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Home , Epimorphism, Inclusion map, Isomorphism, Monomorphism, Reflective subcategory

Introduction to Theory

David Holgate & Ando Razafindrakoto

University of Stellenbosch

June 2011

Contents

Chapter 1. Categories 1 1. Introduction 1 2. Categories 1 3. 4 4. Universal Properties 5 5. 6

Chapter 2. and Natural Transformations 8 1. Functors 8 2. Natural Transformations 12 3. Categories and the Yoneda 13 4. Representable Functors 14

Chapter 3. Special 16 1. and 16 2. Split and extremal monomorphisms and epimorphisms 17

Chapter 4. Adjunctions 19 1. Galois connections 19 2. 20

Chapter 5. Limits and Colimits 23 1. Products 23 2. Pullbacks 24 3. Equalizers 26 4. Limits and Colimits 27 5. Functors and Limits 28

Chapter 6. 30 1. Subcategories 30 2. Reflective Subcategories 31

3 1. Categories

1. Introduction

Sets and their notation are widely regarded as providing the “language” we use to express our mathematics. Naively, the fundamental notion in theory is that of membership or being an element. We write a ∈ X if a is an element of the set X and a set is understood by knowing what its elements are.

Category theory takes another perspective. The category theorist says that we understand what mathematical objects are by knowing how they interact with other objects. For example a G is understood not so much by knowing what its elements are but by knowing about the from G to other groups. Even a set X is more completely understood by knowing about the functions from X to other sets, f : X → Y .

For example consider the single element set A = {?}. If we understand A by considering elements, we could say that A has an element (∃ a ∈ A) and this element is unique (a, b ∈ A ⇒ a = b). On the other hand, using functions we could say that for all sets X there exists a unique X → A.

When understanding groups or sets, for example, category theorists looks at the homomorphisms f G →h H (not the groups) or the functions X → Y (not the sets) as fundamental. We study the “arrows” and the question we ask is, “How much of our mathematics can we formulate and understand using only arrows?”

The answer to this question is both elegantly simple and remarkably deep. It is given by – “the of arrows”.

2. Categories

One of the simplest algebraic structures is a (M, ◦, e), a set M with an associative binary multiplication and two-sided identity. (A monoid can be thought of as a “group without inverses”. For a, b ∈ M the product a ◦ b ∈ M and for any a ∈ M, a ◦ e = a = e ◦ a.) It is this that we adopt for our algebra of arrows.

The principal difference between a monoid and a category is that multiplication of arrows, which is modeled on the composition of functions, is only possible if one arrow follows another. Thus 1 Section 2. Categories Page 2

for the arrows below f

) # h A B c g we can multiply f ◦ g, g ◦ f, h ◦ g and f ◦ h. But g ◦ h and h ◦ f, for example, are not defined. A category is like a monoid, but with a of “nodes”.

Definition 2.1. A category C (with components (Ob, hom, ◦, 1)) consists of:

(1) A collection Ob C of C-objects.

(2) For each pair A, B ∈ Ob C a set homC(A, B) of C-arrows or C-morphisms from A to f B. A f ∈ homC(A, B) is usually denoted f : A → B or A → B.

(3) For any A, B, C ∈ Ob C and pair of arrows f ∈ homC(A, B) and g ∈ homC(B,C), the

composition g ◦ f ∈ homC(A, C)

f g A / B / C :

g◦f

(4) For each A ∈ Ob C an identity arrow 1A ∈ homC(A, A)

Subject to the following conditions:

• Composition ◦ is associative.

• For any arrow f ∈ homC(A, B), f ◦ 1A = f = 1B ◦ f.

• If the pairs of objects (A, B) 6= (C,D) then homC(A, B) ∩ homC(C,D) = ∅.

f Implicit in the definition of hom-sets is the fact that any arrow A → B in C begins at an object, the domain of f (dom(f) = A) and ends at another object, the of f (cod(f) = B). The composition g ◦ f is well-defined only if dom(g) = cod(f).

Examples 2.2. (1) Any monoid (M, ◦, e) is a category C – an algebra of arrows. Thus the elements of M are the arrows and the category has only one object, which we call

?. Then homC(?, ?) = M and 1? = e.

? a∈M g (2) A pre-ordered set (X, ≤) is a set equipped with a reflexive and transitive binary

≤. Such a (X, ≤) is a category C. In this case Ob C = X and for a, b ∈ X, homC(a, b) has exactly one element iff a ≤ b. Section 2. Categories Page 3

These two examples demonstrate two aspects of a category. On the one hand a category can be viewed as a generalised monoid with composition of arrows the algebraic operation. On the other hand a category is a generalised pre-ordered set with the arrows showing how the objects of the category relate to each other.

(3) There are a number of categories which group together particular mathematical struc- tures. Can you name the obvious arrows or morphisms of the categories with the fol- lowing objects? What are the identities and what is composition? • Set – objects sets. • Grp – objects groups. • Top – objects topological spaces. • Met – objects metric spaces.

• VecK – objects finite dimensional vector spaces over a field K. • Mon – objects . • Pre – objects pre-ordered sets. (4) Let K be a field. We define a category Mat(K) with objects the natural N and arrows A : n → m, n × m matrices with coefficients in K. Composition is multiplication and for each n ∈ N, 1n is the n ×n identity matrix. (How is this category

related to the category VecK above?)

2.1. Some remarks on notation. One of the most useful and pleasant aspects of category theory is the notation. Arrow diagrams provide pictures which explain definitions and proofs. This provides a concise and clear overview of categorical reasoning and a well drawn arrow diagram can make a proof or definition very easy to follow. (As they say, “A picture is worth a thousand words.”)

A diagram such as the one below commutes if all the possible arrow compositions with the same domain and codomain are equal.

f A / B ~ ~ ~ d ~ g ~ k ~ ~ ~  ~ h  C / D

That is if k ◦ f = h ◦ g, d ◦ f = g and h ◦ d = k. In other words a is a collection of equations. Section 3. Isomorphisms Page 4

Often we are lazy and omit the ◦ when writing composition, unless the context requires some clarity. So the above equations could be written as kf = hg, df = g and hd = k.

The broken line for the arrow d is a notation used to indicate the mathematical sentence “there exists an arrow d”. Thus the above commutative diagram would be understood to say, “If kf = hg then there exists an arrow d such that df = g and hd = k.”

Exercises 2.3. (1) Show that there are two possible categories with one object and two morphisms. Can you represent these categories using a set and functions on the set? (2) Show that there are exactly 11 different categories with three morphisms. (3) Consider the category generated by the two arrows f : X → Y and g : Y → X satisfying the relations

fgfg = fg gfgf = idX . Show that the category has at most 5 arrows. (4) Express the associativity of the composition and the property of identity arrows in the definition of a category in terms of commuting diagrams. (5) For a set X, show that the P(X) can be viewed as a category where a morphism A → B means A ⊆ B. (6) Given a category C and a fixed object A ∈ Ob C, the category of objects over A, written (C ↓ A), is defined as follows: (a) Objects are pairs (X, f) where f : X → A is a morphism in C. (b) Morphisms h :(X, f) → (Y, g) are C-morphisms h : X → Y such that gh = f. Verify that (C ↓ A) is a category. (It will help to draw diagrams.)

3. Isomorphisms

Two mathematical objects are considered to be isomorphic if they are indistinguishable in some sense. For instance the sets {a, b} and {x, y} are isomorphic because they are both two-element sets. Similarly the real intervals (0, 1) and (0, 2) are topologically isomorphic, the one is a continuous scaling of the other. From the point of view of category theory such isomorphic objects cannot be distinguished by using morphisms or arrows.

Definition 3.1. The morphism f : A → B is an if there exists a g : B → A such that gf = 1A and fg = 1B. The morphism g is an inverse of f.

Two objects A and B are isomorphic if there is an isomorphism f : A → B. In this case we write A ∼= B. Section 4. Universal Properties Page 5

Proposition 3.2. (1) Inverses are unique, i.e. if both g and h are inverses of f, then g = h. For this reason if f has an inverse we write it as f −1. (2) Inverses are isomorphisms, i.e. if f has an inverse f −1 then f −1 is an isomorphism. (3) The composition of two isomorphisms is again an isomorphism. (We say that isomor- phisms are closed under composition or stable under composition.)

Exercise 3.3. What are the isomorphisms in each of the categories given in Examples 2.2?

Because isomorphic objects in a category C cannot be distinguished by the arrows of C, we usually consider isomorphic objects to be “the same” or “essentially equal”. Our intention is to use the arrows to understand the objects of a category, so if they cannot draw a distinction then neither do we.

A notable exception to the paragraph above is when we consider a group G as a category. Obviously every a ∈ G is an isomorphism in the category. A category (typically with more than one object) in which every arrow is an isomorphism is called a .

4. Universal Properties

Many categorical properties and definitions are given given by a categorical diagram together with the existence of a unique arrow. This is best illustrated with an example.

Definition 4.1. An object 1 in a category C is a terminal object if for every A ∈ Ob C there exists a unique arrow, tA : A → 1. Given in a diagram the definition is:

∃!tA ∀ A ______/ 1

We say that being a terminal object is a because for any A ∈ Ob C there is a unique arrow tA to 1. This is a very simple definition given by a universal property, and we will encounter many more. What is typical of such categorical definitions is that they always assert that for any given diagram (in this case the diagram consisting of just the two objects A and 1) there exists a unique arrow which completes the diagram in some way. (In this case the arrow links A to 1.)

The fact that the morphism tA is unique with a given property, namely having domain A and codomain 1 is very powerful. Any other arrow from A to 1 must be equal to tA. The power of the uniqueness can be seen in the following proposition.

Proposition 4.2. If 1 and 10 are terminal objects in a category C, then 1 ∼= 10. Section 5. Duality Page 6

Another more sophisticated example of a universal property is the following. Let B = {ei | i ∈ I} be the basis for a vector V over a field K and let u : I → V be the that maps

i 7→ ei. Then the diagram below illustrates the universal property that for any other function f : I → W to a W over K there is a unique F : V → W which makes the diagram commute. u I / V @ @@ } @@ } @@ } ∀f @ ~} ∃!F W (Can you say how F is defined? Can you prove that it is unique?) In such a definition we call u : I → V a universal arrow as it is the arrow u which has the universal property in this case.

5. Duality

For any category C we get another category, called the of C, by simply reversing the direction of the arrows in C. For instance if our category is the ordered set (N, ≤) then the dual category would be the ordered set (N, ≥). This might seem trivial, but duality is a very useful concept within category theory.

Definition 5.1. The dual of a category C is a category Cop with:

• Ob Cop = Ob C, op • for any A, B ∈ Ob C , homCop (A, B) = homC(B,A), op • for any A, B, C ∈ Ob C and pair of arrows f ∈ homCop (A, B) and g ∈ homCop (B,C), g ◦op f = f ◦ g where the second composition is in C, and op op • for each A ∈ Ob C , 1A = 1A where the second identity is in C.

We usually omit the “op” on composition and identities, understanding that Cop is the category obtained from C by “turning the arrows around”.

CCop

f g f g A / B / CABo o C : d

g◦f f◦g

Exercises 5.2. (1) Let G be a group regarded as a one-object category. What is the dual of G? Prove that G is isomorphic to Gop. (2) Find a monoid which is not isomorphic to its dual. Section 5. Duality Page 7

We can dualise not only a category, but also any categorical definition. For instance the dual of Definition 4.1 gives:

Definition 5.3. An object 0 in a category C is an initial object if for every A ∈ Ob C there

exists a unique arrow, iA : 0 → A. Given in a diagram the definition is:

∃!iA ∀ A o______0

Now we proved a property P of terminal objects in Proposition 4.2. Since P holds in any category C, it holds not only in C but also in Cop. But if P holds in Cop, then it must follow that the dual of P holds in C. (Why?) In other words the dual of P holds in any category.

We have just proved the following.

Proposition 5.4. If 0 and 00 are initial objects in a category C, then 0 ∼= 00.

Duality gives us “two for the price of one”. Any categorical definition has a dual definition, and if a theorem holds in every category then the dual of that theorem holds in every category too.

Exercises 5.5. (1) What are the initial and terminal objects in Top, Grp and CRng (the category of commutative rings with homomorphisms)? (2) What are the initial and terminal objects in a pre-ordered set (X, ≤) considered as category? 2. Functors and Natural Transformations

Seeing as in category theory the arrows are the most important, we surely need to ask what the arrows between categories are. And once we have defined this, what about arrows between arrows? In fact it was precisely to be able to talk about these higher level arrows that the language of category theory was originally introduced.

1. Functors

Definition 1.1. A functor F : A → B between two categories A and B is an assignment which maps objects of A to objects of B and arrows of A to arrows of B such that for A, B and f, g in A:

• if f ∈ homA(A, B) then F f ∈ homB(F A, F B), • F (f ◦ g) = (F f) ◦ (F g), and

• F (1A) = 1FA.

In other words F is a “” which preserves the algebraic structure of the categories.

Examples 1.2. (1) If (M, ◦, eM ) and (N, ∗, eN ) are monoids viewed as categories, then a functor F : M → N is a function from M to N such that for any a, b ∈ M, F (a ◦ b) = (F a) ∗ (F b), i.e. a monoid homomorphism. (2) If (X, ≤) and (Y, v) are pre-ordered sets viewed as categories, then a functor F : X → Y is a function from X to Y such that for any a, b ∈ X, a ≤ b ⇒ F a v F b, i.e. an order preserving map.

(3) Many categories such as Grp, Top and VecK are collections of sets with an additional structure. The morphisms are functions which respect the structure of the category’s objects. For such categories we can “forget” the structure and the special property of the morphisms thinking of them just as sets and functions. This gives a forgetful functor U : C → Set in each instance. (4) The power set functor P : Set → Set maps a function f : X → Y to the function Pf : PX → PY which sends A ⊆ X to Pf(A) := {f(a) | a ∈ A}.

(5) For any category C and C-object X, the hom-set functor homC(X, −): C → Set is

defined by: homC(X, −)(f : A → B) := homC(X, f): homC(X,A) → homC(X,B)

where homC(X, f)(h) := f ◦ h. (Draw some diagrams to help you understand this example.)

8 Section 1. Functors Page 9

Exercises 1.3. (1) Define constant and identity functors and verify that they are functors. (2) Prove that the following definitions can be seen as functors. (In each case, specify the domain and the codomain of the functor.) • The of a vector space over a field K.

• The determinant of a matrix U ∈ GLn(R), where R is a . (3) Prove that there is no functor Z : Grp → Grp such that Z(G) is the center of G for

all groups G. (Hint: consider the chain S2 → S3 → S2.) (4) Show that the power set functor P : Set → Set is a functor. (5) For any set X we construct the free monoid on X by taking all finite “words” over X,

FX := {x1x2 . . . xn : n ∈ N, xi ∈ X}. (Note that even if X is finite, FX is an infinite set.) Multiplication is given by “concatenation”

(x1x2 . . . xn) ? (y1y2 . . . ym) = x1 . . . xny1 . . . ym

and the identity is the empty word, ∅. Verify that (F X, ?, ∅) is a monoid. Then define F on functions as well and prove that this gives a functor, the so-called free monoid functor F : Set → Mon. (6) A contravariant functor from A to B is a functor that goes from Aop to B. Can you give a ‘contravariant power set functor’ from Set to Set? Verify that it is a functor.

(7) Verify for any category C and C-object X, the hom-set functor homC(X, −): C → Set is a functor.

(8) There is a similarly defined ‘contravariant hom-set functor’ homC(−,X) for any C-object X. Define this functor and verify that it is a functor.

We have identity functors and can compose two functors to get another functor, thus the collection of categories and functors between them forms a new category Cat. (We have to be careful here. Can you see where we might run into trouble?)

F G A / B / C ;

G◦F

In particular two categories A and B are isomorphic, written A ∼= B, if there are functors F and

G such that GF = 1A and FG = 1B.

F # 1A A B 1B 5 c h

G Section 1. Functors Page 10

However, isomorphisms between categories are very restrictive. There are more useful properties for functors used to compare categories.

Definition 1.4. Let F : A → B be a functor.

(1) F is an embedding if F is injective on morphisms, i.e. F f = F g ⇒ f = g. (2) F is faithful if the restriction of F to hom-sets is injective, i.e. for all A, B ∈ Ob A the assignment illustrated below is injective.

homA(A, B) / homB(F A, F B)

f / F f (3) F is full if for all A, B ∈ Ob A the assignment illustrated above is surjective, i.e. F is surjective on hom-sets.

Exercises 1.5. (1) Let 1 be the category with one object and one morphism. What does it say about C if the unique functor F : C → 1 is full? And what if it is faithful? (2) Are forgetful functors U : C → Set full and/or faithful? (3) Is the free monoid functor F : Set → Mon free and/or faithful?

Proposition 1.6. Let F : A → B be a functor.

(1) F preserves isomorphisms, i.e. if f is an isomorphism in A, then F f is an isomorphism in B. (2) If F is full and faithful then F reflects isomorphisms, i.e. if F f is an isomorphism in B then f is an isomorphism in A.

Definition 1.7. A functor F : A → B is an equivalence if F is full, faithful and essentially surjective (also termed isomorphism dense) i.e. for all B ∈ Ob B there exists A ∈ Ob A with FA ∼= B.

We say that two categories A and B are equivalent if there is an equivalence F : A → B.

Because we do not distinguish between isomorphic objects, to say that two categories are equiv- alent is saying that from the point of view of category theory they are the same.

Exercises 1.8. (1) Let K be a field. Show that the category of all finite-dimensional vector spaces over K with linear transformations is equivalent to Mat(K). (2) Let FinOrd be the category of finite ordinals with functions between them (a finite ordinal is of the form [n] = {0, 1, 2, . . . , n − 1}) and let FinSet be the category of finite sets with functions. Show that these two categories are equivalent. Section 1. Functors Page 11

The most interesting equivalences between categories are dual equivalences. We say that A is dually equivalent to B if A is equivalent to Bop. Often simply turning the arrows around can have surprising results. What is interesting is that in a dual equivalence one category is often algebraic and the other geometric, somehow is the dual of algebra!

Exercise 1.9. We will prove that the category Set is dually equivalent to the category CABool of complete atomic Boolean and homomorphisms.

A (L, ∧, ∨) is a a set L with two commutative, associative, binary operations ∧ and ∨ (called meet and join) such that for any a, b ∈ L:

• a ∧ a = a and a ∨ a = a (idempotence) • a ∧ (a ∨ b) = a = a ∨ (a ∧ b) (absorbtion)

A lattice homomorphism is a function f : L → M between two lattices that preserves the operations, i.e. for any a, b ∈ L, f(a ∧ b) = f(a) ∧ f(b) and f(a ∨ b) = f(a) ∨ f(b).

(1) Prove that all lattices and lattice homomorphisms form a category. What are the isomorphisms in this category? (2) You can define an order relation on L by:

a ≤ b ⇔ a ∧ b = a.

Prove that this order relation is a partial order (reflexive, transitive and antisymmetric) and that a ∧ b = inf{a, b} and a ∨ b = sup{a, b} with respect to ≤. (3) The lattice L is called • Distributive if for a, b, c ∈ L it is true that

a ∧ (b ∨ c) = (a ∧ b) ∨ (a ∧ c).

• Bounded if there are elements 0, 1 ∈ L such that 0 ≤ a and a ≤ 1 for all a ∈ L. • Complemented if L is bounded and for any a ∈ L there is an element ¬a ∈ L such that a ∧ ¬a = 0 and a ∨ ¬a = 1. A distributive, complemented lattice is called a . Usually it is denoted by (L, ∧, ∨, 0, 1, ¬).A Boolean homomorphism is a lattice homomorphism between two Boolean algebras that preserves 0 and 1. Prove that the following are true in the category of Boolean algebras: (a) Join distributes over meet, i.e. a ∨ (b ∧ c) = (a ∨ b) ∧ (a ∨ c). (b) Homomorphisms preserve ¬, i.e. f(¬a) = ¬f(a). Section 2. Natural Transformations Page 12

(c) The de Morgan’s laws hold

¬(a ∧ b) = (¬a) ∨ (¬b) and ¬(a ∨ b) = (¬a) ∧ (¬b).

(4) The lattice L is called • Complete if for any A ⊆ L the supremum sup A (also termed the join W A) with respect to ≤ exists. Prove that a complete lattice (a) has all possible infima (meets) as well as suprema (joins); (b) is necessarily bounded. (5) An element a of a (complete) lattice L is an atom of the lattice if a 6= 0 and for any b ∈ L it is true that b ≤ a ⇒ b = 0 or b = a. We say that L is atomic if every element of L is the join of all atoms below it. That is for any b ∈ L _ b = {a | a is an atom, a ≤ b}

Now prove that the category CABool of complete atomic Boolean algebras with com- plete homomorphisms (a complete homomorphism preserves all meets and joins) is dually equivalent to Set the and functions.

2. Natural Transformations

Proceeding in the spirit of category theory we now ask what “arrows between arrows” might be.

Definition 2.1. Let F,G : A → B be functors. A σ : F =⇒ G between

F and G assigns to each object A ∈ Ob A a B-morphism σA : FA → GA so that for any arrow f : A → B in A the square below commutes.

σA A FA / GA

f F f Gf    BFB / GB σB

We call σ a natural isomorphism if σA is an isomorphism for each A ∈ Ob A.

Because a natural transformation is an arrow between two functors, you will sometimes encounter the following notation to illustrate a natural transformation σ : F =⇒ G.

F $ A σ B  : G Section 3. Functor Categories and the Yoneda embedding Page 13

Exercises 2.2. (1) Let S be a fixed set. Denote by XS the set of all functions from S to X. Prove that (−)S : Set → Set is a functor. Also prove that (−) × S : Set → Set is a functor (where (X) × S is the X × S). The composition F = (−)S × S is thus a functor. Show that the evaluation map S eX : X × S → X given by eX (f, x) = f(x) for all x ∈ S gives rise to a natural

transformation from F to the identity functor 1Set : Set → Set. (2) Let F and G be two functors from a category C to a (P, ≤) (con- sidered as a category). Prove that there is a unique natural transformation α : F =⇒ G if and only if for any object C ∈ C F (C), ≤ G(C). (3) Show that H × (−): Grp → Grp is a functor (where H × (K) is the group prod- uct H × K) for any group H and that each morphism f : H → K defines a natural transformation φ : H × (−) =⇒ K × (−). (4) Let G and H be groups regarded as one-object categories. Let f, g : G → H be functors. Show that there is a natural transformation f =⇒ g if and only if f and g are conjugate; i.e, if and only if there is h ∈ H such that for all b ∈ G g(b) = hf(b)h−1. (5) Recall that a on a set X is a σ : X → X and a on X is a partial order ≤ such that either x ≤ y or y ≤ x for all x, y ∈ X. Let Sym(X) be

the set of on X (the underlying set of the group SX ) and Ord(X) the set of total orders on X. Let B be the category of finite sets with as morphisms. (a) Give a definition of Sym on morphisms of B so that Sym becomes a functor Sym : B → Set. Do the same for Ord. Both your definitions should be canonical (no arbitrary choices). (b) Show that there is no natural transformation Sym =⇒ Ord. [Hint: consider the identity permutation.] (c) If X has n-elements, how many elements do the sets Sym(X) and Ord(X) have? (d) Conclude that Sym(X) ∼= Ord(X) for all X ∈ B, but the isomorphism is not natural.

3. Functor Categories and the Yoneda embedding

Since natural transformations are arrows between functors, we can now define a where the objects are functors and the arrows are natural transformations. Of course we need to check that the composition of natural transformations is again a natural transformation and that we have identity natural transformations. (Do this now.) Section 4. Representable Functors Page 14

Definition 3.1. Let A and B be categories. Denote by [A, B] the category whose objects are functors A → B and whose arrows are natural transformations between these functors. This category is also denoted BA.

Exercise 3.2. Show that σ is an isomorphism in the functor category BA iff σ is a natural isomorphism.

The next two results show that remarkably any category can be represented as a functor category. (If you have seen Cayley’s Theorem in , then this might be a familiar idea.)

Lemma 3.3 (Yoneda). Let F : A → Set be a functor and A ∈ Ob A. Then there is a bijection

Y {σ | σ : homA(A, −) =⇒ F } / FA given by Y (σ) := σA(1A).

op Theorem 3.4. Any category A can be embedded into the functor category SetA . The embedding sends an A-morphism f to a natural transformation σf between the contravariant hom-functors

X homA(−,X) E f / σf   Y homA(−,Y ) The natural transformation σf is defined for each object A ∈ Aop by the function between f hom-sets σA : homA(A, X) → homA(A, Y ) which maps g 7→ f ◦ g.

Exercises 3.5. (1) Write out the details of the bijection and embedding in the above two results in the case when the category A is a group, and when the category A is a pre-ordered set.

4. Representable Functors

As the previous section shows, hom-functors play a central role in category theory. For this reason we are interested in those functors which, although they might not immediately seem to be, are in fact hom-functors into Set.

Definition 4.1. A functor F : C → Set is representable if it is naturally isomorphic to a hom-set functor homC(A, −): C → Set.

Exercises 4.2. (1) Are the following functors representable? Section 4. Representable Functors Page 15

• The “contravariant power set functor” P : Setop → Set ;

• The identity functor 1Set : Set → Set; • The forgetful functor U : Top → Set. ∼ ∼ 0 (2) Let F : C → Set be a functor. Show that if F = homC(−,A) and F = homC(−,A ) for A, A0 ∈ C, then A ∼= A0; that is, objects that represent the same functor are isomorphic.

Representable functors have a number of nice properties as we will see in later sections. They also help us to develop our intuition in category theory. For instance if we want to invent a good definition for an “injective” arrow in a category C, we can say that f : A → B is “injective” in

C if for any X in C, homC(X, f): homC(X,A) → homC(X,B) is an in Set. (What property of f does this give?)

Similarly we can ask what a “constant” arrow might be, or we could have used this approach to say that 1 is a terminal object in C if homC(X,A) is a one element set. 3. Special Morphisms

Seeing as arrows or morphisms are central in category theory, we are naturally interested in arrows with special properties. Isomorphisms were introduced in the first chapter. In this chapter we look briefly at a few other special morphism types.

1. Monomorphisms and Epimorphisms

As discussed at the end of the previous chapter, if we want to define an “injective” map in a general category, one option is to ask that when we“hom” the map into Set we get an injective function. This gives the following definition.

Definition 1.1. A morphism f : A → B in C is a if for any pair of morphisms u and v, f ◦ u = f ◦ v ⇒ u = v.

u f / X / A / B v Examples 1.2. (1) Any isomorphism is a monomorphism. (2) The monomorphisms in Set are the injective functions. (3) If C is a pre-ordered set considered as a category then every arrow in C is a monomor- phism. (4) In many categories of structured sets the monomorphisms are injective functions which are arrows in the category. For example, in Grp the monomorphisms are injective group homomorphisms, and in Top the monomorphisms are injective continuous maps.

Any categorical definition has a dual definition. Applying this principle to Definition 1.1 we get.

Definition 1.3. A morphism f : A → B in C is an if for any pair of morphisms u and v, u ◦ f = v ◦ f ⇒ u = v.

f u / A / B / X v Examples 1.4. (1) Any isomorphism is an epimorphism. (2) The epimorphisms in Set are the surjective functions. (3) If C is a pre-ordered set considered as a category then every arrow in C is an epimorphism. (4) As for monomorphisms, in many categories of structured sets the epimorphisms are surjective functions which are arrows in the category. This is again the case in Grp in Top. 16 Section 2. Split and extremal monomorphisms and epimorphisms Page 17

(5) In the category Ring of rings (with ) and ring homomorphisms, the inclusion Z → Q is an epimorphism. (6) In the category Met of metric spaces and continuous maps, the inclusion Q → R is an epimorphism.

As the examples suggest, as a general rule epimorphisms are more difficult to characterise than monomorphisms in the categories in which we often work. (Although of course an epimorphism in C is just a monomorphism in Cop.)

Proposition 1.5. (1) Faithful functors reflect monomorphisms. (2) Representable functors preserve monomorphisms.

Exercises 1.6. (1) Prove that the monomorphisms and epimorphisms in Set are injective and surjective functions as claimed. (2) Use your knowledge of the epimorphisms in Set and the Proposition above to prove that the monomorphisms and epimorphisms in Top are the injective and surjective continuous maps. (3) Show that if f is a monomorphism and an epimorphism in a category C then it need not be an isomorphism. (4) The example of Q → R being an epimorphism in Met is simply one instance of the more general fact that any dense continuous map is an epimorphism in Met. Try to prove this. (5) Can you dualise Proposition 1.5? Explain. (6) Use the equivalence between Setop and CABool to characterize the monomorphisms and epimorphisms in CABool.

2. Split and extremal monomorphisms and epimorphisms

Monomorphisms and epimorphisms provide one categorical possibility for describing “” and “quotients” in a general category C. As the examples show, however, the definitions are not really strong enough. For this reason there are a number of further definitions of special types of monomorphism and epimorphism. We mention just two.

Definition 2.1. A morphism f : A → B is called:

(1) a split monomorphism (or a section) if there is a morphism g : B → A such that g ◦f =

1A. Section 2. Split and extremal monomorphisms and epimorphisms Page 18

(2) an extremal monomorphism if f is a monomorphism and if f = g ◦ e where e is an epimorphism, then e is an isomorphism.

f A / B @@ ~? @@ ~~ e @@ ~~ g @ ~~ C (Another way of stating this definition is to say that f cannot be factored through an epimorphism, except trivially.)

Exercises 2.2. (1) Give the dual definitions to those in Definition 2.1, i.e. define a split epimorphism and an extremal epimorphism. (2) Prove that split monomorphisms are extremal monomorphisms. (What is the dual result? Do you need to prove it?) (3) Prove that in Set monomorphisms, split monomorphisms and extremal monomorphisms are the same, i.e. injective functions. (4) What are the extremal monomorphisms and extremal epimorphisms in Top? 4. Adjunctions

If asked what the most fruitful notion is within category theory, most category theorists would probably agree that it is that of adjoint functor. In this context Saunders MacLane, one of the founders of category theory, is remembered for his slogan, ”Adjoint functors arise everywhere”.

1. Galois connections

Before getting to the definition of adjoint functors, we give a special case, namely when the functors are between two partially ordered sets viewed as categories. (Remember that a functor in this case is just an order preserving map.)

Definition 1.1. Let (X, ≤) and (Y, ≤) be partially ordered sets and let f and g be order preserving functions.

f / X o Y g

We say that the pair (f, g) is a Galois connection between X and Y if for all x ∈ X and y ∈ Y ,

f(x) ≤ y ⇔ x ≤ g(y).

The map f is called the left Galois adjoint and g the right Galois adjoint.

Theorem 1.2. Let (f, g) be a Galois connection between partially ordered sets (X, ≤) and (Y, ≤). Then,

(1) x ≤ g(f(x)) for all x ∈ X, (2) f(g(y)) ≤ y for all y ∈ Y , (3) f preserves suprema, i.e. for A ⊆ X, f(sup A) = sup{f(a) | a ∈ A}, (4) g preserves infima, i.e. for B ⊆ Y , g(inf B) = inf{g(b) | b ∈ B}.

Remark 1.3. Conditions (1) and (2) in the theorem above together are equivalent to the fact that (f, g) is a Galois connection. Moreover, if the partially ordered sets are complete (that is if all suprema and infima exist in X and Y ) then if (3) holds we define g(y) := sup{x ∈ X | f(x) ≤ y} and (f, g) is a Galois connection. Dually (4) can be used to define f from g.

Exercises 1.4. (1) Prove the statements made in Remark 1.3 about Galois connections. 19 Section 2. Adjoint Functors Page 20

(2) Let f : X → Y be a function between two sets. Define the maps f(−): P(X) → P(Y ) and f −1(−): P(Y ) → P(X) by f(A) := {f(a) | a ∈ A} and f −1(B) := {x ∈ X | f(x) ∈ B} for A ⊆ X and B ⊆ Y . Prove that

f(−) / P(X) o P(Y ) f −1(−)

is a Galois connection. (Which is the left and which is the right Galois adjoint?)

2. Adjoint Functors

Galois connections are a special instance of a more general categorical situation. Order preserving maps between partially ordered sets are functors and in a partially ordered set a ≤ b iff there is an arrow a → b. Hence the statement

f(x) ≤ y ⇔ x ≤ g(y) is the same as saying that f(x) → y ⇔ x → g(y). Extending this to functors between general categories

F / A o B G we ask that there is a “natural” bijection between arrows FA → B and A → GB.

Definition 2.1. Let F and G be functors between two categories A and B,

F / A o B G

We say that F is left adjoint to G if there is a natural transformation η : 1A =⇒ GF such that for all A ∈ Ob A and for all arrows f : A → GB (where B ∈ Ob B) there exists a unique arrow ∗ ∗ f : FA → B such that (Gf ) ◦ ηA = f.

ηA A / GF A FA CC w w CC w w CC w w ∀f CC w Gf ∗ w ∃!f ∗ ! {w {w GB B

Remark 2.2. (1) If F is left adjoint to G, then G is right adjoint to F and we talk of the adjunction between A and B. We write F a G.

(2) The natural transformation η : 1A =⇒ GF in Definition 2.1 is called the unit of the adjunction. Section 2. Adjoint Functors Page 21

(3) The definition of the adjoint situation (the natural bijection between arrows FA → B and A → GB) as given in Definition 2.1 could equivalently be given by the following (dual) diagram:

εB Bo F GA GB aCC w; w; CC w w CC w w ∀f CC w F f ∗ w ∃!f ∗ w w FA A

The natural transformation ε : FG =⇒ 1B is called the co-unit of the adjunction.

Examples 2.3. (1) An equivalence between categories F : A → B is both a left and a right adjoint. (How do we define the other functor?) (2) A category A has a terminal object iff the constant functor K : A → 1, from A to the category with one object and one arrow, has a right adjoint. (What if K has a left adjoint?) (3) The free monoid functor F : Set → Mon (Exercises 1.3 in Chapter 2) is right adjoint to the forgetful functor U : Mon → Set. (4) The functor D : Set → Top which sends a function f : X → Y to the same map be- tween the topological spaces f :(X, P(X)) → (Y, P(Y )) is left adjoint to the forgetful functor U : Top → Set. Does U have a right adjoint?

Exercises 2.4. (1) Find examples of adjunctions that are not in the notes. (2) Show that left adjoints preserve initial objects; that is, if F : A → B is a left adjoint to a functor G : B → A and 0 is an initial object in A, then F 0 is an initial object in B. Dualize the statement. (3) Let H,K be groups considered as one object categories and f : H → K a group ho- momorphism. Prove that the following statements are equivalent: (a) f is an isomorphism; (b) f has a left adjoint; (c) f has a right adjoint. (4) Consider an order-preserving (monotone) function f : A → B between two partially or- dered sets as a functor and show that: (a) f is a right adjoint if and only if for each b ∈ B the set {a ∈ A | b ≤ f(a)} has a smallest element. (b) If A = B = N (with the usual order), then (i) f is a right adjoint if and only if it is unbounded. (ii) f is a left adjoint if and only if it is unbounded and f(0) = 0. (c) If A = B = Z (with the usual order), then f is a right adjoint if and only if it is a left adjoint. Section 2. Adjoint Functors Page 22

(5) Let X and Y be sets and f : X → Y a function. Show that the following operations are adjunctions: f(−): P(X) → P(Y ) with f(A) := {f(a) | a ∈ A} for any A ∈ P(X). and f −1(−): P(Y ) → P(X) with f −1(B) := {a ∈ X | f(a) ∈ B} for any B ∈ P(Y ). Replace the two sets X,Y by groups (resp. topological spaces), f : X → Y by a (resp. a continuous map) and P(X) by the set of all of X (resp. the set of all subspaces of X). Prove that the property of adjunctions still holds. (6) Show that: (a) if F,G : A → B are both left adjoints to a functor H : B → A then F ∼= G (nat- urally isomorphic). Dualize the statement.

(b) If F1 : A → B and F2 : B → C are right adjoints, then so is F2 ◦ F1 : A → C. (7) Let F a G be adjoint functors. Define the unit η and co-unit ε of the adjunction. Prove

the triangle identities, (εF ) ◦ (F η) = 1F and (Gε) ◦ (ηG) = 1G. (8) The diagonal functor ∆ : Set → Set2 is defined by ∆(A) = (A, A) for all sets A. (How is it defined on functions?) Find a left adjoint and a right adjoint to ∆. 5. Limits and Colimits

One of the beauties of category theory is the insight it gives of how many definitions and con- structions in diverse areas of mathematics are actually the same. This is most evident when studying limits.

1. Products

We are familiar with “products” of various mathematical objects. The cartesian product of sets, the of groups, the (Tychonoff) product of topological spaces are some examples. These products are all instances of a categorical product which we define by means of a universal property.

Definition 1.1 (Binary product). Let A and B be two objects in a category C. The product of A and B is an object A × B with two arrows A × B →πA A and A × B →πB B with the following universal property:

πA πB A o A × B / B bF < FF O xx FF xx FF !h xx p FF xx q F xx Q

p q For any object Q and arrows Q → A and Q → B there exists a unique h : Q → A × B such that the diagram above commutes.

The fact that we use a distinct notation for the binary product suggests that the product as defined in Definition 1.1 is unique ( isomorphism, of course). This is true of any categorical definition given by a universal property. Still we can prove it in this case.

Proposition 1.2. Binary products are . That is if A and B are C-objects and π0 π0 both A ←πA A×B →πB B and A ←A A⊗B →B B satisfy the universal property in Definition 1.1, then 0 0 there is a (unique) isomorphism h : A × B → A ⊗ B such that πA ◦ h = πA and πB ◦ h = πB.

Exercises 1.3. (1) Show that, as stated at the beginning of this section, the cartesian product of two sets, the direct product of two groups and the (Tychonoff) product of two topological spaces are all binary categorical products. (2) Let (X, ≤) be a partially ordered set viewed as a category. Show that the product x × y = inf{x, y} for x, y ∈ X. 23 Section 2. Pullbacks Page 24

Definition 1.1 can easily be extended from a binary product to a more general product.

Definition 1.4 (Product). Let {Ai | i ∈ I} be a set of C-objects. The product of {Ai}i∈I is a Q family of arrows {πi : j∈I Aj → Ai}i∈I such that for any other family {fi : Q → Ai}i∈I there exists a unique h such that the diagram below commutes for each i ∈ I.

Q πi j∈I Aj / Ai cF @ F ÐÐ F ÐÐ !h F ÐÐ f F ÐÐ i Q

As for the binary product, products are essentially unique. The examples and exercises above can easily be extended to general products.

2. Pullbacks

Alongside products one of the most commonly encountered universal constructions in category theory is a pullback. Whereas a (binary) product was defined as a product of objects, a pullback is a universal construction for two arrows.

Definition 2.1. A commutative square f ◦ g0 = g ◦ f 0 is a pullback square if for any other commutative square f ◦ u = g ◦ v there exists a unique h so that the diagram below commutes.

Q ? u ? h ? ? g0 # P / A

v f 0 f

   / B g C

We say that f 0 is the pullback of f along g, or symmetrically that g0 is the pullback of g along f.

As for products, pullbacks are unique (up to isomorphism). Thus we can speak of the pullback of a pair of arrows. Section 2. Pullbacks Page 25

Examples 2.2. (1) Let f : X → Z and g : Y → Z be functions in Set. If P = {(x, y) ∈ X × Y | f(x) = g(y)} and f 0 and g0 are defined by f 0(x, y) = y and g0(x, y) = x, then

g0 P / X

f 0 f   / Y g Z is a pullback square. (2) Let X be a set, and consider (P(X), ⊆) as a category. Then the square

P / A

  B / C is a pullback in (P(X), ⊆) iff P = A ∩ B.

These two examples highlight two possible ways of viewing a pullback, as a generalised product or as a generalised intersection.

Exercises 2.3. (1) Verify that in the category Pos the diagram below is a pullback if P =

{(x, y) ∈ X × Y : f(x) = g(y)} with order relation (x1, y1) ≤ (x2, y2) ⇔ x1 ≤ x2 0 0 and y1 ≤ y2, and g and f are the projection maps.

g0 P / X

f 0 f   / Y g Z (2) Let the square below be a pullback and 1 a terminal object in C. Prove that (P, (f 0, g0)) is the product of A and B in C.

f 0 P / A

g0 g   B / 1 f (3) Prove that f : A → B is a monomorphism iff the square below is a pullback.

1A A / A

1A f   A / B f Section 3. Equalizers Page 26

(4) Prove that if the square below is a pullback and if f is a monomorphism, then so is f 0. (We say that monomorphisms are stable under pullback.)

f 0 P / A

g0 g   B / C f

Do you expect epimorphisms to be stable under pullback? What about isomorphisms or split monomorphisms? Verify your conjectures.

3. Equalizers

The third universal construction we study is that of an equalizer. Together with products and pullbacks, equalizers are fundamental building blocks of universal constructions in category theory.

Definition 3.1. Let u, v : A → B. We say that e : E → A is the equalizer of u and v if u ◦ e = v ◦ e and for any f : Q → A such that u ◦ f = v ◦ f there exists a unique h for which e ◦ h = f. u e / E / A / B _? ? v ?  ?  ∃!h ?  ∀f Q

Example 3.2. In Set, the equalizer of two functions u, v : A → B is given by E = {a ∈ A | u(a) = v(a)} with e : E → A given by the .

In many categories of structured sets equalizers are constructed in this way, with an appropriate structure on them.

Exercises 3.3. (1) Prove that if e is the equalizer of u and v then e is a monomorphism. (A monomorphism which is an equalizer is called a regular monomorphism.) (2) Prove that any regular monomorphism is also an extremal monomorphism. (3) Prove that regular monomorphisms are stable under pullback. (4) You can construct a pullback

f 0 P / A

g0 g   B / C f Section 4. Limits and Colimits Page 27

by first taking the product of A and B and then an equalizer. What is the equalizer of? Prove that you get a pullback square. (5) You can also use a product and a pullback to construct the equalizer of any pair of arrows u, v : A → B. Try to do this.

4. Limits and Colimits

4.1. Limits. The universal constructions of products, pullbacks and equalizers are all exam- ples of categorical limits. In each case we have an incomplete diagram which we complete in a universal way. This can be formalised in the following definitions.

Definition 4.1. A diagram in a category C is a functor D : I → C where the category I has only a set of objects (we call I a small category).

We write i ∈ I for the objects of I and t : i → j for the arrows in I.

Definition 4.2. Let D : I → C be a diagram in C. The family {fi : L → Di}i∈I is a of D

in C if (Dt) ◦ fi = fj for any t : i → j in I, and if there is any other family {gi : Q → Di}i∈I for

which (Dt)◦gi = gj for any t : i → j in I, then there is a unique h : Q → L such that fi ◦h = gi for all i ∈ I.

fi gi fi L / Di and Q / Di L / Di @@ @@ +3 O ~? @@ @@ ~~ @@ Dt @@ Dt !h ~ f @ gj @ ~~ gi j @  @  ~~ Dj Dj Q

Exercises 4.3. (1) Verify that products, pullbacks and equalizers are all limits by identifying the appropriate diagram in each case. (2) What is an empty limit, if it exists? i.e. If D : ∅ → C is the (unique) empty functor from the empty category to a category C, what is the limit of D? (3) Let D : I → C be a diagram. Define a new category D∗ as follows: Objects are sets

of C-arrows {fi : A → Di}i∈I and arrows between two such sets {fi : A → Di}i∈I and

{gi : B → Di}i∈I are C-arrows α : A → B such that gi · α = fi for each i ∈ I. (a) Verify that D∗ is a category. (b) Prove that if the limit of D exists then it is a terminal object in D∗. (c) Conclude from this that limits for diagrams, if they exist, are essentially unique.

In Exercises 3.3 we saw that that you can use products and equalizers to construct pullbacks. This is a special case of the following theorem. Section 5. Functors and Limits Page 28

Theorem 4.4. The limit of any diagram D : I → C can be constructed using products and equalizers.

For this reason we will often ask whether a category C has products and equalizers, meaning that in C it is always possible to find the product of a set of objects and the equalizer of two parallel arrows. If a category has products and equalizers then it has all limits. Such categories are called complete categories and are nice because they allow for universal constructions of the type discussed in this chapter.

4.2. Colimits. All the work done in this chapter can be dualised. If we do this then we would define co-products, pushouts (and regular epimorphisms), co-equalizers and in general colimits.

We will not spend time investigating colimits in these notes but they are very useful constructions and so you are encouraged to look for examples and the dual of results proved about limits.

5. Functors and Limits

In algebra the morphisms are chosen so as to preserve the algebraic operations. In analysis the morphisms are chosen to preserve limits. In the case of categories we chose our arrows between categories, i.e. the functors, to preserve the algebraic structure of categories. However, seeing as we now also have limits in categories it is likely that functors which preserve them will be interesting.

In the chapter on adjoints we noted in Theorem 1.2 that right Galois adjoints preserve infima. We also saw that in a partially ordered set viewed as a category a product is an infimum. This is a special case of the following result.

Theorem 5.1. Let G : B → A be right adjoint to F : A → B then G preserves limits. That is if {fi : L → Di}i∈I is the limit of a diagram D : I → B in B, then {Gfi : GL → GDi}i∈I is the limit of the diagram GD : I → A in A.

The result is in fact even stronger in that preservation of limits characterises adjoint functors. A functor is a right adjoint iff it preserves limits and dually a functor is a left adjoint iff it preserves colimits. However there needs to be a technical restriction on the categories involved, the so- called solution set condition, which ensures that when calculating limits we only have a set of arrows between certain objects. Section 5. Functors and Limits Page 29

Exercises 5.2. (1) Prove that hom-functors preserve all limits. (And thus that any repre- sentable functor preserves limits.) (2) Use the characterisation of monomorphisms in terms of pullbacks and the exercise above to draw a conclusion about representable functors and monomorphisms. (This was proved by a different argument earlier.) 6. Subcategories

1. Subcategories

Definition 1.1. A category A is a of a category B if:

(1) Ob A ⊆ Ob B,

(2) for each A and B in A, homA(A, B) ⊆ homB(A, B), and (3) identities and composition in A are as in B.

If homA(A, B) = homB(A, B) for all A and B in A, then we say that A is a full subcategory of B.

f f If A is a subcategory of B then the inclusion functor I : A → B which maps A → B to A → B is an embedding. A is a full subcategory iff I is a full functor.

Examples 1.2. In most instances where our category is a category of structured sets the sub- categories we consider are full subcategories. For instance AbGrp as a subcategory of Grp or CompHaus as a subcategory of Top. There are also cases where the subcategories might not be

full, for instance Metu the category of metric spaces and uniformly continuous maps inside Metc the category of metric spaces and continuous maps. Bool the category of Boolean algebras and boolean homomorphisms is a non-full subcategory of Pos.

We need to be careful when working with non-full subcategories as universal constructions in the subcategory and supercategory will in general be different.

Exercise 1.3. Let F : A → B and G : B → A be an adjunction with unit η : 1A =⇒ GF and

co-unit ε : FG =⇒ 1B. Let Fix(GF ) be the full subcategory of A whose objects are those

A ∈ A for which ηA is an isomorphism, and dually Fix(FG) ⊆ B.

Prove that the adjunction (F, G, η, ε) restricts to an equivalence (F 0,G0, η0, ε0) between Fix(GF ) and Fix(FG).

In this way, any adjunction restricts to an equivalence between full subcategories. Take some examples of adjunctions and work out what this equivalence is.

30 Section 2. Reflective Subcategories Page 31

2. Reflective Subcategories

Often a subcategory represents a property, for instance AbGrp is the subcategory of Grp which represents the property that the group operation is commutative. In many instances this “prop- erty” is well behaved and then the subcategory has nice categorical properties relative to the supercategory.

Definition 2.1. A is a reflective subcategory of B if the inclusion functor I : A → B has a left adjoint.

The fact that the inclusion functor I acts as an identity means that we can simplify the adjunction ∗ diagrams to say that A is reflective in B if for all B ∈ Ob B there is an arrow ηB : B → B where B∗ ∈ Ob A and for all f : B → A where A ∈ Ob A there exists a unique f ∗ : B∗ → A in A so ∗ that f ◦ ηB = f.

ηB B / B∗ @ @@ } @@ } f @ } f ∗ @ ~} A

The left adjoint to I is called a reflector, while the unit map from the adjunction ηB : B → IRB is called the B-reflection arrow.

Exercises 2.2. (1) Prove that if A is a reflective subcategory of B, then any two reflectors F,G : B → A for A are naturally isomorphic.

(2) Denote by Metu the category of metric spaces with uniformly continuous functions and

by CMetc the category of complete metric spaces with uniformly continuous functions.

Prove that CMetc is a full subcategory of Metu. Recall that for any (X, d) there is a X∗ such ∗ ∗ that X is dense in X . Show that this gives a functor (−) : Metc → CMetc.

Show that CMetc is reflective in CMetc. (3) Show that a full subcategory A of a partially ordered set B, considered as category, is reflective in B if and only if for each element b ∈ B the set {a ∈ A | b ≤ a} has a smallest element. (4) Consider the partially ordered set of natural numbers N as a category. Verify that a A of N, considered as a full subcategory of N, is (a) reflective in N if and only if A is infinite; (b) coreflective in N if and only if 0 ∈ A. Section 2. Reflective Subcategories Page 32

Conclude that the intersection of any nonempty family of coreflective subcategories of N is corefelective in N and that every full subcategory of N is an intersection of two reflective full subcategories of N. (5) Show that no finite monoid, considered as a category, has a proper reflective subcategory. Let A be the monoid of all maps from N into N. Consider A as a category. Show that the subcategory of A consisting of all maps f : N → N with f(0) = 0, is reflective in A. (6) Let A be a subcategory of B and B be a subcategory of C. Prove that: (a) A is a subcategory of C; (b) If A is reflective in B and B is reflective in C, then A is reflective in C; (c) If A is reflective in C and B is a full subcategory of C, then A is reflective in B; (d) If A is reflective in C, then A need not be reflective in B. (Give a counterexample.)