage category is locally small 4 ny collage denes the unique functor 2 making the diagram 0 2 commute Conversely, any functor H C 2 denes a unique collage with = C and = H 46

47 3 Collages and Commas Denition 33 Given a pair of collages, N, a morphism Φ from to N, written Φ N, is dened by a functor Φ N, collage functor, satisfying the following equivalent conditions: Φ is identity on + ; 2 the triangle + Φ N N commutes; 3 the two triangles N Φ N N commute; 4 the triangle Φ N commutes 2 N Proposition 34 Given a pair of categories and, all locally small collages and morphisms among them dene the category [ ] with the obvious identities and the composition Indeed, [ ] is fully embedded into the coslice category under + Proof Self explanatory Denition 35 Let and N Y B be collages Given a pair of functors P Y and Q B, a collage cell Φ P Q N is dened by a functor Φ N, collage functor, satisfying the following equivalent conditions: Φ is identity on + ; 2 the diagram commutes; P Φ Y NY N B N B Q 47