The Freyd's Adjoint Functor Theorem

Home , Adjoint functors, Category (mathematics), Comma category, Complete category, Coproduct, Diagonal functor

Universitédu Québec àMontréal Département de mathématiques Marco A. PérezB. [email protected]

THE FREYD’S ADJOINT FUNCTOR THEOREM

January 2011

Contents

Introduction i

1 Universal Arrows 1 1.1 Comma categories and universal arrows ...... 1 1.2 Limits and colimits ...... 3

2 Limits and Adjunction 11 2.1 Adjoint functors ...... 11 2.2 Preservation of limits ...... 17 2.3 The Freyd’s Adjoint Functor Theroem ...... 20

Bibliography 25

Introduction

Adjoint functors are pairs of functors, G : D −→ C and F : C −→ D, which stand in a particular relationship with one another, called an adjunction. Speciﬁcally, (F,G) is an adjunction if there exists a natural isomorphism θ : HomD(F (·), ·) −→ HomC(·,G(·)). Under these conditions, G is said to have a left adjoint, namely F . Of course, not every functor admits a left adjoint. For instance, if X is a set having two or more elements, then the functor X × − : Set −→ Set does not have a left adjoint. A natural question that comes to us is under which conditions it is possible to determine the existence of a left adjoint F for a given functor G. Peter Freyd answered that question, for the particular case where the category D is complete. If D is a complete category, then the functors with left adjoints can be characterized by the adjoint functor theorem:

Theorem. Given a small and complete category D, a functor G : D −→ C has a left adjoint if and only if it preserves all limits and satisﬁes the following Solution Set Condition. For each object C ∈ Ob(C) there is a set I and an I-indexed family of arrows fi : C −→ G(Di) such that every arrow h : C −→ G(D) can be written as a composite h = G(t) ◦ fi for some index i and some t : Di −→ D.

In these notes we shall give a prove of this result due to Saunders Mac Lane. First, we study universal arrows and characterize them as initial objects in certain comma categories. Then we study the concept of a limit for a functor F : J −→ C. Limits give rise to a special type of categories known as complete categories. In these categories there exists a lot of universal constructions, such as products, equalizers and pullbacks. We begin the last chapter recalling the notion of adjunction. Adjoint functors have a deep relation with limits, in fact, every functor having a left adjoint preserves limits. Before proving the Freyd’s theorem, we ﬁrst study the case of the existence of an initial object in a category and then use the fact that each universal arrow deﬁned by the unit of a left adjoint is an initial object in a suitable comma category.

i ii Chapter 1

Universal Arrows

1.1 Comma categories and universal arrows

A comma category is a construction in category theory, introduced in 1963 by F. W. Lawvere, which provides another way of looking at morphisms: instead of simply relating objects of a category to one another, morphisms become objects in their own right. We shall see there are certain guarantees about the existence of limits and colimits in the context of comma categories. Let F : C −→ D be a functor and D ∈ Ob(D). We deﬁne (D ↓ F ) the comma category of objects F -under D as follows:

(1) the objects of (D ↓ F ) are all pairs (u, C) where C ∈ Ob(C) and u : D −→ F (C) is an arrow of D;

(2) the arrows of (D ↓ F ), h :(u1,C1) −→ (u2,C2), are arrows h : C1 −→ C2 of C making the following triangle commute:

D E yy EE u1 yy EE u2 yy EE yy EE |yy E" F (C1) / F (C2) F (h)

Similarly, the comma category (F ↓ D) of objects F -over D is deﬁned as follows:

(1) the objects of (F ↓ D) are all pairs (u, C) where C ∈ Ob(C) and u : F (C) −→ D is an arrow of D;

1 (2) the arrows of (F ↓ D), h :(u1,C1) −→ (u2,C2), are arrows h : C1 −→ C2 of C such that the triangle

D < bE yy EE u1 yy EE u2 yy EE yy EE yy E F (C1) / F (C2) F (h) commutes.

Example 1.1.1. Let U : Grp −→ Set be the forgetful functor and X a set. An object of (X ↓ U) is a function X −→ U(G) from X into the underlying set of G, where G is a group.

If G : D −→ C is a functor and C ∈ Ob(C), a universal arrow from C to G is a pair (D, µ) consisting of an object D of D and an arrow µ : C −→ G(D) of C, such that for every arrow f : C −→ G(D0) of C there is a unique arrow h : D −→ D0 of D such that the diagram

µ D C / G(D) C CC CC CC ∃!h CC G(h) f CC CC C! D0 G(D0) commutes. The arrow µ is also called G-free. Similarly, a universal arrow from G to C is a pair (D, ν) where D is an object of D and ν : G(D) −→ C is an arrow of C, such that for every arrow f : G(D0) −→ C of C there exists a unique arrow h : D0 −→ D of D such that the diagram ν D G(D) / C O = O {{ {{ {{ ∃!h G(h) {{ {{ f {{ {{ D0 G(D0) commutes. The arrow ν is also called G-cofree.

Proposition 1.1.1.

(1) If (D1, µ1) and (D2, µ2) are universal arrows from C to G, then D1 and D2 are isomor- phic.

(2) If (D1, ν1) and (D2, ν2) are universal arrows from G to C, then D1 and D2 are isomor- phic.

2 Proof: We only prove (1). Part (2) can be proven in a similar way.

Since (D1, µ1) is a universal arrow from C to G, there exists a unique arrow h : D1 −→ D2 0 of D such that G(h) ◦ µ1 = µ2. Similarly, there exists a unique arrow h : D2 −→ D1 such 0 that G(h ) ◦ µ2 = µ1. So we get

0 G(h ) ◦ (G(h) ◦ µ1) = µ1 0 (G(h ) ◦ (G(h)) ◦ µ1 = µ1 0 (G(h ◦ h)) ◦ µ1 = µ1.

On the other hand, idD1 is the only arrow of D satisfying G(idD1 ) ◦ µ1 = µ1. Hence 0 0 h ◦ h = idD1 . Similarly, h ◦ h = idD2 . Therefore, h : D1 −→ D2 is an isomorphism.

From the deﬁnitions of comma categories and universal arrows, the following proposition is immediate.

Proposition 1.1.2.

(1) µ : C −→ G(D) is a universal arrow from C to G if and only if (D, µ) is an initial object in the comma category (C ↓ G).

(2) ν : G(D) −→ C is a universal arrow from G to C if and only if (D, ν) is a terminal object in the comma category (G ↓ C).

1.2 Limits and colimits

The abstract notion of a limit captures the essential properties of universal constructions such as products, equalizers and pullbacks. The dual notion of a colimit generalizes constructions such as disjoint unions, direct sums, coproducts and pushouts. Let J be a small category, i.e., Ob(J ) and Hom(J ) are sets. Let C be a category. Consider the category [J , C] whose objects are all functors F : J −→ C and whose arrows are all natural transformations α : F −→ G. Deﬁne a functor ∆ : C −→ [J , C] as follows:

(1) ∆(C): J −→ C is the constant functor

∆(i) = C for every i ∈ Ob(J ), and

∆(a) = idC for every arrow a : i −→ j of J ;

3 (2) for every arrow f : C1 −→ C2 of C, ∆(f) : ∆(C1) −→ ∆(C2) is the natural transformation deﬁned as the J -indexed set

∆(f) = {∆(f)i = f : C1 −→ C2}i∈Ob(J ).

The functor ∆ is known as the diagonal functor.A limit for a functor F : J −→ C is a universal arrow (L, ν) from ∆ to F . In this situation, ν : ∆(L) −→ F is a natural transformation, i.e., a J -indexed family of arrows νi : L −→ F (i) of C, with i ∈ Ob(J ), such that the triangle νi L / F (i) ?? ?? ?? ?? F (a) νj ?? ?? ? F (j) commutes, for every arrow a : i −→ j of J . Moreover, if β : ∆(L0) −→ F is another natural 0 transformation, i.e., a J -indexed family of arrows (βi : L −→ F (i))i∈Ob(J ) of C satisfying 0 F (a) ◦ βi = βj for every arrow a : i −→ j, then there exists a unique arrow h : L −→ L of C such that β = ν ◦ ∆(h), i.e., βi = νi ◦ h for every i ∈ Ob(J ). In pictures, we have the commutative diagram L0

∃!h

β βi L j < ÓÓ << ÓÓ << νi ÓÓ << νj ÓÓ << ÓÓ << ÓÓ << ÑÓÓ < F (i) / F (j) F (a) By Proposition 1.1.1, the object L is unique up to isomorphisms.

Example 1.2.1.

(1) Terminal objects: Let J be the empty category. Then any F : J −→ C is the empty functor. Thus, every pair (L, ν : ∆(L) −→ F ) is just the object L. If the previous pair is the limit for F , then for any other pair (L0, β : ∆(L0) −→ F ), i.e., any other object L0, there exists a unique arrow h : L0 −→ L of C. Hence, L is a terminal object in C.

(2) Products: Let J = I be a discrete small category, i.e., a small category whose only arrows are the identity arrows. A functor F : I −→ C is just a I-indexed set of objects {Ci = F (i)}i∈I . Suppose (P, ρ : ∆(P ) −→ F ) is the limit for F , then ρ is just a family 0 of arrows ρi : P −→ Ci of C. If (βi : P −→ Ci)i∈I is another family of arrows of C, i.e., a

4 natural transformation β : ∆(P 0) −→ F , then there exists a unique arrow h : P 0 −→ P of C such that ρi ◦h = βi, for every i ∈ I. In pictures, we have the commutative triangle

ρi P / Ci O ~> ~~ ∃!h ~~ ~~β ~~ i ~~ P 0

In this case, (P, (ρi : P −→ Ci)i∈I ) is called the product of the family {Ci}i∈I and is Q denoted P = i Ci. The arrows ρi are called projections. (3) Equalizers: Let J = {1, 2} be a two-object category having two parallel arrows a, b : 1 −→ 2. Let X = F (1), Y = F (2), f = F (a) and g = F (b). Suppose there exists the limit (K, ν : ∆(K) −→ F ) for the functor F , which is just the pair of arrows {f, g}. Then ν is just a pair of arrows u = ν1 : K −→ X and ν2 : K −→ Y satisfying 0 f ◦ u = g ◦ u = ν2. Moreover, if v : K −→ X is another arrow satisfying f ◦ v = g ◦ v, then there exists a unique arrow h : K0 −→ K such that v = u ◦ h. In pictures, we have the commutative diagram

u f K / X / Y O < g yy yy ∃!h yy yyv yy yy K0

In this case, the arrow u : K −→ X is called the equalizer of f and g.

Dually, given a small category J , a colimit for a functor F : J −→ C is a universal arrow (C, µ) from F to ∆, i.e., a family of arrows µi : F (i) −→ C of C such that the µi = µj ◦ F (a), 0 for every arrow a : i −→ j of J ; and if (αi : F (i) −→ C )i∈Ob(J ) is another family of arrows of C satisfying αj ◦ F (a) = αi for every arrow a : i −→ j, then there exists a unique arrow 0 h : C −→ C of C such that αi = h ◦ µi. In pictures, we have the commutative diagram

F (a) F (i) / F (j) FF x FF xx FF xx µ FF xxµ i FF xx j FF xx F xx " C { αi αj

∃!h Ð C0 By Proposition 1.1.1, the object C is unique up to isomorphisms.

5 Example 1.2.2.

(1) Initial objects: If F : J −→ C is the empty functor, then the colimit for F is an initial object in C.

(2) Coproducts: The coproduct of a family {Ci}i∈I of objects of C, where I is a set, is the colimit for the functor F : I −→ C, where the set I can be considered as a discrete category, satisfying F (i) = Ci for each i ∈ I. Speciﬁcally, it is an object C of C and an 0 I-indexed family of arrows ιi : Ci −→ C such that if αi : Ci −→ C is another I-indexed 0 family of arrows then there exists a unique arrow h : C −→ C such that αi = h ◦ ιi, for every i ∈ I. In pictures, we have the commutative diagram

ιi Ci / C AA AA AA ∃!h αi AA AA A C0 ` The object C is denoted C = i Ci and the arrows ιi are called inclusions. (3) Coequalizers: The coequalizer of two parallel arrows f, g : X −→ Y of C is the colimit of the functor F : J −→ C, where J = {1, 2} is a two-object category with two parallel arrows a, b : 1 −→ 2, satisfying X = F (1), Y = F (2), f = F (a) and g = F (b). It is an arrow u : Y −→ K such that u ◦ f = u ◦ g, and if v : Y −→ K0 is another arrow of C satisfying v ◦ f = v ◦ g then there exists a unique arrow h : K −→ K0 of C such that v = h ◦ u. In pictures, we have the commutative diagram

f u X / Y / K g / D DD DD DD ∃!h v DD DD D" K0

A category C is said to be complete if every functor F : J −→ C has a limit in C, where J is a small category. Dually, C is cocomplete if every functor F : J −→ C has a colimit in C.

Theorem 1.2.1. A category C is complete if and only if it has products and equalizers. Dually, C is cocomplete if and only if it has coproducts and coequalizers.

6 Proof: By Example 1.2.1, if C is complete then it has products and equalizers. Now suppose that C is a category in which there exist products and equalizers. Let J be a small category and F : J −→ C a functor. For every arrow a of J , we denote o(a) the domain of a and t(a) the codomain of a. Since Ob(J ) and Hom(J ) are sets, we can take products   Y P = F (i), (pi : P −→ F (i))i∈Ob(J ) , i∈Ob(J )   Y Q = F (t(a)), (qi : Q −→ F (t(a)))a∈Hom(J ) . a∈Hom(J )

Let fa = pt(a) : P −→ F (t(a)) for each a ∈ Hom(J ). Since the arrows qa are universals, we have that there exists a unique arrow f : P −→ Q such that the triangle

qa Q / F (t(a)) O < xx xx f xx xx fa xx xx P commutes. Similarly, there exists a unique arrow g : P −→ Q and a commutative triangle

qa Q / F (t(a)) O < xx xx g xx xx ga xx xx P

po(a) F (a) where ga is the composite arrow P −→ F (o(a)) −→ F (t(a)). Let u : L −→ P be the equalizer of f and g. We shall prove that (L, (pi ◦ u : L −→ F (i))i∈Ob(J )) is the limit of F : J −→ C.

(1) Let a : i −→ j be a morphism of J . We have

F (a) ◦ (pi ◦ u) = (F (a) ◦ pi) ◦ u = ga ◦ u = (qa ◦ g) ◦ u = qa ◦ (g ◦ u)

= qa ◦ (f ◦ u) = (qa ◦ f) ◦ u = fa ◦ u = pj ◦ u, i.e., the following triangle commutes

pi◦u L / F (i) A AA AA AA F (a) pj ◦u AA AA A F (j)

7 0 (2) Let (hi : L −→ F (i))i∈Ob(J ) be another family of arrows of C satisfying F (a) ◦ hi = hj, for each arrow a : i −→ j of J . Since the arrows pi are universal, there exists a unique arrow v : L0 −→ P of C such that the triangle

pi P / F (i) O |= || v || ||h || i || L0 commutes. On the other hand,

qa ◦ (f ◦ v) = (qa ◦ f) ◦ v = fa ◦ v = pt(a) ◦ v = ht(a)

qa ◦ (g ◦ v) = (qa ◦ g) ◦ v = ga ◦ v = (F (a) ◦ po(a)) ◦ v

= F (a) ◦ (po(a) ◦ v) = F (a) ◦ ho(a) = ht(a). We have that the triangle qa Q / F (t(a)) O < xx xx f◦v g◦v xx xxht(a) xx xx L0

commutes. But ht(a) factors uniquely through qa, so we get f ◦ v = g ◦ v. Since the arrow u is universal, there exists a unique arrow h : L0 −→ L such that the diagram

u f L / P // Q O = g {{ {{ h {{ {{v {{ {{ L0

commutes. Moreover (pi ◦ u) ◦ h = pi ◦ (u ◦ h) = pi ◦ v = hi, i.e. the triangle

pi◦u L / F (i) O |= || h || ||h || i || L0

commutes. It is only left to show that h is the only arrow of C satisfying hi = 0 0 (pi ◦ u) ◦ h, for every i ∈ Ob(J ). Suppose there exists another arrow h : L −→ L 0 such that (pi ◦ u) ◦ h = hi. Then we have the following commutative triangle

pi P / F (i) O |= || u◦h0 v || ||h || i || L0

8 0 It follows v = u ◦ h , since hi factors uniquely through pi. Similarly, from the commutative diagram u f L / P // Q O = g {{ {{ h h0 {{ {{v {{ {{ L0 we can conclude that h = h0.

Example 1.2.3. By the previous theorem, one can get some complete and cocomplete categories, for example Set, Grp and VctK.

9 10 Chapter 2

Limits and Adjunction

2.1 Adjoint functors

Let F : C −→ D and G : D −→ C be functors. Consider the product category Cop × D. If we assume that C and D are small, then HomD(FX,Y ) and HomC(X, GY ) are objects of Set, op 0 0 for every pair (X,Y ) ∈ Ob(C × D). If f ∈ HomCop (X,X ) and g ∈ HomD(Y,Y ), then we 0 deﬁne the map HomD(F (f), g) : HomD(FX,Y ) −→ HomD(FX ,Y ) as the function

HomD(F (f), g)(h) = g ◦ h ◦ F (f), for every arrow h : FX −→ Y of D.

HomD(F (f),g)(h) FX0 / Y 0 O

F (f) g

/ FX h Y

0 0 Similarly, we can deﬁne a function HomC(f, G(g)) : HomC(X, GY ) −→ HomC(X , GY ) by setting HomC(f, G(g))(h) = G(g) ◦ h ◦ f, for every arrow h : X −→ GY of C.

HomC(f,G(g))(h) X0 / GY 0 O

f G(g)

/ X h GY

11 op Proposition 2.1.1. HomD(F (·), ·) and HomC(·,G(·)) are functors from C × D to Set.

Proof: We only show that HomD(F (·), ·) is a functor. Let f1 ∈ HomCop (X1,X2), f2 ∈ HomCop (X2,X3), g1 ∈ HomD(Y1,Y2) and g2 ∈ HomD(Y2,Y3). Consider h : FX1 −→ Y1 an arrow of D. We have

HomD(F (f2 ◦ f1), g2 ◦ g1)(h) = (g2 ◦ g1) ◦ h ◦ F (f2 ◦ f1)

= (g2 ◦ g1) ◦ h ◦ (F (f1) ◦ F (f2))

= g2 ◦ (g1 ◦ h ◦ F (f1)) ◦ F (f2)

= g2 ◦ HomD(F (f1), g1)(h) ◦ F (f2)

= HomD(F (f2), g2)(HomD(F (f1), g1)(h)).

Hence HomD(F (f2◦f1), g2◦g1) = HomD(F (f2), g2)◦HomD(F (f1), g1). Now let (idX , idY ): (X,Y ) −→ (X,Y ) be the identity arrow of the object (X,Y ) of Cop×D. Let h : FX −→ Y be an arrow of D. We have

HomD(F (idX ), idY )(h) = idY ◦ h ◦ F (idX ) = h ◦ idFX = h.

So we get HomD(F (idX ), idY ) = idHomD(F X,Y ).

An adjunction from C to D is a triple (F, G, θ), where F : C −→ D and G : D −→ C are functors, and θ : HomD(F (·), ·) −→ HomC(·,G(·)) is a natural isomorphism. The functor F is said to be the left adjoint of G, and G the right adjoint of F .

Example 2.1.1. Consider the categories Set and VctK. Let U : VctK −→ Set be the forgetful functor. Deﬁne a functor V : Set −→ VctK as follows:

(1) V (X) is the vector space with basis X, i.e., the vectors of V (X) are formal ﬁnite linear combinations with scalar coeﬃcients;

(2) if f : X −→ Y is a function, then V (f): V (X) −→ V (Y ) is the map given by V (f)(x) = f(x) at each basis element, and extended by linearity.

Each function g : X −→ U(W ) extends to a unique linear map f : V (X) −→ W , given P P by f( rixi) = ri(g(xi)). This correspondence g 7→ f has an inverse θ : f 7→ f|X , the restriction of f to X. Hence, we get a bijection

=∼ θX,W : VctK(V (X),W ) −→ Set(X,U(W )).

12 It is easy to check that θ deﬁnes a natural isomorphism. Therefore, (V, U, θ) is an adjunction from Set to VctK. For every pair (X,Y ) ∈ Ob(Cop × D), we have a function

θXY : HomD(FX,Y ) −→ HomC(X, GY ). Setting Y = FX we get the following arrow of C:

ηX := θX,F X (idFX ): X −→ GF X. Similarly, if we set X = GY then we get the following arrow of D: −1 Y := θGY,Y (idGY ): F GY −→ Y

Proposition 2.1.2. η : IdC −→ GF and : FG −→ IdD deﬁne natural transformations.

Proof: We only give a proof for η. A similar argument can be used for . Given f : X −→ X0 an arrow of C, we must verify that the square

ηX X / GF X

f GF (f) X0 / GF X0 ηX0

0 op commutes. Consider the arrow (idX ,F (f)) : (X,FX) −→ (X,FX ) of C × D. Then we have functions

0 HomD(idFX ,F (f)) : HomD(FX,FX) −→ HomD(FX,FX )

0 HomC(idX , GF (f)) : HomC(X, GF X) −→ HomC(X, GF X ). Since θ is a natural transformation, we have the commutative square

θX,F X HomD(FX,FX) / HomC(X, GF X)

HomD(idFX ,F (f)) HomC(idX ,GF (f))

0 0 HomD(FX,FX ) / HomC(X, GF X ) θX,F X0

It follows

HomC(idX , GF (f)) ◦ θX,F X (idFX ) = θX,F X0 ◦ HomD(idFX ,F (f))(idFX )

GF (f) ◦ ηX ◦ idX = θX,F X0 (F (f) ◦ idFX ◦ idFX )

GF (f) ◦ ηX = θX,F X0 (F (f)).

13 0 0 0 op Now consider the arrow (f, idFX0 ):(X ,FX ) −→ (X,FX ) of C × D. Then we have the commutative square

θ 0 0 X0,F X0 0 0 HomD(FX ,FX ) / HomC(X , GF X )

HomD(F (f),idFX0 ) HomC(f,idGF X0 )

0 0 HomD(FX,FX ) / HomC(X, GF X ) θX,F X0

It follows

HomC(f, idGF X0 ) ◦ θX0,F X0 (idFX0 ) = θX,F X0 ◦ HomD(F (f), idFX0 )(idFX0 )

idGF X0 ◦ ηX0 ◦ f = θX,F X0 (idFX0 ◦ idFX0 ◦ F (f))

ηX0 ◦ f = θX,F X0 (F (f)).

Hence GF (f) ◦ ηX = θX,F X0 (F (f)) = ηX0 ◦ f.

We can say more about η and . In fact, they are universal arrows. Before stating this fact in detail, we need to prove ﬁrst the following:

Lemma 2.1.1. For a functor G : D −→ C, a pair (D, µ : C −→ G(D)) is a universal arrow from C to G if and only if the function sending each arrow f : D −→ D0 of D into 0 0 0 G(f) ◦ µ : C −→ G(D ) is a bijection from HomD(D,D ) to HomC(C,G(D )).

0 0 Proof: The function ϕ : HomD(D,D ) −→ HomC(C,G(D )) given by ϕ(f) = G(f) ◦ µ is a bijection if and only if for every arrow g : C −→ G(D0) of C there exists a unique arrow f : D −→ D0 of D such that g = ϕ(f), i.e., if and only if the triangle

µ C / G(D) C CC CC CC G(f) g CC CC C! G(D0)

commutes for one and only one arrow f : D −→ D0 of D.

14 Proposition 2.1.3.

(1) For each C ∈ Ob(C), ηC : C −→ GF (C) is a universal arrow from C to G.

(2) For each D ∈ Ob(D), D : FG(D) −→ D is a universal arrow from F to D.

Proof: We only prove (1). By the previous lemma, we only need to show that the function ϕ : HomD(F (C),D) −→ HomC(C,G(D)) given by ϕ(f) = G(f) ◦ ηC is a bijection. Let f : F (C) −→ D be an arrow of D. Since the square

θC,F C HomD(FC,FC) / HomC(C, GF C)

HomD(F (f),idFC ) HomC(idC ,G(f)) HomD(FC,D) / HomC(C, GD) θC,D

commutes, we have that

HomC(idC ,G(f)) ◦ θC,F C (idFC ) = θC,D ◦ HomD(idFC , f)(idFC )

G(f) ◦ ηC ◦ idC = θC,D(f ◦ idFC ◦ idFC )

G(f) ◦ ηC = θC,D(f)

ϕ(f) = θC,D(f).

Notice that θ is a natural isomorphism and hence ϕ = θC,D is a bijection.

Theorem 2.1.1 (Pointwise Construction of Adjoints).

(1) A functor G : D −→ C has a left adjoint if and only if for each C ∈ Ob(C) there is an object F0(C) of D and a universal arrow ηC : C −→ G(F0(C)) from C to G. (2) A functor F : C −→ D has a right adjoint if and only if for each D ∈ Ob(D) there is an object G0(D) of C and a universal arrow D : F (G0(D)) −→ D from F to D.

Proof: We only prove (1). If (F, G, θ) is an adjunction then we can set F0 = F and ηC = θC,F C (idFC ). By the previous proposition, ηC is a universal arrow from C to G, for each C ∈ Ob(C). Now suppose that for each object C of C there exist an object F0(C) of D and a universal arrow ηC : C −→ G(F0(C)) from C to G. We deﬁne a functor F : C −→ D by setting

15 (1) F (C) = F0(C) for each C ∈ Ob(C), and (2) if f : C −→ C0 is an arrow of C then F (f) is the only arrow of D such that the triangle ηC C / G(F0(C)) GG GG GG GG GG G(F (f)) ηC0 ◦f GG GG # 0 G(F0(C )) commutes.

Let f : C −→ C0 and g : C0 −→ C00 be two arrows of C. We know that F (g ◦ f): 00 F (C) −→ F (C ) is the only arrow of D such that G(F (g ◦ f)) ◦ ηC = ηC00 ◦ (g ◦ f). On the other hand,

G(F (g) ◦ F (f)) ◦ ηC = [G(F (g)) ◦ G(F (f))] ◦ ηC = G(F (g)) ◦ [G(F (f)) ◦ ηC ]

= G(F (g)) ◦ [ηC0 ◦ f] = [GF (g) ◦ ηC0 ] ◦ f

= [ηC00 ◦ g] ◦ f = ηC00 ◦ (g ◦ f).

It follows F (g ◦ f) = F (g) ◦ F (f). Similarly, one can prove that F (idC ) = idFC . It is only left to construct a natural isomorphism

θ : HomD(F (·), ·) −→ HomC(·,G(·)). Let C ∈ Ob(C) and D ∈ Ob(D). We deﬁne a function

θC,D : HomD(FC,C) −→ HomC(C, GD) by θC,D(h) = G(h) ◦ ηC , for every arrow h : FC −→ D of D. We have

HomC(f, G(g)) ◦ θC,D(h) = HomC(f, G(g))(G(h) ◦ ηC ) = G(g) ◦ (G(h) ◦ ηC ) ◦ f

= (G(g) ◦ G(h)) ◦ ηC ◦ f = G(g ◦ h) ◦ ηC ◦ f,

θC0,D0 ◦ HomD(F (f), g)(h) = θC0,D0 (g ◦ h ◦ F (f)) = G(g ◦ h ◦ F (f)) ◦ ηC0

= G(g ◦ h) ◦ G(F (f)) ◦ ηC0 = G(g ◦ h) ◦ ηC ◦ f. Hence, we get the following commutative square

θC,D HomD(FC,D) / HomC(C, GD)

HomD(F (f),g) HomC(f,G(g))

0 0 0 0 HomD(FC ,D ) / HomC(C , GD ) θC0,D0

By Lemma 2.1.1, each θC,D is bijective. Therefore, θ is a natural isomorphism.

16 2.2 Preservation of limits

A functor H : C −→ D is said to preserve limits of functors F : J −→ C when if (L, (νi : L −→ F (i))i∈Ob(J )) is the limit for F , then (H(L), (H(νi): H(L) −→ HF (i))i∈Ob(J )) is the limit for HF .

Example 2.2.1. If C is a small category, then the functor HomC(C, ·) preserves limits.

Suppose that (L, (νi : L −→ F (i))i∈Ob(J )) is the limit for a functor F : J −→ C. Consider the family of functions (HomC(C, νi))i∈Ob(J ). First, if a : i −→ j is an arrow of J and f : C −→ L is an arrow of C, then we have

HomC(C,F (a)) ◦ HomC(C, νi)(f) = HomC(C,F (a))(νi ◦ f) = F (a) ◦ (νi ◦ f) = (F (a) ◦ νi) ◦ f

= νj ◦ f = HomC(C, νj)(f), i.e., the triangle

HomC(C,νi) HomC(C,L) / HomC(C,F (a)) OO OOO OOO OOO OOO HomC(C,F (a)) Hom (C,νj )OO C OOO OOO OO' HomC(C,F (j)) commutes. Now let (βi : X −→ HomC(C,F (i)))i∈Ob(J ) be another family of functions such that HomC(C,F (a)) ◦ βi = βj, for every arrow a : i −→ j. Then for each x ∈ X, we have βj(x) = F (a) ◦ βi(x), where each βi(x): C −→ F (i) is an arrow of C. Since (νi : L −→ F (i))i∈Ob(J ) is the limit for F , then there exists a unique arrow hx : C −→ L of C such that βi(x) = νi ◦ hx. Thus, we get a well deﬁned function h : X −→ HomC(C,L) given by h(x) = hx, for each x ∈ X. Clearly, we have that the commutative triangle

HomC(C,νi) HomC(C,L) / HomC(C,F (i)) O n7 nnn nnn nnn h nnn nn βi nnn nnn nnn X n

It is only left to show that h is the only function satisfying HomC(C, νi) ◦ h = βi. Let 0 0 h : X −→ HomC(C,L) be another function such that HomC(C, νi) ◦ h = βi. Then for each 0 x ∈ X, we have νi ◦ h (x) = βi(x). On the other hand, hx is the only arrow of C satisfying 0 0 νi ◦ hx = βi(x). So we get hx = h (x), for each x ∈ X, i.e., h = h .

17 Theorem 2.2.1. If (F, G, θ) is an adjunction from C to D, then G preserves limits and F preserves colimits.

Proof: We prove that G preserves limits. Let (L, (νi : L −→ T (i))i∈Ob(J )) be the limit for a functor T : J −→ D. Consider the family of arrows (G(νi): GL −→ GT (i))i∈Ob(J ) of C.

(1) If a : i −→ j is an arrow of J , then we have

GT (a) ◦ G(νi) = G(T (a) ◦ νi) = G(νj),

since G is a functor and (νi : L −→ T (i))i∈Ob(J ) is the limit for T . Hence, the triangle

G(νi) G(L) / GT (i) H HH HH HH GT (a) G(νj ) HH H# GT (j) commutes.

(2) Let (fi : X −→ GT (i))i∈Ob(J ) be another family of arrows of C with GT (a)◦fi = fj. Recall we have a natural isomorphism

θ : HomD(F (·), ·) −→ HomC(·,G(·)).

−1 Note that fi ∈ HomC(X, GT (i)). Thus, set gi = θX,T (i)(fi) ∈ HomD(FX,T (i)). Consider the commutative square

θX,T (i) HomD(FX,T (i)) / HomC(X, GT (i))

HomD(idFX ,T (a)) HomC(idX ,GT (a))

HomD(FX,T (j)) / HomC(X, GT (j)) θX,T (j)

where a : i −→ j is an arrow of J . We get

−1 −1 HomD(idFX ,T (a)) ◦ θX,T (i)(fi) = θX,T (j) ◦ HomC(idX , GT (a))(fi) −1 −1 T (a) ◦ θX,T (i)(fi) ◦ idFX = θX,T (j)(GT (a) ◦ fi ◦ idX ) −1 −1 T (a) ◦ θX,T (i)(fi) = θX,T (j)(fj)

T (a) ◦ gi = gj.

18 Hence, we obtain the following commutative triangle in D:

gi FX / T (i) D DD DD DD T (a) gj DD D" T (j) It follows there exists a unique arrow of D, say g : FX −→ L, making the following triangle commute, for each i ∈ Ob(J ):

νi L / T (i) O < yy g yy yygi yy yy FX

Now set h = θX,L(g) ∈ Hom(X, GL). Since the square

θX,L HomD(FX,L) / HomC(X, GL)

HomD(idFX ,νi) HomC(idX ,G(νi))

HomD(FX,T (i)) / HomC(X, GT (i)) θX,T (i) commutes, it follows that

HomC(idX ,G(νi)) ◦ θX,L(g) = θX,T (i) ◦ HomD(idFX , νi)(g)

G(νi) ◦ θX,L(g) ◦ idX = θX,T (i)(νi ◦ g ◦ idFX )

G(νi) ◦ h = θX,T (i)(gi)

G(νi) ◦ h = fi, i.e., the triangle

G(νi) GL / GT (i) O ; ww ww h ww ww fi ww X commutes. It is only left to show that h : X −→ GL is the only arrow of C satisfying 0 G(νi) ◦ h = fi. Suppose there is another arrow of h : X −→ GL of C satisfying 0 −1 0 G(νi) ◦ h = fi. Consider the arrow θX,L(h ) ∈ HomD(FX,L). Since the above square commutes, we get −1 0 −1 0 HomD(idFX , νi) ◦ θX,L(h ) = θX,T (i) ◦ HomC(idX ,G(νi))(h ) −1 0 −1 0 νi ◦ θX,L(h ) = θX,T (i)(G(νi) ◦ h ) −1 0 νi ◦ θX,L(h ) = gi.

19 On the other hand, g is the only arrow of D satisfying νi ◦ g = gi. Hence, we have −1 0 0 g = θX,L(h ), i.e., h = θX,L(g) = h.

Therefore, (GL, (G(νi): GL −→ GT (i))i∈Ob(J )) is the limit for the functor GT .

2.3 The Freyd’s Adjoint Functor Theroem

Lemma 2.3.1 (Existence of an initial object). Let D be a small and complete category. Then D has an initial object if and only if it satisﬁes the following Solution Set Condition: There exists a set I and an i-indexed family (Di)i∈I of objects of D such that for every D ∈ Ob(D) there is an i ∈ I and an arrow Di −→ D of D.

Proof: If D has an initial object D0 then set I = {0}. Then for every D ∈ Ob(D) there exists one and only one arrow D0 −→ D, i.e., {D0} satisfies the Solution Set Condition. Now suppose that D satisfies the Solution Set Condition. Let (P, (pi : P −→ Di)i∈I ) be the product of the family (Di)i∈I . Let J be the small category whose only object is P and HomJ (P,P ) = HomD(P,P ). Define a functor F : J −→ D by F (P ) = P and F (f) = f, for every f ∈ HomJ (P,P ). Let (L, (ν : L −→ P )) be the limit for F . Then the triangle ν L / P ?? ?? ?? F (f) ν ?? ? P

commutes for every f ∈ HomJ (P,P ). It follows that f ◦ν = g◦ν, for every f, g : P −→ P in D. Let ν0 : L0 −→ P be another arrow of D such that f ◦ ν0 = ν0 for every f. Then there exists a unique arrow h : L0 −→ L of D such that ν0 = ν ◦ h; i.e., if ν0 : L0 −→ P 0 0 0 is another arrow such that f ◦ ν = g ◦ ν for every f, g ∈ HomD(P,P ), then ν factors uniquely through ν. Hence, ν : L −→ P is the equalizer of the set HomD(P,P ). We shall see that L is an initial object in D. Let D ∈ Ob(D). There is an i ∈ I and an arrow fi : Di −→ D. Then fi ◦ pi ◦ ν is an arrow from L to D. Now we prove that such an arrow is unique. Let f, g : L −→ D be two arrows of D and µ : K −→ L the equalizer pi of f and g. There is an i ∈ I and an arrow Di −→ K. Then s : P −→ Di −→ K is s µ ν an arrow from P to K. So we get the arrow P −→ K −→ L −→ P in HomD(P,P ). Since ν is the equalizer of HomD(P,P ), we have that (ν ◦ µ ◦ s) ◦ ν = idP ◦ ν. It follows ν ◦(µ◦s◦ν) = ν ◦idL. Hence µ◦s◦ν = idL since equalizers are monic. We have that µ is a monic arrow having a right inverse. It follows µ is an isomorphism. Hence f ◦ µ = g ◦ µ implies that f = g.

20 Lemma 2.3.2. If D is a complete category and G : D −→ C is a functor that preserves products and equalizers, then the comma category (C ↓ G) is complete, for each C ∈ Ob(C).

Proof: By Theorem 1.2.1 it is only left to prove that (C ↓ G) has products and equalizers.

(1)( C ↓ G) has products: Let (ui,Di)i∈I be a family of objects in (C ↓ G). Since D is complete, there exists the product of the family (Di)i∈I , say (P, (pi : P −→ Di)i∈I ). On the other hand, G preserves products, so

(G(P ), (G(pi): G(P ) −→ G(Di))i∈I )

is the product of (G(Di))i∈I . Since the arrows G(pi) are universal, there exists a unique arrow u : C −→ G(P ) such that the triangle

G(pi) G(P ) / G(Di) O : uu u uu uuu uu i uu C u

commutes. Thus, (u, P ) is an object of (C ↓ G) and pi :(u, P ) −→ (ui,Di) is an arrow of (C ↓ G). We show that (u, P ) is the product of (ui,Di)i∈I . Let

0 (fi :(v, P ) −→ (ui,Di))i∈I

be another family of arrows in (C ↓ G). Since the arrows pi are universal in D, there exists a unique arrow h : P 0 −→ P such that the triangle

pi P / Di O }> }} h }} }} fi }} P 0

commutes. Recall that u is the only arrow of C satisfying G(pi) ◦ u = ui. Also, G(pi)◦G(h)◦v = G(pi ◦h)◦v = G(fi)◦v = ui, for every i ∈ I. It follows u = G(h)◦v 0 and so h :(v, P ) −→ (u, P ) is an arrow of (C ↓ G) such that pi ◦ h = fi, i.e., the triangle pi (u, P ) / (ui,Di) O t: tt tt h tt tt fi tt (v, P 0) commutes. Now suppose that there exists another arrow h0 :(v, P 0) −→ (u, P ) of 0 (C ↓ G) such that pi ◦ h = fi. Since h is the only arrow of D satisfying pi ◦ h = fi, we have h0 = h.

21 (2)( C ↓ G) has equalizers: Let f, g :(s, X) −→ (r, S) be two parallel arrows on (C ↓ G). Then G(f) ◦ s = r = G(g) ◦ s. Since D is complete, there exists the equalizer of f : X −→ Y and g : X −→ Y as arrows of D, say u : K −→ X. But G preserves equalizers, then G(u) is the equalizer of G(f) and G(g). Since G(f) ◦ s = G(g) ◦ s, there exists a unique arrow λ : C −→ G(K) of C such that G(u) ◦ λ = s. Thus, u :(λ, K) −→ (s, X) is an arrow of (C ↓ G). We prove that u is the equalizer of f and g, as arrows of (C ↓ G). Let v :(λ0,K0) −→ (s, X) be another arrow of (C ↓ G) such that f ◦v = g ◦v. Since u : K −→ X is the equalizer of f and g as arrows of D, there exists a unique arrow h : K −→ K such that the triangle u K / X O }> }} h }} }} v }} K0 commutes. Then we have G(u) ◦ (G(h) ◦ λ0) = G(u ◦ h) ◦ λ0 = G(v) ◦ λ0 = s. On the other hand, λ is the only arrow of C satisfying G(u)◦λ = s. It follows G(h)◦λ0 = λ, i.e., h :(λ0,K0) −→ (λ, K) is an arrow of (C ↓ G). If h0 :(λ0,K0) −→ (λ, K) is another arrow satisfying u ◦ h0 = v, then h0 = h since h is the only arrow of D making the diagram u f K / X // Y O < g yy yy h yy yy v yy K0 commute.

Theorem 2.3.1 (Freyd). Given a small and complete category D, a functor G : D −→ C has a left adjoint if and only if it preserves all limits and satisﬁes the following Solution Set Condition. For each object C ∈ Ob(C) there is a set I and an I-indexed family of arrows fi : C −→ G(Di) such that every arrow h : C −→ G(D) can be written as a composite h = G(t) ◦ fi for some index i and some t : Di −→ D.

Proof: Suppose G : D −→ C has a left adjoint F : C −→ D. By Theorem 2.2.1, G preserves all limits in D. Consider the unit η : IdC −→ GF . For each C ∈ Ob(C), we get a universal arrow ηC : C −→ GF (C) (see Proposition 2.1.3). Let I = {C}, DC = F (C)

22 and fC = ηC . Let h : C −→ G(D) be an arrow in C of C. Since ηC is universal, there exists a unique arrow t : F (C) −→ D of D such that the triangle

ηC C / GF (C) EE EE EE EE G(t) h EE E" G(D)

commutes, i.e., G(t) ◦ fC = h. Hence, G satisﬁes the Solution Set Condition. Now suppose that G is a functor which preserves limits and satisﬁes the Solution Set Con- dition. By Theorem 2.1.1, we only need to construct a universal arrow from C to G, ηC : C −→ G(F0(C)), for each C ∈ Ob(C). By Proposition 1.1.2, such an arrow (ηC ,F0(C)) is an initial object in the comma category (C ↓ G). By Lemma 2.3.2, (C ↓ G) is complete. On the other hand, there exists an I-indexed family of arrows fi : C −→ G(Di) such that every arrow h : C −→ G(D) can be written as a composition h = G(t)◦fi for some i ∈ I and some arrow t : Di −→ D of D. In other words, there exists a set I and an I-indexed family of objects (fi,Di) of (C ↓ G) such that for every object (h, D) of (C ↓ G) there is an i ∈ I and an arrow t :(fi,Di) −→ (h, D) of (C ↓ G). We have that (C ↓ G) is a small and complete category satisfying the Solution Set Condition given in Lemma 2.3.1. Therefore, (C ↓ G) has an initial object (ηC ,F0(C)).

Using the dual statements of both Lemma 2.3.1 and 2.3.2, on can prove the dual version of the Freyd’s theorem.

Theorem 2.3.2 (Freyd). Given a small and complete category C, a functor F : C −→ D has a right adjoint if and only if it preserves all colimits and satisﬁes the following Solution Set Condition. For each object D ∈ Ob(D) there is a set I and an I-indexed family of arrows gi : F (Ci) −→ D such that every arrow h : F (C) −→ D can be written as a composite h = gi ◦ F (t) for some index i and some t : C −→ Ci.

23 24 Bibliography

[1] Mac Lane, S. Categories for the Working Mathematician. Springer-Verlag. New York (1997).

[2] Saor´ın,M. Teor´ıade Categor´ıas. (Notes in preparation). Universidad de Murcia.