Universit´edu Qu´ebec `aMontr´eal D´epartement de math´ematiques Marco A. P´erezB. [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 2 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 2 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 2 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 2 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 2 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 9!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 {{ {{ {{ 9!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 2 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 transforma- tion defined as the J -indexed set ∆(f) = f∆(f)i = f : C1 −! C2gi2Ob(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 2 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))i2Ob(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 2 Ob(J ). In pictures, we have the commutative diagram L0 9!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 fCi = F (i)gi2I .