Category Theory. Travis Dirle. December 12, 2017

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1 Category Theory

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3 Category Theory Travis Dirle December 12, 2017

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5 Contents 1 Categories 1 2 Construction on Categories 7 3 Universals and Limits 11 4 Adjoints 23 5 Limits 31 6 Generators and Projectives 37 7 Categories of Fractions 43 8 Flat Functors and Cauchy Completeness 47 9 Bicategories and Distributors Internal Category Theory Abelian Categories Regular Categoriies Algebraic Theories Monads and Algebras Enriched Category Theory Fibred Categories Locales Sheaves 111 i

6 CONTENTS 19 Grothendieck Toposes The Classifying Topos Elementary Toposes Internal Logic of a Topos 139 ii

7 Chapter 1 Categories Definition A universe is a set U with the following properties: i) x y and y U x U, ii) I U and i I x i U i I x i U, iii) x U P(x) U, iv) x U and f : x y surjective function y U, v) N U, where N denotes the set of finite ordinals. Proposition i) x U and y x y U, ii) x U and y U {x, y} U, iii) x U and y U x y U, iv) x U and y U x y U. Definition A category A consists of: a collection ob(a ) of objects; for each A, B ob(a ), a collection A (A, B) of maps/arrows/morphisms from A to B for each A, B, C ob(a ), a function A (B, C) A (A, B) A (A, C) (g, f) g f, called composition; for each A ob(a ), an element 1 A of A (A, A), called the identity on A, satisfying the following axioms: associativity: for each f A (A, B), g A (B, C) and h A (C, D), we have (h g) f = h (g f); identity laws: for each f A (A, B), we have f 1 A = f = 1 B f. 1

8 CHAPTER 1. CATEGORIES Definition If f A (A, B), we call A the domain and B the codomain of f. Every map in every category has a definite domain and codomain. Definition A map f : A B in a category A is an isomorphism if there exists a map g : B A in A such that gf = 1 A and fg = 1 B. We call g the inverse of f and write g = f 1 which is unique. The objects of a category need not be remotely like sets and also, the maps in a category need not be remotely like functions. Definition A category that has no maps, except for the identities, is called a discrete category. Definition Let A and B be categories. A functor F : A B consists of: a function ob(a ) ob(b), written as A F (A); for each A, A A, a function A (A, A ) B(F (A), F (A )), written as f F (f), satisfying the following axioms: F (f f) = F (f ) F (f) whenever A f A f A ; F (1 A ) = 1 F (A) whenever A A. Definition A functor F : A B is faithful (respectively, full) if for each A, A A, the function is injective (respectively, surjective). 2 A (A, A ) B(F (A), F (A )) f F (f)

9 CHAPTER 1. CATEGORIES Note the roles of A and A in the definition. Faithfulness does not say that if f 1 and f 2 are distinct maps in A then F (f 1 ) F (f 2 ). F is faithful if for each A, A and g : F (A) F (A ), there is at most one map from A to A that F sends to g. It is full if for each such A, A and g, there is at least one map from A to A that F sends to g. Definition Let A be a category. A subcategory I of A consists of a subclass ob(i ) of ob(a ) together with, for each S, S ob(i ), a subclass I (S, S ) of A (S, S ), such that I is closed under composition and identities. It is a full subcategory if I (S, S ) = A (S, S ) for all S, S ob(i ). A full category therefore consists of a selection of the objects, with all of the maps between them. So a full subcategory can be specified simply by saying what its objects are. Whenever I is a subcategory of A, there is an inclusion functor. It is automatically faithful, and it is full iff I is a full subcategory. Note that the image of a functor need not be a subcategory. Definition Let A and B be categories and let F, ( G : A B be) functors. A natural transformation α : F G is a family F (A) α A G(A) of maps in B such that for every map A f A in A, the square F (A) F (f) F (A ) α A α A A A G(A) G(f) G(A ) commutes. The maps α A are called the components of α. Definition For any two categories A and B, there is a category whose objects are the functors from A to B and whose maps are the natural transformations between them. This is called the functor category from A to B, and written as [A, B] or B A. Definition A natural isomorphism between functors from A to B is an isomorphism in [A, B]. 3

10 CHAPTER 1. CATEGORIES Lemma Let α : F G be a natural transformation between functors F, G : A B. Then α is a natural isomorphism if and only if α A : F (A) G(A) is an isomorphism for all A A. Definition Given functors F, G : A B, we say that they are naturally isomorphic if there exists a natural isomorphism from F to G. Also, if F and G are naturally isomorphic. F (A) = G(A) naturally in A This terminology can be understood as follows. If F (A) = G(A) naturally in A then certainly F (A) = G(A) for each individual A, but more is true: we can choose isomorphisms α A : F (A) G(A) in such a way that the naturality axiom is satisfied, i.e. the above commutative diagram. There are many examples of categories and functors such that F (A) = G(A) for all A A, but not naturally in A. Definition An equivalence between categories A and B consists of a pair F : A B and G : B A of functors together with a pair of natural isomorphisms η : 1 A G F ɛ : F G 1 B. If there exists an equivalence between A and B, we say that A and B are equivalent, and write A B. We also say that the functors F and G are equivalences. Definition A functor F : A B is essentially surjective on objects if for all B B, there exists A A such that F (A) = B. Proposition A functor is an equivalence if and only if it is full, faithful and essentially surjective on objects. 4

11 CHAPTER 1. CATEGORIES Corollary Let F : C D be a full and faithful functor. Then C is equivalent to the full subcategory C of D whose objects are those of the form F (C) for some C C. Definition A category A is small if the collection of all maps in A is small, i.e. if it is a set, otherwise it is called large. We call A locally small if for each A, B A, the collection A (A, B) is small. So small implies locally small. A category is small if and only if it is locally small and its class of objects is small. It is essentially small if it is equivalent to some small category. 5

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13 Chapter 2 Construction on Categories Definition Every category A has an opposite/dual category A op, defined by reversing the arrows. Formally, ob(a op ) = ob(a ) and A op (B, A) = A (A, B) for all objects A and B. Identities in A op are the same as in A. Composition in A op is the same as in A, but with the arguments reversed. Definition Given categories A and B, there is a product category A B, in which ob(a B) = ob(a ) ob(b), (A B)((A, B), (A, B )) = A (A, A ) B(B, B ). Put another way, an object is a pair (A, B) and a map (A, B) (A, B ) is a pair (f, g) where f : A A in A and g : B B in B. Definition Let A and B be categories. A contravariant functor from A to B is a functor A op B. An ordinary functor A B is sometimes called a covariant functor from A to B, for emphasis. Functors C D correspond one-to-one with functors C op D op, and (A op ) op = A, so a contravariant functor from A to B can also be described as a functor A B op. Definition An equivalence of the form A op B is sometimes called a duality between A and B. One says that they are dual. 7

14 CHAPTER 2. CONSTRUCTION ON CATEGORIES Definition Given categories and functors B A P C the comma category written as (P Q) or (P Q), is the category defined as follows: objects are triples (A, h, B) with A A, B B, and h : P (A) Q(B) in C ; maps (A, h, B) (A, h, B ) are pairs (f : A A, g : B B ) of maps such that the square commutes. Q P (A) P (f) P (A ) h h Q(B) Q(g) Q(B ) Definition The slice category of A over A, denoted by A /A, is the category whose objects are maps into A and whose maps are commutative triangles. More precisely, an object is a pair (X, h) with X A and h : X A in A, and a map (X, h) (X, h ) in A /A is a map f : X X in A making the triangle X f X h commute. Slice categories are a special case of comma categories. Dually, there is a coslice category A/A = (A 1 A ), whose objects are the maps out of A. A h Definition A congruence on a category A is an equivalence relation on arrows such that i) f g implies dom(f) = dom(g) and cod(f) = cod(g) ii) f g implies bfa bga for all arrows a : A X and b : Y B, where dom(f) = X = dom(g) and cod(f) = Y = cod(g). 8

15 CHAPTER 2. CONSTRUCTION ON CATEGORIES Definition Let be a congruence on the category A, and define the congruence category A by: ob(a ) = ob(a ) and arrows is the set {< f, g >: f g} with 1 A =< 1 A, 1 A > and < f, g > < f, g >=< f f, g g >. Definition We define the quotient category A / as follows: ob(a / ) = ob(a ) and arrows are (Hom A )/. Definition Suppose we have a functor F : A B, then F determines a congruence F on A by setting f F g iff dom(f) = dom(g), cod(f) = cod(g), and F (f) = F (g). We write ker(f ) = A F category of F. for this congruence category and call it the kernel Definition A group in a category C consists of objects and arrows that satisfy the same commutative diagrams that typical groups satisfy. That is, for all (generalized) elements, x, y, z : Z G, the following equations hold: m(m(x, y), z) = m(x, m(y, z)) m(x, u) = x = m(u, x) m(x, ix) = u = m(ix, x) Definition A homomorphism h : G H of groups in C consists of an arrow in C that preserves structure and whose properties can be shown in commutative diagrams. With these identities and composites, we thus have a category of groups in C, denoted by Group(C ). For example, a group in the usual sense is a group in the category Sets. Also, the groups in Group are exactly the abelian groups. Definition Enlarging Group to include also categories with more that one object, but still having inverses for all arrows, gives us a category called groupoids. Definition A strict monoidal category is a category C equipped with a binary operation : C C C which is funtorial and associative, A (B C) = (A B) C, together with a distinguished object I that acts as a unit, I C = C = C I. A strict monoidal category is exactly the same thing as a monoid in Cat. 9

16 CHAPTER 2. CONSTRUCTION ON CATEGORIES Definition A monoidal category consists of a categor C equipped with a functor : C C C and a distinguished object I, together with natural isomorphisms α ABC : A (B C) (A B) C, λ A : I A A, ρ A : A I A. These satisfy commutative diagrams as well. A monoidal category is thus a category that is strict monoidal up to natural isomorphism. 10

17 Chapter 3 Universals and Limits Definition Let A be a category. A presheaf on A is a functor A op Set. Definition Let A be a category. An object I A is initial if for every A A, there is exactly one map I A. An object T A is terminal if for every A A, there is exactly one map A T. A category need not have an initial object, but if it does, it is unique up to isomorphism. Lemma Let I and I be initial objects in a category. Then there is a unique isomorphism I I. In particular, I = I. Definition Let A be a locally small category and A A. we define a functor H A = A (A, ) : A Set as follows: for objects B A, put H A (B) = A (A, B); for maps B g B in A, define by H A (g) = A (A, g) : A (A, B) A (A, B ) p g p for all p : A B. Sometimes H A (g) is written as g or g. 11

18 CHAPTER 3. UNIVERSALS AND LIMITS Definition Let A be a locally small category. A functor X : A Set is representable if X = H A for some A A. A representation of X is a choice of an object A and an isomorphism between H A and X. Definition Let A be a locally small category. The functor H : A op [A, Set] is defined on objects A by H (A) = H A and on maps f by H (f) = H f. More precisely, a map A f A induces a natural transformation H f : H A H A whose B-component is the function H A (B) = A (A, B) H A (B) = A (A, B) p p f Definition Let A be a locally small category and A A. We define a functor H A = A (, A) : A op Set as follows: for objects B A, put H A (B) = A (B, A); for maps B g B in A, define H A (g) = A (g, A) = g = g : A (B, A) A (B, A) by for all p : B A. p p g Definition Let A be a locally small category. A functor X : A op Set is representable if X = H A for some A A. A representation of X is a choice of object A A and an isomorphism between H A and X. 12

19 CHAPTER 3. UNIVERSALS AND LIMITS Definition Let A be a locally small category. The Yoneda embedding of A is the functor H : A [A op, Set ] defined on objects A by H (A) = H A and on maps f by H (f) = H f. More precisely, a map A f A induces a natural transformation H f : H A H A, whose B-component is H A (B) = A (B, A) H A (B) = A (B, A ) p f p As a summary: For each A A, we have a functor A HA Set. Putting them all together gives a functor A op H [A, Set ]. For each A A, we have a functor A op Set. Putting them all together gives a functor A H [A op, Set ]. H A The second pair of functors is the dual of the first. Definition Let A be a locally small category. The functor Hom A : A op A Set is defined as follows: Hom A (A, B) = A (A, B) and (Hom A (f, g))(p) = g p f, whenever A f p g A B B. We see that Hom A slightly differently. carries the same information as H (or H ), presented Definition Let A be an object of a category. A generalized element of A is a map with codomain A. A map S A is a generalized element of A of shape S. Theorem (Yoneda) Let A be a locally small category. Then [A op, Set ](H A, X) = X(A) naturally in A A and X [A op, Set ]. 13

20 CHAPTER 3. UNIVERSALS AND LIMITS Informally, then, the Yoneda lemma says that for any A A and presheaf X on A : A natural transformation H A X is an element of X(A). Corollary Let A be a locally small category and X : A op Set. Then a representation of X consists of an object A A together with an element u X(A) such that: for each B A and x X(B), there is a unique map x : B A such that (X x)(u) = x. Recall that by definition, a representation of X is an object A A together with a natural isomorphism α : H A X. The above corollary states that such pairs (A, α) are in natural bijection with pairs (A, u) satisfying the last condition. Pairs (B, x) are sometimes called elements of the presheaf X. The Yoneda lemma tells us that x amounts to a generalized element of X of shape H B. An element u satisfying the above condition, is sometimes called a universal element of X. So the corollary says that a representation of a presheaf X amounts to a universal element of X. Corollary Let A be a locally small category and X : A Set. Then a representation of X consists of an object A A together with an element u X(A) such that: for each B A and x X(B), there is a unique map x : A B such that (X x)(u) = x. Corollary For any locally small category A, the Yoneda embedding is full and faithful. H : A [A op, Set ] Informally, this says that for A, A A, a map H A H A of presheaves is the same thing as a map A A in A. Lemma Let J : A B be a full and faithful functor and A, A A. Then i) a map f in A is an isomorphism if and only if the map J(f) in B is an isomorphism; 14

21 CHAPTER 3. UNIVERSALS AND LIMITS ii) for any isomorphism g : J(A) J(A ) in B, there is a unique isomorphism f : A A in A such that J(f) = g; iii) the objects A and A of A are isomorphic if and only if the objects J(A) and J(A ) of B are isomorphic. Corollary Let A be a locally small category and A, A A. Then H A = HA A = A H A = H A. Definition Let A be a category and X, Y A. A product of X and Y consists of an object P and maps X p 1 P p 2 Y with the property that for all objects and maps X f 1 A f 2 Y in A, there exists a unique map f : A P such that X f 1 p 1 A P f f 2 p 2 Y commutes. The maps p 1 and p 2 are called the projections. Definition Let A be a category, I a set, and (X i ) i I a family of objects of A. A product of (X i ) i I consists of an object P and a family of maps ( ) P p i X i with the property that for all objects A and families of maps ( ) A f i X i there exists a unique map f : A P such that p i f = f i for all i I. We call the maps f i the components of the map (f i ) i I. i I i I 15

22 CHAPTER 3. UNIVERSALS AND LIMITS Some examples of products (each in a different category) are min{x, y} for x, y (R, ), X Y for X, Y P(S), or gcd(x, y) in the poset (N, ). A fork in a category consists of objects and maps such that sf = tf f : A s, t : X Y Definition Let A be a category and let s, t : X Y be objects and maps in A. An equalizer of s and t is an object E together with a map E i X such that i : E s, t : X Y is a fork, and with the property that for any fork, there exists a unique map f : A E such that A commutes. f f E i X An equalizer describes the set of solutions of a single equation, but by combining equalizers with products, we can also describe the solution set of any system of simultaneous equations. Definition Let A be a category, and take objects and maps Y X s Z in A. A pullback of this diagram is an object P A together with maps p 1 : P X and p 2 : P Y such that t P p 2 Y p 1 X s Z commutes, and with the property that for any commutative square 16 A f 2 Y f 1 X s Z t t

23 CHAPTER 3. UNIVERSALS AND LIMITS in A, there is a unique map f : A P such that the resulting diagram commutes. Namely, p 1 f = f1 and p 2 f = f2. The first square above is called the pullback square Definition Let A be a category and I a small category. A functor I A is called a diagram in A of shape I. Definition Let A be a category, I a small category, and D : I A a diagram in A. i) A cone on D is an object A A (the vertex of the cone) together with a family ( ) A f I D(I) I I of maps in A such that for all maps I u J in I, the triangle A f I D(I) Du f J D(J) commutes. ( ) ii) A limit of D is a cone L P I D(I) with the property that for any I I cone on D, there exists a unique map f : A L such that p I f = f I for all I I. The maps p I are called the projections of the limit. In general, the limit of a diagram D is the terminal object in the category of cones on D, and is therefore an extremal example of a cone on D. The word limit can be understood as meaning on the boundary. Definition i) Let I be a small category. A category A has limits of shape I if for every diagram D of shape I in A, a limit of D exists. ii) A category has all limits (or properly, has small limits) if it has limits of shape I for all small categories I. Definition A finite limit is a limit of shape I for some finite category I. For instance, binary products, terminal objects, equalizers and pullbacks are all finite limits. 17

24 CHAPTER 3. UNIVERSALS AND LIMITS Proposition Let A be a category. i) If A has all products and equalizers then A has all limits. ii) If A has binary products, a terminal object and equalizers then A has finite limits. Definition Let A be a category. A map X f Y in A is monic (or a monomorphism if for all objects A and maps x, x : A X, f x = f x x = x In Set, a map is monic if and only if it is injective. In categories of algebras such as Grp, Vect k, Ring, etc, it is also true that the monic maps are exactly the injections. Lemma A map f : X Y is monic if and only if the square is a pullback. X 1 X 1 X f Y f Definition Let A be a category and I a small category. Let D : I A be a diagram in A, and write D op for the corresponding functor I op A op. A cocone on D is a cone on D op, and a colimit of D is a limit of D op. Explicitly, a cocone on D is an object A A (the vertex of the cocone) together with a family ( ) D(I) f I A I I of maps in A such that for all maps I u J in I, the diagram D(I) f I Du f J A D(J) 18

25 CHAPTER 3. UNIVERSALS AND LIMITS commutes. A colimit of D is a cocone ( ) D(I) p I C I I with the property that for any cocone on D, there is a unique map f : C A such that f pi = f I for all I I. We write (the vertex of) the colimit as lim I D, and call the maps p I coprojections. Definition A sum/coproduct is a colimit over a discrete category. (That is, it is a colimit of shape I for some discrete category I.) Definition A coequalizer is a colimit of shape E (shape of a fork/equalizer). A coequalizer is a generalization of a quotient by an equivalence relation. Definition A pushout of a diagram X s Y is (if it exists) a commutative square t Z X s Y t Z that is universal as such. In other words still, a pushout in a category A is a pullback in A op. It is a colimit. Definition Let A be a category. A map X f Y in A is epic/epimorphism if for all objects Z and maps g, g : Y Z, An epic in A is a monic in A op. g f = g f g = g. In categories of algebras, any surjective map is certainly epic. In some categories, the coverse holds as well. However, there are examples where this fails, 19

26 CHAPTER 3. UNIVERSALS AND LIMITS like in Ring, the inclusion Z Q is epic but not surjective. This is also an example of a map that is monic and epic but not an isomorphism. We have that any isomorphism in any category is both monic and epic. Definition A split mono (epi) is an arrow with a left (right) inverse. Given arrows e : X A and s : A X such that es = 1 A, the arrow s is called a section/splitting of e, and the arrow e is called a retraction of s. The object A is called a retract of X. The condition that every epimorphism splits or every surjection has a section is the categorical version of the axiom of choice. Definition An object P is said to be projective if for any epi e : E X and arrow f : P X there is some (not necessarily unique) arrow f : P E such that e f = f. One says that f lifts across e i.e. the diagram commutes: P f f E X e Definition i) Let I be a small category. A functor F : A ( B preserves ) limits of shape I if for all diagrams D : I A and all cones A p I D(I) on D, ( ) A p I D(I) I I ( ) F (A) F p I F D(I) I I is a limit cone on D in A is a limit cone on F D in B. ii) A functor F : A B preserves limits if it preserves limits of shape I for all small categories I. iii) Reflection of limits is defined as in i), but with in place of. I I Definition A functor F : A B creates limits (of shape I) if whenever D : I A is a diagram( in A, ) i) for any limit cone B q I F D(I) on the diagram F D, there is a ( ) I I unique cone A p I D(I) on D such that F (A) = B and F (p I ) = q I for all I I; 20 I I

27 ii) this cone ( ) A p I D(I) I I CHAPTER 3. UNIVERSALS AND LIMITS is a limit cone on D. Lemma Let F : A B be a functor and I a small category. Suppose that B has, and F creates, limits of shape I. Then A has, and F preserves, limits of shape I. Since Set has all limits, it follows that all our categories of algebras have all limits, and that the forgetful functors preserve them. Definition Consider a functor G : D A with colimit (L, (p D ) D D ). That colimit is absolute when for every functor F : A B, (F L, (F p D ) D D ) is the colimit of F G. Definition A functor G : C D is final when the following conditions are satisfied for every category A and every functor F : D A : i) if the limit (L, (p D ) D D ) of F exists, then (L, (p GC ) C C ) is the limit of F G; ii) if the limit (L, (q C ) C C ) of F G exists, then the limit of F exists as well. 21

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29 Chapter 4 Adjoints Definition Let F : A B be a functor and B an object of B. A reflection of B along F is a pair (R B, η B ) where i) R B is an object of A and η B : B F (R B ) is a morphism of B, ii) if A A and b : B F (A) is a morphism of B, there exists a unique morphism a : R B A in A such that F (a) η B = b. Proposition Let F : A B be a functor and B an object of B. When the reflection of B along F exists, it is unique up to isomorphism. Proposition Consider a functor F : A B and assume that, for every B B, the reflection of B along F exists and such a reflection (R B, η B ) has been choosen. In that case, there exists a unique functor R : B A satisfying the two properties i) B B R(B) = R B, ii) (η B : B F RB) B B is a natural transformation. Definition A functor R : B A is left adjoint to the functor F : A B when there exists a natural transformation η : 1 B F R such that for every B B, (RB, η B ) is a reflection of B along F. In an analogous way a functor R : B A is right adjoint to F when there exists a natural transformation ɛ : F R 1 B such that for each B B, (RB, ɛ B ) is a coreflection of B along F, where coreflection is the dual notion of reflection. 23

30 CHAPTER 4. ADJOINTS Definition Let F : A B and G : B A be functors. We say that F is left adjoint to G, and G is right adjoint to F, and write F G, if B(F (A), B) = A (A, G(B)) naturally in A A and B B. An adjunction between F and G is a choice of natural isomorphism. Natually in A and B means that there is a specified bijection for each A and B, and that it satisifies a naturality axiom. To state it, we need some notation. Given objects A A and B B, the correspondence between maps F (A) B and A G(B) is denoted by a horizontal bar, in both directions: ( F (A) g B ) ( A g G(B) ), ( ) ( ) F (A) f B A f G(B). So f = f and ḡ = g. We call f the transpose of f, and similarly for g. The naturality axiom has two parts: ) ( (F (A) g B q B = A ḡ ) G(B) G(q) G(B ) (that is, q g = G(q) ḡ) for all g and q, and ) ( ) (A p f A G(B) = F (A ) F (p) F (A) f B for all p and f. The concept of left and right adjoint are dual to each other. Adjunctions can be composed as well. Definition For each A A, we have a map ( ) ( ) A η A GF (A) = F (A) 1 F (A). Dually, for each B B, we have a map ( ) ( ) F G(B) ɛ B B = G(B) 1 G(B). These define natural transformations η : 1 A G F, ɛ : F G 1 B. called the unit and counit of the adjunction, respectively. 24

31 CHAPTER 4. ADJOINTS Definition Given an adjunction F G with unit η and counit ɛ, the triangles F F η F GF ɛf 1 F F G ηg GF G Gɛ 1 G G commute. These are called the triangle identities. They are commutative diagrams in the functor categories [A, B] and [B, A ], respectively. The unit and counit determines the whole adjunction, even though they appear to know only the transpose of identities. Lemma Let F G be an adjunction, with unit η and counit ɛ. Then ḡ = G(g) η A for any g : F (A) B, and f = ɛ B F (f) for any f : A G(B). Theorem Take functors F : A B and G : B A. There is a one-to-one correspondence between: i) adjunctions ( between F and G (with ) F on the left and G on the right); η ɛ ii) pairs 1 A GF, F G 1 B of natural transformations satisfying the triangle identities. Recall that by definition, an adjunction between F and G is a choice of isomorphism for each A and B, satisfying the naturality equations. Corollary We have that F G if and only if there exist natural transformations 1 η GF and F G ɛ 1 satisfying the triangle identities. Lemma Take and adjunction F G and an object A A. Then the unit map η A : A GF (A) is an initial object of (A G). 25

32 CHAPTER 4. ADJOINTS Theorem Take categories and functors F : A B and G : B A. There is a one-to-one correspondence between: i) adjunctions between F and G (with F on the left and G on the right); ii) natural transformations η : 1 A GF such that η A : A GF (A) is initial in (A G) for every A A. Corollary Let G : B A be a functor. Then G has a left adjoint if and only if for each A A, the category (A G) has an initial object. Lemma Adjunctions give rise to representable functors in the following way. Let F : A B and G : B A with F G between locally small categories. Then the functor A (A, G( )) : B Set (that is, the composite B G A Set) HA is representable. Proposition Any set-valued functor with a left adjoint is representable. Definition Consider two functors F : A B and G : A C. The left Kan extension of G along F, if it exists, is a pair (K, α) where K : B C is a functor, α : G K F is a natural transformation, satisfying the following universal property: if (H, β) is another pair with H : B C a functor, β : G H F a natural transformation, there exists a unique natural transformation γ : K H satisfying the equaltiy (γ F ) α = β. We shall use the notation Lan F G to denote the left Kan extension of G along F. The notation Ran F G is used for the dual notion of right Kan extension. We write (γ F ) instead of γ 1 F. 26

33 CHAPTER 4. ADJOINTS Theorem Consider two functors F : A B and G : A C, with A small and C cocomplete. Under these conditions, the left Kan extension of G along F exists. Proposition Consider a full and faithful functor F : A B with A a small category. Let C be a cocomplete category. Given a functor G : A C, the canonical natural transformation G (Lan F G) F is an isomorphism. Proposition Consider a functor G : A C, with A a small category. Write 1 for the category with a single object and a single arrow, and F : A 1 for the corresponding functor. The functor G has a colimit if and only if the left Kan extension Lan F G of G along F exists. Proposition Consider a functor F : A B between small categories. The following are equivalent: i) F has a right adjoint G; ii) Lan F 1 A exists and, for every functor L : A C, the isomorphism L Lan F 1 A = LanF L holds; iii) Lan F 1 A exists and the isomorphism F Lan F 1 A = LanF F holds. Definition A functor F : A B satisfies the solution set condition with respect to an object B B when there exists a set S B A of objects such that A A b : B F A A S B a : A A b : B F A F (a) b = b Theorem (Adjoint functor theorem) Consider a complete category A and a functor F : A B. The following are equivalent: i) F has a left adjoint functor. ii) The following conditions hold: a) F preserves small limits; b) F satisfies the solution set condition for every object B B. 27

34 CHAPTER 4. ADJOINTS Proposition (Adjoint functor theorem for ordered sets) Let A be an ordered set, B a complete ordered set, and G : B A an order-preserving map. Then G has a left adjoint G preserves meets Theorem (Special adjoint functor theorem) Consider a functor F : A B and suppose the following conditions are satisfied: i) A is complete; ii) F preserves small limits; iii) A is well-powered; iv) A has a cogenerating family. Under these conditions, F has a left adjoint functor. Definition Let C be a category. A weakly initial set in C is a set S of objects with the property that for each C C, there exist an element S S and a map S C. Theorem (General Adjoint Functor Theorem) Let A be a category, B a complete category, and G : B A a functor. Suppose that B is locally small and that for each A A, the category (A G) has a weakly initial set. Then G has a left adjoint G preserves limits. Definition A full subcategory A of a category B is replete when, with every A A, A also contains every object B B isomorphic to A. Definition A reflective subcategory of a category B is a full replete subcategory A of B whose inclusion i : A B in B admits a left adjoint r : B A, called the reflection. Definition A localization of a category B with finite limits is a reflective subcategory A of B whose reflection preserves finite limits. 28

35 CHAPTER 4. ADJOINTS Definition An essential localization of a category B is a reflective subcategory A of B whose reflection itself admits a left adjoint. 29

36 CHAPTER 4. ADJOINTS 30

37 Chapter 5 Limits Definition A category C is filtered when i) C C ii) C 1, C 2 C C 3 C f : C 1 C 3 g : C 2 C 3, iii) C 1, C 2 C f, g : C 1 C 2 C 3 C h : C 2 C 3 h f = h g. By a filtered colimit we mean the colimit of a functor defined on a filtered category. We say that a category A has filtered colimits when for every small filtered category C and every functor F : C A, the colimit of F exists. Theorem Consider a small filtered category C and a finite category D. Given a functor F : C D Set to the category of sets and mappings, the following mixed interchange property holds: colim C D ( lim D D F (C, D)) = lim D D (colim C D F (C, D)). Definition Given categories I and A and an object A A, there is a functor A : I A with constant value A on objects and 1 A on maps. This defines, for each I and A, the diagonal functor : A [I, A ] Now, given a diagram D : I A and an object A A, a cone on D with vertex A is simply a natural transformation A D, and we write Cone(A, D) for the set of cones on D with vertex A, we therefore have Cone(A, D) = [I, A ]( A, D). 31

38 CHAPTER 5. LIMITS Proposition Let I be a small category, and D : I A a diagram. Then there is a one-to-one correspondence between limit cones on D and representations of the functor Cone(, D) : A op Set with the representing objects of Cone(, D) being the limit objects (that is, the vertices of the limit cones) of D. Briefly put: a limit of D is a representation of [I, A ](, D). The proposition formalizes the thought that cones on a diagram D correspond one-to-one with maps into lim I D. It implies that if D has a limit then Cone(A, D) ( ) = A A, lim D I naturally in A. Corollary Limits are unique up to isomorphism. Proposition Let I be a small category and A a category with all limits of shape I. Then lim I defines a functor [I, A ] A, and this functor is right adjoint to the diagonal functor. This functor is defined as such: choose for each D [I, A ] a limit cone D, and call its vertex lim I D. For each map α : D D, we have a canonical map lim I α : lim D lim D. Thus naturally in A and D. [I, A ]( A, D) = Cone(A, D) = A (A, lim D) Lemma Let I be a small category, A a locally small category, D : I A a diagram, and A A. Then naturally in A and D. Cone(A, D) = lim I A (A, D) Here, A (A, D) is the functor A (A, D) : I Set 32

39 CHAPTER 5. LIMITS I A (A, D(I)) Proposition (Representables preserve limits) Let A be a locally small category and A A. Then A (A, ) : A Set preserves limits. This proposition tells us that ( A A, lim D) = lim I A (A, D). I To dualize this, a limit in A op is a limit in A, so A (, A) transforms colimits in A into limits in Set: ( A lim D, A) = lim I A (D, A). I Note the right hand side is a limit, not a colimit. Definition Let A and I be categories. For each A A, there is a functor ev A : [A, I ] I called evaluation at A. X X(A), Theorem (Limits in functor categories) Let A and I be small categories and I a locally small category. Let D :I [ A, I ] be a diagram, and suppose that for each A A, the diagram D( )(A) : I I has a limit. Then there is a cone on D whose image under ev A is a limit cone on D( )(A) for each A A. Moreover, any such cone on D is a limit cone. Limits in a functor category are computed pointwise (meaning the objects of A). For example, given two functors X, Y [A, I ], their product can be computed by first taking the product X(A) Y (A) in I for each point A, then assembling them to form a functor X Y. The pointwise character of this construction is precisely expressed by the formula ( lim F (D))(C) = lim (F (D)(C)). D D D D In other words, the value of the limit lim D D F (D) at an object C is the limit of the values of F (D) at C. 33

40 CHAPTER 5. LIMITS Theorem Consider a complete category A and a small category C. Under these conditions, the category Fun(C, A ) is complete and limits in it are computed pointwise. Theorem Consider a small category C and a functor F from C to Set. In the category Fun(C, Set), F can be presented as the colimit of a diagram just constituted of representable functors and representable natural transformations. Corollary Let I and A be small categories, and I a locally small category. If I has all limits (respectively, colimits) of shape I then so does [ A, I ], and for A A, the evaluation functor ev A : [A, I ] I preserves them. Take categories I,J and I. There are isomorphisms of categories [I, [J, I ]] = [I J, I ] = [J, [I, I ]]. Under these isomorphisms, a functor D :I J I corresponds to the functors D mapping I D(I, ) and D mapping J D(, J). Proposition (Limits commute with limits) Let I and J be small categories. Let I be a locally small category with limits of shape I and shape J. Then for all D : I J I, we have lim lim D = lim D = lim lim D, J I I J I J and all these limits exist. In particular, I has limits of shape I J. Corollary Let A be a small category. Then [A op, Set ] has all limits and colimits, and for each A A, the evaluation functor ev A : [A op, Set ] Set preserves them. Corollary The Yoneda embedding preserves limits, for any small category. 34

41 CHAPTER 5. LIMITS Definition Let A be a category and X a presheaf on A. The category of elements E(X) of X is the category in which: i) objects are pairs (A, x) with A A and x X(A); ii) maps (A, x ) (A, x) are maps f : A A in A such that (Xf)(x) = x. There is a projection functor P : E(X) A defined by P (A, x) = A and P (f) = f. Theorem (Density) Let A be a small category and X a presheaf on A. Then X is the colimit of the diagram that is, X = lim I (H P ). E(X) P A H [A op, Set] Theorem Let F G be an adjunction. Then F preserves colimits and G preserves limits. The previous theorem is often used to prove that a functor does not have an adjoint. Definition A category is complete (or properly, small complete) if it has all limits. Theorem A category C is complete precisely when each family of objects has a projuct and each pair of parallel arrows has an equalizer. Proposition For a category C, the following conditions are equivalent: i) C is finitely complete; ii) C has a terminal object, binary products and equalizers; iii) C has a terminal object and pullbacks. 35

42 CHAPTER 5. LIMITS Definition A category D is finitely generated when i) D has finitely many objects, ii) there are finitely many arrows f 1,..., f n such that each arrow of D is the composite of finitely many of these f i. Proposition Let F : D A be a functor, with A finitely complete and D finitely generated. Then the limit of F exists. Definition A category A is cartesian closed if it has finite products and for each B A, the functor B : A A has a right adjoint. We write the right adjoint as ( ) B, and, for C A, call C B an exponential. Theorem For any small category A, the presheaf category is cartesian closed. 36

43 Chapter 6 Generators and Projectives Definition Consider a category A and an object A A. Two monomorphisms f : R A and g : S A are equivalent when there exists an isomorphism r : R S such that g r = f. An equivalence class of monomorphisms with codomain A is called a subobject of A. The dual notion is that of a quotient of A. Definition A category A is well-powered when the subobjects of every object constitute a set. In Set, the subobjects of a set X are in bijection with the subsets of X. In Gr, they are in bijection with subgroups. Given an object A of a category C, let us consider the class Mono(A) of all monomorphisms with codomain A. A monomorphism r : R A is smaller than a monomorphism s : S A when there exists a (mono)morphism t : R S such that s t = r. Performing the quotient on Mono(A) which identifies isomorphic monomorphisms, we obtain a partial order on the class Sub(A) of subobjects of A. We recall that C is well-powered when, for each A C, Sub(A) is a set. Definition Consider an object A of a category C. By the intersection of a family of subobjects of A, we mean their infimum in Sub(A). By the union of a family of subobjects of A, we mean their supremum in Sub(A). 37

44 CHAPTER 6. GENERATORS AND PROJECTIVES Proposition Consider an object A C and suppose Sub(A) is a set. The following are equivalent: i) the intersection of every family of subobjects of A exists; ii) the union of every family of subobjects of A exists. Definition In a category, an epimorphism is called regular when it is the coequalizer of a pair of arrows. Definition An epimorphism f : A B in a category is called extremal when it does not factor through any proper subobject of B; i.e., given f = i p with i a monomorphism, i is necessarily an isomorphism. Definition In a category A, an epimorphism f : A B is called a strong epimorphism when, for every commutative square z u = v f, with z : X Y a monomorphism, there exists a (unique) arrow w : B X such that w f = u, z w = v. Proposition In a category A, i) the composite of two strong epimorphisms is a strong epimorphism, ii) if a composite f g is a strong epi, f is a strong epi, iii) a morphism which is both a mono and a strong epi, is an isomorphism, iv) every regular epi is strong, v) every strong epi is extremal. Proposition Let F : A B be a functor admitting a left adjoint functor G : B A. The functor F preserves strong monomorphisms, and the functor G preserves strong epimorphisms and regular epimorphisms. Definition A category C is finitely well-complete when i) C is finitely complete, ii) given an object C C, the intersection of an arbitrary class of subobjects of C always exists. 38

45 CHAPTER 6. GENERATORS AND PROJECTIVES Proposition In a finitely well-complete category, every morphism f factors as f = i p, where i is a monomorphism and p is a strong epimorphism. Definition A category C has strong-epi-mono-factorizations when every morphism f of C factors as f = i p, with p a strong epimorphism and i a monomorphism. The monomorphism i is also called the image of f. Definition Let C be a category. A family (G i ) i I of objects of C is called a family of generators when, given any two parallel morphisms u, v : A B in C, i I g : G i A u g = v g u = v. Generators are important because of the following property: every object can be recaptured as a quotient of a coproduct of generators. Proposition Let C be a category with coproducts and (G i ) i I a family of objects of C. The following are equivalent: i) (G i ) i I is a family of generators; ii) for every object C C, the unique morphism γ C : (domain of f) C i I,f C (G i,c) such that γ C s f = f is an epimorphism. Definition Let C be a category with coproducts and (G i ) i I a family of objects of C. The family (G i ) i I is a strong family of generators when, for every object C C, the morphism γ C is a strong epimorphism. The family is a regular family of generators when, for every object C C, the morphism γ C is a regular epimorphism. When the family is reduced to a single element {G}, we say that G is a strong or a regular generator, according to the case. 39

46 CHAPTER 6. GENERATORS AND PROJECTIVES Definition Let C be a category and (G i ) i I a family of objects of C. Let us write G for the full subcategory of C generated by the G i s and G /C for the full subcategory of C /C generated by the objects of the form f : G i C. The family is a dense family of generators when for every object C C, the colimit of the functor Γ C : G /C G, (f : G i C) G i, is precisely (C, (f) f G /C ). When the family is reduced to a single element {G}, G is called a dense generator. Proposition In a category with coproducts, every dense family of generators is regular and every regular family of generators is strong. Definition i) A family of functors (F i : A B i ) i I is collectively faithful when given morphisms f, g : A A in A ( i I F i (f) = F i (g)) (f = g). ii) A family of functors (F i : A B i ) i I collectively reflects isomorphisms when, given a morphism f : A A in A, ( i I F i (f) is an isomorphism ) (f is an isomorphism ). Definition Let C be a category (with finite limits). A family (G i ) i I of objects of C is a strong family of generators when the family of functors C (G i, ) : C Set collectively reflects isomorphisms. When the family is reduced to a single object {G}, G is called a strong generator. Definition An object P of a category C is projective when, given a strong epimorphism p : X Y and a morphism f : P Y, there exists a factorization g : P X such that p g = f. Proposition For an object P of a category C, the following condition are equivalent: i) P is projective; ii) the functor C (P, ) : C Set preserves epimorphisms. 40

47 CHAPTER 6. GENERATORS AND PROJECTIVES Definition A category C has enough projectives when every object is a strong quotient of a projective object. 41

48 CHAPTER 6. GENERATORS AND PROJECTIVES 42

49 Chapter 7 Categories of Fractions A graph is, roughly speaking, a category without a composition law. Definition A graph G consists of i) a class G whose elements are called the objects (or vertices) of the graph. ii) for each pair (A, B) G G, a set G (A, B) whose elements are called the morphisms (or arrows) from A to B. The graph G is small when G itself is a set. Definition A morphism of graphs F : F G between two graphs consists of i) a mapping F : F G, ii) for each pair (A, B) F F of objects, a mapping F (A, B) G (F A, F B). Obviously, every category is a graph (just forget composition). Definition Let G be a graph. A path in G is a nonempty finite sequence (A 1, f 1, A 2, f 2,..., A n ) alternating objects and arrows in G ; each arrow f i has domain A i and codomain A i+1. Definition Let G be a graph. A commutativity condition on G is a pair of paths both defined from some given object A to some given object B. We now formally add some inverse arrows of a given category. 43

50 CHAPTER 7. CATEGORIES OF FRACTIONS Definition Consider a category C and a class Σ of arrows of C. The category of fractions C [Σ 1 ] is said to exist when a category C [Σ 1 ] and a functor φ : C C [Σ 1 ] can be found, with the following properties: i) f Σ φ(f) is an isomorphism; ii) if D is a category and F : C D is a functor such that for all morphisms f Σ, F (f) is an isomorphism, there exists a unique functor G : C [Σ 1 ] D such that G φ = F. Proposition Consider a category C and a set Σ of arrows of C. The category of fractions C [Σ 1 ] exists. Moreover when C is small, C [Σ 1 ] is small as well. Definition Consider a category C and a class Σ of morphisms of C. The class Σ admits a right calculus of fractions when the following holds: i) C C 1 C Σ; ii) given s : A B and t : B C, (s Σ and t Σ) (t s Σ); iii) if f : A B is in C and s : C B is in Σ, there exist g : D C in C and t : D A in Σ such that f t = s g; iv) if f, g : A B are in C and s : B C is in Σ with the property s f = s g, there exists t : D A in Σ with the property f t = g t. Definition Let C be a category and Σ C a class of morphisms such that the category of fractions φ : C C [Σ 1 ] exists. The class Σ is saturated when for every morphism f C φ(f) is an isomorphism iff f Σ. Definition Consider two arrows f : A B, g : C D in a category C. We say that f is orthogonal to g and write f g when, given arbitrary morphisms u, v such that v f = g u there exists a unique morphism w such that w f = u, g w = v. An epimorphism f is strong when, for every monomorphism g, f g. 44

51 CHAPTER 7. CATEGORIES OF FRACTIONS Definition Given an arrow f : A B and objects X, Y of a category C : i) we say that f is orthogonal to X and write f X when for every morphism a : A X, there exists a unique morphism b : B X such that b f = a; ii) we say that Y is orthogonal to f and write Y f when for every morphism c : Y B there exists a unique morphism d : Y A such that f d = c. Definition Let C be a category and Σ a class of morphisms of C. By the orthogonal subcategory of C determined by Σ, we mean the full subcategory C Σ of C whose objects are those X C such that f X for every f Σ. Theorem Let C be a cocomplete category in which every object is presentable. Given a set Σ of morphisms of C, the corresponding orthogonal subcategory C Σ is reflective in C. Definition Let C be a cocomplete category and E a class of morphisms of C. The class E is closed under colimits when given a small category D, two functors F, G : D C and a natural transformation α : F G, if all the morphisms α D : F D GD are in E, then the corresponding factorization colimα D : colimf D colimgd is in E as well. Definition By a factorization system on a category B we mean a pair (E, M) where both E and M are classes of morphisms of B and i) every isomorphism belongs to both E and M, ii) both E and M are closed under composition, iii) e E m M e m, iv) every morphism f B can be factored as f = m e, with e E and m M. Definition Consider a finitely complete category B. A univeral closure operation on B consists in giving, for every subobject S B in B, another subobject S B called the closure of S in B; these assignments have to satisfy the following properties, where S, T are subobjects of B and f : A B is a morphism of B; 45

52 CHAPTER 7. CATEGORIES OF FRACTIONS i) S S; ii) S T S T ; iii) S = S; iv) f 1 (S) = f 1 (S). Definition Consider a finitely complete category B provided with a universal closure operation. i) A subobject S B is dense when S = B; ii) a subobject S B is closed when S = S. Definition Consider a finitely complete category B with strong-epimono factorizations. Given a universal closure operation on B, a morphism f : A B is bidense when its image is dense and the equalizer of its kernel pair is dense. 46

53 Chapter 8 Flat Functors and Cauchy Completeness Definition Consider two finitely complete categories A, B. A functor F : A B is left exact when it preserves finite limits. Theorem Let A be a small category. The category Lex(A, Set) of left exact functors is reflective in the category Fun(A, Set) of all functors. Definition For an arbitrary category A, a functor F : A Set is flat when the category Elts(F ) of elements of F is cofiltered. Given an arbitrary functor F : A B, F is flat when for each object B B, the functor B(B, F ) : A Set is flat. Proposition Given a category A, every representable functor is flat. A (A, ) : A Set Proposition Let F : A B be a functor with a left adjoint. Then F is flat. 47