Categories
In this chapter and the next one on universal algebra we consider two unifying concepts that permit us to study simultaneously certain aspects of a large number of mathematical structures. The concept we shall study in this chapter is that of category, and the related notions of functor and natural transformation. These were introduced in 1945 by Eilenberg and MacLane to provide a precise meaning to the statement that certain isomorphisms are “natural.” A typical example is the natural isomorphism between a finitedimensional vector space V over a field and its double dual V**, the space of linear functions on the space V* of linear functions on V. The isomorphism of V onto V** is the linear map associating with any vector x∈V the evaluation function f 
f(x) defined for all f∈V*. To describe the “naturality” of this isomorphism, Eilenberg and MacLane had to consider simultaneously all finite-dimensional vector spaces, the linear transformations between them, the double duals of the spaces, and the corresponding linear transformations between them. These considerations led to the concepts of categories and functors as preliminaries to defining natural transformation. We shall discuss a generalization of this example in detail in section 1.3.
The concept of a category is made up of two ingredients: a class of objects and a class of morphisms between them. Usually the objects are sets and the morphisms are certain maps between them, e.g., topological spaces and continuous maps. The definition places on an equal footing the objects and the morphisms. The adoption of the category point of view represents a shift in emphasis from the usual one in which objects are primary and morphisms secondary. One thereby gains precision by making explicit at the outset the morphisms that are allowed between the objects collected to form a category.
The language and elementary results of category theory have now pervaded a substantial part of mathematics. Besides the everyday use of these concepts and results, we should note that categorical notions are fundamental in some of the most striking new developments in mathematics. One of these is the extension of algebraic geometry, which originated as the study of solutions in the field of complex numbers of systems of polynomial equations with complex coefficients, to the study of such equations over an arbitrary commutative ring. The proper foundation of this study, due mainly to A. Grothendieck, is based on the categorical concept of a scheme. Another deep application of category theory is K. Morita’s equivalence theory for modules, which gives a new insight into the classical Wedderburn-Artin structure theorem for simple rings and plays an important role in the extension of a substantial part of the structure theory of algebras over fields to algebras over commutative rings.
A typical example of a category is the category of groups. Here one considers “all” groups, and to avoid the paradoxes of set theory, the foundations need to be handled with greater care than is required in studying group theory itself. One way of avoiding the well-known difficulties is to adopt the Godel-Bernays distinction between sets and classes. We shall follow this approach, a brief indication of which was given in the Introduction.
In this chapter we introduce the principal definitions of category theory— functors, natural transformations, products, coproducts, universals, and adjoints—and we illustrate these with many algebraic examples. This provides a review of a large number of algebraic concepts. We have included some nontrivial examples in order to add a bit of seasoning to a discussion that might otherwise appear too bland.
1.1 DEFINITION AND EXAMPLES OF CATEGORIES
DEFINITION 1.1. A category C consists of
1. A class ob C of objects (usually denoted as A, B, C, etc.).
2. For each ordered pair of objects (A,B) a set homc(A,B) (or simply hom(A,B) if C is clear) whose elements are called morphisms with domain A and codomain B (or from A to B).
3. For each ordered triple of objects (A,B,C), a map (f,g)gf of the product set hom (A, B) × hom (B, C) into hom (A, C).
It is assumed that the objects and morphisms satisfy the following conditions:
C1. If (A, B) ≠ (C,D), then hom (A,B) and hom(C,D) are disjoint.
C2. (Associativity). If f∈hom (A, B) g∈hom(B,C), and h∈hom(C, D), then (hg)f = h(gf). (As usual, we simplify this to hgf.)
C3. (Unit). For every object A we have an element 1A∈hom (A, A) such that flA = f for every f∈hom(A,B) and lAg = g for every g ∈ hom (B, A). (1A is unique.)
If f ∈ hom (A, B) we write f : A → B or A 
B (sometimes A 
B), and we call f an arrow from A to B. Note that gf is defined if and only if the domain of g coincides with the codomain of f and gf has the same domain as f and the same codomain as g.
The fact that gf = h can be indicated by saying that
is a commutative diagram. The meaning of more complicated diagrams is the same as for maps of sets (BAI, pp. 7–8). For example, the commutativity of
means that gf = kh, and the associativity condition (hg)f = h(gf) is expressed by the commutativity of
The condition that 1A is the unit in hom (A, A) can be expressed by the commutativity of
for all f ∈ hom (A,B) and all g ∈ hom (B,A).
We remark that in defining a category it is not necessary at the outset that the sets hom(A,B) and hom (C, D) be disjoint whenever (A,B) ≠ (C,D). This can always be arranged for a given class of sets hom(A,B) by replacing the given set hom (A,B) by the set of ordered triples (A,B,f) where f ∈ hom (A,B). This will give us considerably greater flexibility in constructing examples of categories (see exercises 3–6 below).
We shall now give a long list of examples of categories.
EXAMPLES
1. Set, the category of sets. Here ob Set is the class of all sets. If A and B are sets, hom (A, B) = BA, the set of maps from A to B. The product gf is the usual composite of maps and 1A is the identity map on the set A. The validity of the axioms C1, C2, and C3 is clear.
2. Mon, the category of monoids, ob Mon is the class of monoids (BAI, p. 28), hom (M, N) for monoids M and N is the set of homomorphisms of M into N, gf is the composite of the homomorphisms g and f and 1M is the identity map on M (which is a homomorphism). The validity of the axioms is clear.
3. Grp, the category of groups. The definition is exactly like example 2, with groups replacing monoids.
4. Ab, the category of abelian groups, ob Ab is the class of abelian groups. Otherwise, everything is the same as in example 2.
A category D is called a subcategory of the category C if ob D is a subclass of obC and for any A,B∈obD, homD(A,B) ⊂ homc(A,B). It is required also (as part of the definition) that 1A for A ∈ob D and the product of morphisms for D is the same as for C. The subcategory D is called full if homD(A,B) = homC(A,B) for every A,B∈D. It is clear that Grp and Ab are full subcategories of Mon. On the other hand, since a monoid is not just a set but a triple (M,p, 1) where p is an associative binary composition in M and 1 is the unit, the category Mon is not a subcategory of Set. We shall give below an example of a subcategory that is not full (example 10).
We continue our list of examples.
5. Let M be a monoid. Then M defines a category M by specifying that obM = {A}, a set with a single element A, and defining hom (A, A) = M, 1A the unit of M, and xy for x, y∈ hom (A, A), the product of x and y as given in M. It is clear that M is a category with a single object. Conversely, let M be a category with a single object: obM = {A}. Then M = hom (A, A) is a monoid. It is clear from this that monoids can be identified with categories whose object classes are single-element sets.
A category is called small if ob C is a set. Example 5 is a small category; 1–4 are not.
An element f∈ hom (A, B) is called an isomorphism if there exists a g∈hom(B,A) such that fg = 1B and gf = 1A. It is clear that g is uniquely determined by f so we can denote it as f–1. This is also an isomorphism and (f–1)–1 =f If f and h are isomorphisms and fh is defined, then fh is an isomorphism and (fh)-1 = h-1f–1. In Set the isomorphisms are the bijective maps, and in Grp they are the usual isomorphisms (= bijective homomorphisms).
6. Let G be a group and let this define a category G with a single object as in example 5. The characteristic property of this type of category is that it has a single object and all arrows ( = morphisms) are isomorphisms.
7. A groupoid is a small category in which morphisms are isomorphisms.
8. A discrete category is a category in which hom (A,B) = Ø if A≠B and hom (A, A) = {1A}. Small discrete categories can be identified with their sets of objects.
9. Ring, the category of (associative) rings (with unit for the multiplication composition). obRing is the class of rings and the morphisms are homomorphisms (mapping 1 into 1).
10. Rng, the category of (associative) rings without unit (BAI, p. 155), homomorphisms as usual. Ring is a subcategory of Rng but is not a full subcategory, since there exist maps of rings with unit that preserve addition and multiplication but do not map 1 into 1. (Give an example.)
11. R-mod, the category of left modules for a fixed ring R. (We assume 1x = x for x in a left R-module M.) ob R-mod is the class of left modules for R and the morphisms are JR-module homomorphisms. Products are composites of maps. If R = A is a division ring (in particular, a field), then R-mod is the category of (left) vector spaces over Δ. In a similar manner one defines mod-R as the category of right modules for the ring R.
12. Let S be a pre-ordered set, that is, a set S equipped with a binary relation a ≤ b such that a ≤ a and a ≤ b and b ≤ c imply a ≤ c. S defines a category S in which ob S = S and for a,b∈S, hom (A, B) is vacuous or consists of a single element according as a ≤ b or a ≤ b. If f ∈ hom (a, b) and g ∈ hom (b, c), then gf is the unique element in hom (a, c). It is clear that the axioms for a category are satisfied. Conversely, any small category such that for any pair of objects A, B, hom (A, B) is either vacuous or a single element is the category of a pre-ordered set as just defined.
13. Top, the category of topological spaces. The objects are topological spaces and the morphisms are continuous maps. The axioms are readily verified.
We conclude this section by giving two general constructions of new categories from old ones. The first of these is analogous to the construction of the opposite of a given ring (BAI, p. 113). Suppose C is a category; then we define Cop by obCop = obC; for A, B∈obCop, homCoP(A,B) = homc(B,A); if f∈homCoP(A,B) and g ∈homcoP(B,C), then g-f (in Cop) =fg (as given in C). 1A is as in C. It is clear that this defines a category. We call this the dual category of C. Pictorially we have the following: If A B in C, then A 
B in Cop and if
is commutative in C, then
is commutative in Cop. More generally, any commutative diagram in C gives rise to a commutative diagram in Cop by reversing all of the arrows.
Next let C and D be categories. Then we define the product category C × D by the following recipe: ob C × D = ob C × ob D; if A, B ∈ ob C and A′, B′ ∈ ob D, then hom C × D ((A ,A′), (B, B′) = homC (A,B) × homD(A′, B′), and 1(A′, A′) =(1A,1A); if f ∈ homC(A,B), g ∈ homC(B, C), f´ ∈ homD (A′, B′), and g´ ∈ homD(B′, C′), then
The verification that this defines a category is immediate. This construction can be generalized to define the product of indexed sets of categories. We leave it to the reader to carry out this construction.
1. Show that the following data define a category Ring*: obRing* is the class of rings; if R and S are rings, then homRing*(R, S) is the set of homomorphisms and anti-homomorphisms of R into S; gf for morphisms is the composite g following f for the maps f and g; and 1R is the identity map on R.
2. By a ring with involution we mean a pair (R,j) where R is a ring (with unit) and j is an involution in R; that is, if j:a a*, then (a + b)* = a*+ b*, (ab)* = b*a*, 1* = 1, (a*)* = a. (Give some examples.) By a homomorphism of a ring with involution (R,J) into a second one (S, k) we mean a map η of R into 5 such that η is a homomorphism of R into S (sending 1 into 1) such that η(ja) = k(ηa) for all a∈R. Show that the following data define a category Rinv: obRinv is the class of rings with involution; if (R,J) and (S,k) are rings with involution, then hom ((R,J), (S,k)) is the set of homomorphisms of (R,J) into (S,k); gf for morphisms is the composite of maps; and 1(R,J) = 1R.
3. Let C be a category, A an object of C. Let obC/A = ∪x∈obchom (X, A) so obC/A is the class of arrows in C ending at A. If f∈hom(B, A) and g ∈ hom (C, A), define hom (f,g) to be the set of u :B C such that
is commutative. Note that hom (f, g) and hom (f′, g′) may not be disjoint for (f, g) ≠ (f′, g′). If u∈hom(fg) and v∈hom(g, h) for h : D → A, then vu ∈ hom (f, h). Use this information to define a product from hom (f, g) and hom (g, h) to hom (f, h). Define 1f = 1B for f: B → A. Show that these data and the remark on page 11 can be used to define a category C/A called the category of objects of C over A.
4. Use Cop to dualize exercise 3. This defines the category C A of objects of C below A.
5. Let C be a category, A1, A2∈ob C. Show that the following data define a category C/{Al, A2} ′ The objects are the triples (B, fl, f2) where f1∈ homc(B, Ai). A morphism h : (B, f1, f2) (C, gl, g2) is a morphism h : B → C in C such that
is commutative. Arrange to have the hom sets disjoint as before. Define l(B, f1, f2) = 1B and the product of morphisms as in C. Verify the axioms C2 and C3 for a category.
6. Dualize exercise 5 by applying the construction to Cop and interpreting in C. The resulting category is denoted as C\{A1, A2}.
7. (Alternative definition of a groupoid.) Let G be a groupoid as defined in example 7 above and let G = 
A B∈obGhom(A,B). Then G is a set equipped with a composition fg that is defined for some pairs of elements (f,g),f,g∈G, such that the following conditions hold:
(i) For any f ∈ G there exist a uniquely determined pair (u, v), u,v∈G such that uf and fv are defined and uf = f = fv. These elements are called the left and right units respectively of f.
(ii) If u is a unit (left and right for some f∈G), then u is its own left unit and hence its own right unit.
(iii) The product fg is defined if and only if the right unit of f coincides with the left unit of g.
(iv) If fg and gh are defined, then (fg)h and f(gh) are defined and (fg)h = f(gh).
(v) If f has left unit u and right unit v, then there exists an element g having right unit u and left unit v such that fg = u and gf = v.
Show that, conversely, if G is a set equipped with a partial composition satisfying conditions (i)-(v), then G defines a groupoid category G in which ob G is the set of units of G; for any objects u, v, hom (u, v) is the subset of G of elements having u as left unit and v as right unit; the product composition of hom (u, v) x hom (v, w) is that given in G.
8. Let G be as in exercise 7 and let G* be the disjoint union of G and a set {0}. Extend the composition in G to G* by the rules that 0a = 0 = a0 for all a∈G* and fg = 0 if f,g∈G and fg is not defined in G. Show that G* is a semigroup (BAI, p. 29).
1.2 SOME BASIC CATEGORICAL CONCEPTS
We have defined a morphism f in a category C to be an isomorphism if f : A → B and there exists a g :B → A such that fg = 1B and gf = 1A. If f : A → B, g : B → A, and gf = 1A, then f is called a section of g and g is called a retraction of f. More interesting than these two concepts are the concepts of monic and epic that are defined by cancellation properties: A morphism f : A → B is called monic (epic) if it is left (right) cancellable in C; that is, if g1 and g2 ∈ hom (C, A) (hom (B, C)) for any C and fg1 = fg2 (g1f = g2f), then g1 = g2. The following facts are immediate consequences of the definitions:
1. If A B and B 
C and f and g are monic (epic), then gf is monic (epic).
2. If A B and B C and gf is monic (epic), then f is monic (g is epic).
3. If f has a section then f is epic, and if f has a retraction then f is monic.
If f is a map of a set A into a set B, then it is readily seen that f is injective (that is, f(a) ≠ f(a′) for a ≠ a′ in A) if and only if for any set C and maps gl,g2 of C into A, fg1 = fg2 implies g1 = g2 (exercise 3, p. 10 of BAI). Thus f ∈ homSet(A,B) is monic if and only if fis injective. Similarly, f is epic in Set if and only iff is surjective (f(A) = B). Similar results hold in the categories R-mod and Grp: We have
PROPOSITION 1.1. A morphism f in R-mod or in Grp is monic (epic) if and only if the map of the underlying set is injective (surjective).
Proof. Let f:A → B in R-mod or Grp. If f is injective (surjective) as a map of sets, then it is left (right) cancellable in Set. It follows that f is monic (epic) in R-mod or Grp. Now suppose the set map f is not injective. Then C = kerf ≠ 0 in the case of R-mod and ≠ 1 in the case of Grp. Let g be the injection homomorphism of C into A (denoted by C 
A), so g(x) = x for every x ∈ C. Then fg is the homomorphism of C into B, sending every x ∈ C into the identity element of B. Next let h be the homomorphism of C into A, sending every element of C into the identity element of A. Then h ≠ g since C ≠ 0 (or ≠ 1), but fg = fh. Hence f is not monic.
Next suppose we are in the category R-mod and f is not surjective. The image f(A) is a submodule of B and we can form the module C = B/f(A), which is ≠0 since f(A) ≠ B. Let g be the canonical homomorphism of B onto C and h the homomorphism of B into C, sending every element of B into 0. Then g ≠ h but gf = hf. Hence f is not epic.
Finally, suppose we are in the category Grp and f is not surjective. The foregoing argument will apply if C = f(A)
B (C is a normal subgroup of B). This will generally not be the case, although it will be so if [B :C] = 2. Hence we assume [B : C] > 2 and we shall complete the proof by showing that in this case there exist distinct homomorphisms g and h of B into the group Sym B of permutations of B such that gf = hf We let g be the homomorphism b bL of B into Sym B where bL is the left translation x bx in B. We shall take h to have the form kg where k is an inner automorphism of Sym B. Thus k has the form ypyp-1 where y ∈ Sym B and p is a fixed element of Sym B. Then h = kg will have the form b pbLpl and we want this to be different from g :b bL. This requires that the permutation p does not commute with every bL. Since the permutations commuting with all of the left translations are the right translations (exercise 1, p. 42 of BAI), we shall have h = kg ≠ g if p is not a right translation. Since translations ≠ 1 have no fixed points, our condition will be satisfied if p is any permutation ≠ 1 having a fixed point. On the other hand, to achieve the condition gf = hf we require that p commutes with every cL, c∈C. To construct a p satisfying all of our conditions, we choose a permutation π of the set C\B of right cosets Cb, b∈B, that is not the identity and has a fixed point. Since |C\B| > 2, this can be done. Let I be a set of representatives of the right cosets Cb. Then every element of B can be written in one and only one way as a product cu, c∈C, u∈I. We now define the map p by p(cu) = cu' where π(Cu) = Cu'. Then p∈SymB, p ≠ 1, and p has fixed points since π ≠= 1 and π has fixed points. It is clear that p commutes with every dL, d∈C. Hence p satisfies all of our requirements and f is not epic. 
What can be said about monies and epics in the category Ring? In the first place, the proof of Proposition 1.1 shows that injective homomorphisms are monic and surjective ones are epic. The next step of the argument showing that monies are injective breaks down totally, since the kernel of a ring homomorphism is an ideal and this may not be a ring (with unit). Moreover, even if it were, the injection map of the kernel is most likely not a ring homomorphism. We shall now give a different argument, which we shall later generalize (see p. 82), to show that monies in Ring are injective.
Let f be a homomorphism of the ring A into the ring B that is not injective. Form the ring A 
A of pairs (a1,a2), ai∈ A, with component-wise addition and multiplication and unit 1 = (1, 1). Let K be the subset of A A of elements (a1,a2) such that f(a1)= f(a2). It is clear that K is a subring of A A and K 
D = {(a,a)|a ∈ A}. Let g1 be the map (a1, a2) al and g2 the map (al, a2) a2 from K to A. These are ring homomorphisms and fgl =fg2, by the definition of K. On the other hand, since K D, we have a pair (a1, a2)∈K with a1 ≠ a2. Then g1(a1,a2) = ax ≠ a2 = g2(al,a2). Hence g1 ≠ g2 which shows that f is not monic.
Now we can show by an example that epics in Ring need not be surjective. For this purpose we consider the injection homomorphism of the ring 
of integers into the field 
of rationals. If g and h are homomorphisms of into a ring R, then gf= hf if and only if the restrictions g| = h|. Since a homomorphism of is determined by its restriction to , it follows that gf = hf implies g = h. Thus f is epic and obviously f is not surjective.
We have proved
PROPOSITION 1.2. A morphism in Ring is monic if and only if it is injective. However, there exist epics in Ring that are not surjective.
The concept of a monic can be used to define subobjects oa an object A of a category C. We consider the class of monicsending in A
We introduce a preorder in the class of these monies by declaring that f ≤ g if there exists a k such that f = gk. It follows that k is monic. We write f ≡ g if f ≤ g and g ≤ f. In this case the element k is an isomorphism. The relation ≡ is an equivalence and its equivalence classes are called the subobjects of A.
By duality we obtain the concept of a quotient object of A. Here we consider the epics issuing from A and define f ≥ g if there exists a k such that f = kg. We have an equivalence relation f ≡ g defined by f = kg where k is an isomorphism. The equivalence classes determined by this relation are called the quotient objects of A.
If the reader will consider the special case in which C = Grp, he will convince himself that the foregoing terminology of subobjects and quotient objects of the object A is reasonable. However, it should be observed that the quotient objects defined in Ring constitute a larger class than those provided by surjective homomorphisms.
EXERCISES
1. Give an example in Top of a morphism that is monic and epic but does not have a retraction.
2. Let G be a finite group, H a subgroup. Show that the number of permutations of G that commute with every hL, h∈H (acting in G), is [G :H]!|H|[G:H] where [G:H] is the index of H in G.
1.3 FUNCTORS AND NATURAL TRANSFORMATIONS
In this section we introduce the concept of a functor or morphism from one category to another nthat commute with ad the concept of maps between functors called natural transformations. Before proceeding to the definitions we consider an example.
Let R be a ring and let U(R) denote the multiplicative group of units ( = invertible elements) of R. The map R U(R) is a map of rings into groups, that is, a map of ob Ring into ob Grp. Moreover, if f :R → S is a homomorphism of rings, then the restriction f |U(R) maps U(R) into U(S) and so may be regarded as a map of U(R) into (U(S). Evidently this is a group homomorphism. It is clear also that if g :S → T is a ring homomorphisM, then (gf)|U(R) = (g|U(S))(f|U(R)). Moreover, the restriction 1 RU(R) is the identity map on U(R).
The map R U(R) of rings into groups and ff|U(R) of ring homomorphisms into group homomorphisms constitute a functor from Ring to Grp in the sense of the following definition.
DEFINITION 1.2. If C and D are categories, a (covariant) functor F from C to D consists of
1. A map A FA of ob C into ob D.
2. For every pair of objects (A,B) of C, a map f F(f) of homc(A, B) into homD (FA,FB).
We require that these satisfy the following conditions:
FI. If gf is defined in C, then F(gf) = F(g)F(f).
F2. F(1A) = 1FA.
The condition F1 states that any commutative triangle
in C is mapped into a commutative triangle in D
A contravariant functor from C to D is a functor from Cop to D. More directly, this is a map F of ob C into ob D and for each pair (A, B) of objects in C, a map F of hom (A,B) into hom (FB,FA) such that F(fg) = F(g)F(f) and F(1A) = lFA. A functor from B × C to D is called a bifunctor from B and C into D. We can also combine bifunctors with contravariant functors to obtain functors from Bop × C to D and from Bop × Cop to D. The first is called a bifunctor that is contravariant in B and covariant in C and the second is a bifunctor that is contravariant in B and C.
EXAMPLES
1. Let D be a subcategory of the category C. Then we have the injection functor of D into C that maps every object of D into the same object of C and maps any morphism in D into the same morphism in C. The special case in which D = C is called the identity functor lc.
2. We obtain a functor from Grp to Set by mapping any group into the underlying set of the group and mapping any group homomorphism into the corresponding set map. The type of functor that discards some of the given structure is called a “forgetful” functor. Two other examples of forgetful functors are given in the next example.
3. Associated with any ring (R, + ,·,0, 1) we have the additive group (R, + ,0) and the multiplicative monoid (R,., 1). A ring homomorphism is in particular a homomorphism of the additive group and of the multiplicative monoid. These observations lead in an obvious manner to definitions of the forgetful functors from Ring to Ab and from Ring to Mon.
4. Let n be a positive integer. For any ring R we can form the ring Mn(R) of n × n matrices with entries in R. A ring homomorphism f: R → S determines a homomorphism (rij) (f(rij)) of Mn(R) into Mn(S). In this way we obtain a functor Mn of Ring into Ring.
5. Let n and R be as in example 4 and let GLn(R) denote the group of units of Mn(R), that is, the group of n × n invertible matrices with entries in R. The maps R GLn(R),f into (rij) (f(rij)) define a functor GLn from Ring to Grp.
6. We define the power functor 
in Set by mapping any set A into its power set (A) and any set map f:A → B into the induced map f of (A) into (B) which sends any subset A1 of A into its image f(Al) ⊂ B(∅ ∅).
7. The abelianizing functor from Grp to Ab. Here we map any group G into the abelian group G/(G, G) where (G, G) is the commutator group of G (BAI, p. 238). If f is a homomorphism of G into a second group H,f maps (G, G) into (H,H) and so induces a homomorphism 
of G/(G, G) into H/(H,H). The map f completes the definition of the abelianizing functor.
8. Let Poset be the category of partially ordered sets. Its objects are partially ordered sets (BAI, p. 456) and the morphisms are order-preserving maps. We obtain a functor from R-mod to Poset by mapping any R-module M into L(M), the set of submodules of M ordered by inclusion. If f: M → N is a module homomorphisM, f determines an order-preserving map of L(M) into L(N). These maps define a functor.
9. We define a projection functor of C × D into C by mapping any object (A, B) of C × D into the object A of C and mapping (fg)∈ hom ((A, B) (A′,B′)) into f∈ hom (A, A′).
10. We define the diagonal functor C → C × C by mapping A (A, A) and f:A → B into (A,A)
(B,B).
11. Consider the categories R-mod and mod-R of left R-modules and right R-modules respectively for the ring R. We shall define a contravariant functor D from R-mod to mod-R as follows. If M is a left R-module, we consider the set M* = homR(M,R) of homomorphisms of M into R regarded as left R-module in the usual way. Thus M* is the set of maps of M into R such that
for x, y ∈ M, r ∈ R. If f, g ∈ M* and s ∈ R, then f + g defined by (f + g) (x) = f(x) + g(x) and fs defined by (fs) (x) = f(x)s are in M*. In this way M* becomes a right R-module and we have the map M M* of obj R-mod into obmod-R. Now let L : M → N be a homomorphism of the R-module M into the R-module N. We have the transposed map L* :N* ’ M* defined as
the composite of g and L :
If 
in R-mod and g ∈ M3*, then (L2L1)*(g) = gL2Ll = (gL2)L1 = L*1*L2*(g). Hence
It is clear that (1M)* = 1M*. It follows that
defines a contravariant functor, the duality functor D from R-mod to mod-R. In a similar fashion one obtains the duality functor D from mod-R to R-mod.
It is clear that a functor maps an isomorphism into an isomorphism: If we have fg = 1B, gf = 1A, then F(f)F(g) = 1FB and F(g)F(f) = 1FA. Similarly, sections are mapped into sections and retractions are mapped into retractions by functors. On the other hand, it is easy to give examples to show that monies (epics) need not be mapped into monies (epics) by functors (see exercise 3 below).
If F is a functor from C to D and G is a functor from D to E, we obtain a functor GF from C to E by defining (GF)A = G(FA) for ,A∈obC and (GF) (f) = G(F(f)) for f∈homc(A,B). In a similar manner we can define composites of functors one or both of which are contravariant. Then FG is contravariant if one of F, G is contravariant and the other is covariant, and FG is covariant if both F and G are contravariant. Example 5 above can be described as the composite UMn where Mn is the functor defined in example 4 and U is the functor from Ring to Grp defined at the beginning of the section. As we shall see in a moment, the double dual functor D2 from R-mod to itself is a particularly interesting covariant functor.
A functor F is called faithful (full) if for every pair of objects A, B in C the map f F(f) of homc(A, B) into homD(FA, FB) is injective (surjective). In the foregoing list of examples, example 1 is faithful and is full if and only if D is a full subcategory of C; examples 2 and 3 are faithful but not full (why?); and example 9 is full but not faithful.
We shall define next the important concept of natural transformation between functors. However, before we proceed to the definition, it will be illuminating to examine in detail the example mentioned briefly in the introduction to this chapter. We shall consider the more general situation of modules. Accordingly, we begin with the category R-mod for a ring R and the double dual functor D2 in this category. This maps a left R-module M into M** = (M*)* and a homomorphism L :M → N into L** = (L*)* :M** → N**. If x ∈ M, g ∈ N*, then L*g ∈ M* and (L*g)(x) = g(Lx). If Φ ∈ M**, L**Φ ∈ N** and (L**Φ) (g) = Φ(L*g). We now consider the map
of M* into R. This is contained in M** = homR(M*,R) and the map ηM:x ηM(x) is an R-homomorphism of M into M**. Now for any homomorphism L :M N, the diagram
is commutative, because if x ∈ M, then ηN(Lx) is the map g g(Lx) of N* into R and for φ = ηM(x) ∈ M**, (L**φ) (g) = (φ(L*g). Hence (L**ηM(x)) (g) = ηM(x) (L*g) = (L*g) (x) = g(Lx).
We now introduce the following definition of “naturality.”
DEFINITION 1.3. Let f and G be functors from C to D. We define a natural transformation η from f to G to be a map that assigns to every object A in C a morphism ηA∈homD(FA,GA) such that for any objects A,B of C and any f∈homc(A,B) the rectangle in
is commutative. Moreover, if every ηA is an isomorphism then η is called a natural isomorphism.
In the foregoing example we consider the identity functor 1R-mod and the double dual functor D2 on the category of left R-modules. For each object M of R-mod we can associate the morphism ηM of M into M**. The commutativity of (1) shows that η :M ηM is a natural transformation of the identity functor into the double dual functor.
We can state a stronger result if we specialize to finite dimensional vector spaces over a division ring 
. These form a subcategory of -mod. If V is a finite dimensional vector space over , we can choose a base (e1,e2,…,en) for V over . Let ei* be the linear function on V such that ei*(ej) =δij Then (e1*, e2*, … en*,) is a base for V* as right vector space over —the dual (or complementary) base to the base (e1…,en). This shows that V* has the same dimensionality n as V. Hence V** has the same dimensionality as V. Since any non-zero vector x can be taken as the element e1 of the base (e1, e2, …, en), it is clear that for any x ≠ 0 in V there exists a g ∈ V* such that g(x) ≠ 0. It follows that for any x ≠ 0 the map ηv(x) :f f(x) is non-zero. Hence ηv:x ηv(x) is an injective linear map of V into V**. Since dim V** = dim V, it follows that ηv is an isomorphism. Thus, in this case, η is a natural isomorphism of the identity functor on the category of finite dimensional vector spaces over onto the double dual functor on this category.
We shall encounter many examples of natural transformations as we proceed in our discussion. For this reason it may be adequate to record at this point only two additional examples.
1. We define the functor n in R-mod by mapping any module M into M(n) the direct sum of n copies of M (BAI, p. 175), and any homomorphism f: M → N into
For any M we define the diagonal homomorphism
Then δ(n):M δM(n) is a natural transformation of 1R-mod into n since we have the commutative diagram
2. We consider the abelianizing functor as defined in example 7 above, but we now regard this as a functor from Grp to Grp rather than from Grp to Ab. (This is analogous to changing the codomain of a function.) Let vG denote the canonical homomorphism of G onto the factor group. Then we have the commutative diagram
which shows that v:G vG is a natural transformation of the identity functor of Grp to Grp to the abelianizing functor.
Let F, G, H be functors from C to D, η a natural transformation of F to G, and ζ a natural transformation from G to H. If A ∈ obC then ηA ∈ homD(FA,GA) and ζA ∈ homD(GA,HA). Hence ζAηA ∈ homD(FA,HA). We have the commutativity of the two smaller rectangles in
which implies the commutativity of the large rectangle with vertices FA, HA, FB, HB. This implies that A ζAηA is a natural transformation from F to H. We call this ζη, the product of ζ and η.
If F is a functor from C to D, we obtain a natural transformation 1F of F into itself by mapping any, A ∈ ob C into lFA ∈ homD(FA, FA). If η is any natural transformation from F to a functor G from C to D, then we evidently have η1F = η = 1 Gη.
Let η be a natural isomorphism of F to G. Then ηA is an isomorphism ηA : FA → GA for every, A ∈ ob C. Hence we have the isomorphism ηA–l : GA → FA. The required commutativity is clear, so A → ηA–l is a natural isomorphism of G to F. We call this the inverse η–1 of η (η–1A = η–1). It is clear that we have η–1η = 1F and ηη–1 = 1G. Conversely, if η is a natural transformation from F to G and η is one from G to F such that ζ η =1G 1F and ηζ = 1G, then η is a natural isomorphism with η–1 = ζ.
If η is a natural isomorphism of a functor E of C to C with the functor lc, then the commutativity of
shows that E(f) = ηBfη–1A, which implies that the map f E(f) of hom (A, B) into hom (EA, EB) is bijective.
EXERCISES
1. Let F be a functor from C to D that is faithful and full and let f∈homc(A,B). Show that any one of the following properties of F(f) implies the same property for f:F(f) is monic, is epic, has a section, has a retraction, is an isomorphism.
2.Let M and N be monoids regarded as categories with a single object as in example 5, p. 12. Show that in this identification, a functor is a homomorphism of M into N and that a natural transformation of a functor F to a functor G corresponds to an element b∈N such that b(Fx) = (Gx)b, x∈M.
3. Use exercise 2 to construct a functor F and a monic (epic) f such that F(f) is not monic (epic).
4. Let G be a group, G the one object category determined by G as in example 6 on p. 12. Show that a functor from G to Set is the same thing as a homomorphism of G into the group Sym S of permutations of a set S, or, equivalently, an action of G on S (BAI, p. 72). Show that two such functors are naturally isomorphic if and only if the actions of G are equivalent (BAI, p. 74).
5. Let B, C, D, E be categories, F and G functors from C to D, K a functor from B to C, and H a functor from D to E. Show that if η is a natural transformation from F to G, then A HηA is a natural transformation from HF to HG for A ∈obC and B ηKB is a natural transformation from FK to GK for B∈obB.
6. Define the center of a category C to be the class of natural transformations of the identity functor 1C to 1C. Let C = R-mod and let C be an element of the center of R. For any M ∈ ob R-mod let ηM(c) denote the map x cx, x ∈ M. Show that η(c) :M ηM(c) is in the center of C and every element of the center of C has this form. Show that c η(c) is a bijection and hence that the center of R-mod is a set.
1.4 EQUIVALENCE OF CATEGORIES
We say that the categories C and D are isomorphic if there exist functors F :C → D (from C to D) and G :D → C such that GF = 1C and FG = 1D. This condition is rather strong, so that in most cases in which it holds one tends to identify the isomorphic categories. Here is an example. Let C = Ab and D = -mod. If M is an abelian group (written additively), M becomes a -module by defining nx for n ∈ , x ∈ M, as the nth multiple of x (BAI, p. 164). On the other hand, if M is a -module, then the additive group of M is an abelian group. In this way we have maps of ob Ab into ob -mod and of ob -mod into ob Ab that are inverses. If f is a homomorphism of the abelian group M into the abelian group N, then f(nx) = nf (x), n ∈ , x ∈M. Hence f is a homomorphism of M as -module into N as -module. Conversely, any -homomorphism is a group homomorphism. It is clear from this that Ab and -mod are isomorphic categories, and one usually identifies these two categories.
Another example of isomorphic categories are R-mod and mod-Rop for any ring R. If M is a left R-module, M becomes a right Rop-module by defining xr = rx for x ∈ M, r ∈ Rop = R (as sets). Similarly, any right Rop-module becomes a left R-module by reversing this process. It is clear also that a homomorphism of RM, M as left R-module, into RN is a homomorphism of MRop, M as right Rop-module, into NRop. We have the obvious functors F and G such that gf = 1R-mod and FG = 1mod – RoP. Hence the two categories are
The concept of isomorphism of categories is somewhat too restrictive; a considerably more interesting notion is obtained by broadening this in the following manner. We define C and D to be equivalent categories if there exist functors F:C → D and G:D → C such that GF 
lc and FG 1D where denotes the natural isomorphism of functors. Evidently isomorphism of categories implies equivalence. It is clear also that the relation of equivalence between categories is what the name suggests: it is reflexive, symmetric, and transitive.
We note that the functor G in the definition of equivalence is not uniquely determined by F. It is therefore natural to shift the emphasis from the pair (F, G) to the functor f and to seek conditions on a functor f :C → D in order that there exists a G:D → C such that (F, G) gives an equivalence, that is, GF 1C and FG 1D. We have seen that GF lc implies that the map fGF(f) of homC(A,B) onto homc(GFA,GFB) is bijective. Similarly, g FG(g) is bijective of homD(A′,B′) onto homD(FGA',FGB'). Now the injectivity of f GF(f) implies the injectivity of the map f F(f) of the set homC(A,B) into the set homD(FA,FB) and the surjectivity of g FG(g) implies the surjectivity of f F(f). Thus we see that the functor F is faithful and full. We note also that given any , A′ ∈ obD, the natural isomorphism FG 1D gives an isomorphism ηA′ ∈ homD(A′, FGA′). Thus if we put A = GA′ ∈ obC, then there is an isomorphism contained in homD(A′, FA) or, equivalently, in homD(FA,A′).
We shall now show that the conditions we have sorted out are also sufficient and thus we have the following important criterion.
PROPOSITION 1.3. Let F be a functor from C to D. Then there exists a functor G :D → C such that (i, G) is an equivalence if and only if F is faithful and full and for every object A′ of D there exists an object A of C such that FA and A′ are isomorphic in D, that is, there is an isomorphism contained in hom D(FA,A′).
Proof. It remains to prove the sufficiency of the conditions. Suppose these hold. Then for any A′ ∈ ob D we choose ,A ∈ ob C such that FA and A′ are isomorphic and we choose an isomorphism ηA′ : A′ → FA. We define a map G of ob D into ob C by A′ A where A is as just chosen. Then ηA′ : A′ → FGA′. Let B' be a second object of D and let f′ ∈ homD (A′, B′). Consider the diagram
Since ηA′ is an isomorphism, we have a unique morphism ηB′f′ηA′–1: FGA′ → FGB′ making a commutative rectangle. Since F is full and faithful, there is a unique f GA′ → GB′ in C such that F(f) = ηf′ηA–1 We define the map G from homD(A–B’) to homC(GA′, GB′) by f′ f. Then we have the commutative rectangle
and G(f') is the only morphism GA' → GB' such that (2) is commutative.
Now let g' ∈ homD(B', C'). Then we have the diagram
in which the two small rectangles and hence the large one are commutative. Since F is a functor, we have the commutative rectangle
On the other hand, we have the commutative rectangle
and G(g′f′): GA′ GC′ is the only morphism for which (4) is commutative. Hence we have G(g′)G(f′) = G(g′ f′). In a similar manner we see that G(1A) = 1GA′. Thus the maps G : A′ GA′, G : f′ G(f′) for all of the hom sets homD(A′, B′) constitute a functor from D to C. Moreover, the commutativity of (2) shows that η′ : A′ ηA′, is a natural isomorphism of 1D to FG.
We observe next that since F is faithful and full, if A ,B ∈ ob C and f′ : FA → FB is an isomorphism, then the morphism f : A → B such that F(f)= f′ is an isomorphism (exercise 1, p. 25). It follows that since ηFA : FA → FGFA is an isomorphism, there exists a unique isomorphism ζA : A → GFA such that F(ζA) = ηFA. The commutativity of (2) for A′ = FA, B′ = FB, and f′ = F(f) where f : A→ B in C implies that
is commutative. Since F is faithful, this implies that
is commutative. Hence ζ : A → ζA is a natural isomorphism of lc into GF.
As an illustration of this criterion we prove the following very interesting proposition.
PROPOSITION 1.4. Let R be a ring, and Mn(R) the ring of nxn matrices with entries in R. Then the categories mod-R and mod-Mn(R) of right modules over R and Mn(R) respectively are equivalent.
Before proceeding to the proof we recall some elementary facts about matrix units in Mn(R) (BAI, pp. 94–95). For i, j = 1,…, n, we define eij to be the matrix whose (i,j)-entry is 1 and other entries are 0, and for any a∈R we let a′ denote diag {a,… ,a}, the diagonal matrix in which all diagonal entries are a.
Then we have the multiplication table
and
Moreover,
and this matrix has a in the (i,j)-position and 0’s elsewhere. Hence if
then
We can now give the
Proof of Proposition 1.4. Let M be a right R-module and let M(n) be a direct sum of n copies of M (BAI, p. 175). If x = (x1, x2, …, xn) ∈ M(n) and A ∈ Mn(R) as in (8), we define xA to be the matrix product
where
and the right-hand side is as calculated in M. Using the associativity of matrix multiplication (in this mixed case of multiplication of “vectors” by matrices) we can verify that M(n) is a right Mn(R)-module under the action we have defined. Thus we have a map M M(n) of ob mod-R into ob mod-Mn(R). If f is a module homomorphism of M into N, then the diagonal homomorphism f(n) : (x1, x2, …, xn) (f(x1), f(x2), …, f(xn)) is a homomorphism of the right Mn(R)-module M(n) intotheright Mn(R)-module N(n). The maps M M(n), f → f(n) constitute a functor F from mod-R to mod-Mn(R). We shall verify that F satisfies the conditions of Proposition 1.3.
1. F is faithful. Clear.
2. F is full: Let g be an Mn(R)-homomorphism of M(n) into N(n) where M and N are right R-modules. Now M(n)e11 is the set of elements (x, 0, …, 0), x ∈ M, and N(n)e11 is the set of elements (y, 0, …, 0), y ∈ N. Since g is an Mn(R)-homomorphism, g(M(n)e11) ⊂ N(n)e11. Hence g(x, 0, …, 0) = (f(x), 0, …, 0). It is clear that f is additive, and f(xa) = f(x)a for a ∈ R follows from g((x, 0, …, 0)a′) = (g(x, 0, …, 0))a′. Hence f is an R-homomorphism of M into N. Now (x, 0, …, 0)e1i = (0, …, 0, 
, 0, …, 0), so g((x, 0, …, 0)e1i) = (g(x, 0, …, 0))e1i implies that g(0, …, 0, , 0, …, 0) = (0, …, 0, f(), 0,…, 0). Then g = f(n) and F is full.
3. Any right Mn(R)-module M′ is isomorphic to a module FM, M a right R-module: The map a a′ is a homomorphism of R into Mn(R). Combining this with the action of Mn(R) on M′, we make M′ a right R-module in which x′a = x′a′, x′∈ M′. Then M = M′e11 is an R-submodule of M′ since e11a′ = a′e11, a∈R. Moreover, x′ei1 = x′ei1,e11∈M for any i. We define a map ηM, :M′ → FM = M(n) by
Direct verification using (9) and the definition of x′ a shows that ηM′ is an Mn(R)-homomorphism. If x′ei1 =0 for 1 ≤ i ≤ n, then x′ = ∑x′eii = ∑x′eileli = 0. Hence ηM′ is injective. Moreover, ηM′, is surjective: If (x1,x2,…,xn)∈ M(n), then xi = x′ie11 = (x′ie11)eil and
Thus ηM′ is an isomorphism.
EXERCISES
1. Let (F, G) be an equivalence of C into D and let f ∈ homc(A, B). Show that any one of the following properties of f implies the same property for F(f) : f is monic, is epic, has a section, has a retraction, is an isomorphism.
2. Are mod-R and mod-Mn(R) isomorphic for n > 1?
There are many basic constructions in mathematics and especially in algebra—such as the (Cartesian) product of sets, the direct product of groups, the disjoint union of sets, and the free product of groups—that have simple characterizations by means of properties of maps. This fact, which has been known for some time, can be incorporated into category theory in the definition and examples of products and coproducts in categories. We shall begin with an example: the direct product of two groups.
Let G = G1 × G2, the direct product of the groups G1 and G2 :G is the group of pairs (g1,g2), gi ∈ Gi with the multiplication defined by
the unit 1 = (11, 12) where 1i is the unit of Gi, and (g1, g2)−1 = (g1−1, g2−1). We have the projections pi : G → Gi, defined by
These are homomorphisms, since P1,((g1,g2)(h1,h2)) = P1(g1h1,g2h2) = g1h1 = (P1(g1,G2))(P1(h1,h2)) and p1(1) = 11 Similar relations hold for p2.
Now let H be another group and let fi :H → Gi be a homomorphism of H into Gi. Then we define a map f of H into G = G1 × G2 by
It is clear that this is a homomorphism and Pif(h) = fi(h). Hence we have the commutativity of
Next let f′ be any homomorphism of H into G such that pif’ =fi, i = 1, 2. Then f′(h) = (f1(h),f2(h)) =f(h) so f′ = f Thus f is the only homomorphism H → G making (14) commutative.
We now formulate the following
DEFINITION 1.4. Let A1 and A2 be objects of a category C. A product of A1 and A2 in C is a triple (A,p1,p2) where A ∈ ob C and Pi ∈ homC(A,Ai) such that if B is any object in C and fi ∈ homC(B, Ai), i = 1, 2, then there exists a unique f ∈ homC(B,A) such that the diagrams
are commutative.
It is clear from our discussion that if G1 and G2 are groups, then (G = G1 × G2,P1,P2) is a product of G1 and G2 in the category of groups. The fact that (G1 × G2,p1,p2) is a product of G1 and G2 in Grp constitutes a characterization of the direct product G1 × G2 and the projections pi. This is a special case of
PROPOSITION 1.5. Let (A,p1,p2) and (A′,p′1p′2) be products of A1 and A2 in C. Then there exists a unique isomorphism h:A→A′ such that Pi = p′1h, i= 1, 2.
Proof. If we use the fact′that (A′, p′1, p2) is a product of A1 and A2, we obtain a unique homomorphism h : A → A′ such that pi = p′ih, i = 1, 2. Reversing the roles of (A, pl, p2) and (A′, p′l, p′2) we obtain a unique homomorphism h′ : A′ → A such that p′i = pih′. We now have pi = pih′h and p′i = pihh′. On the other hand, if we apply definition 1.4 to B = A and fi = pi, we see that 1A is the only homomorphism A → A such that pi = pi1A. Hence h′h = lA and similarly hh′ = 1A. Thus h is an isomorphism and h′ = h−1.
Because of the essential uniqueness of the product we shall denote any product of A1 and A2 in the category C by Al∏2. The concept of product in a category can be generalized to more than two objects.
DEFINITION 1.4′. Let {Aα|α ∈ I} be an indexed set of objects in a category C. We define a product ∏Aα of the Aα to be a set {A,pα|α∈I} where A ∈ ob C, pα ∈ homc(A,Aα) such that if B ∈ ob C and fα ∈ homC(B, Aα), α ∈ I, then there exists a unique f ∈ homc(B, A) such that every diagram
is commutative.
We do not assume that the map a Aα is injective. In fact, we may have all the Aα equal. Also, if a product exists, then it is unique in the sense defined in Proposition 1.5. The proof of Proposition 1.5 carries over to the general case without change. We now consider some examples of products in categories.
EXAMPLES
1. Let {Aα|α ∈ I} be an indexed set of sets. We have the product set A = ∏Aα, which is the set of maps a: I → Aα such that for every α ∈ I, a(α) ∈ Aα. For each α we have the projection pα:a a(α). We claim that {A,pα} is a product of the Aα in the category Set. To see this, let B be a set and for each α ∈ I let fα be a map: B → Aα. Then we have the map f : B → A such that f(b) is α → fα). Then pαf = fα and it is clear that f is the only map: B → A satisfying this condition. Hence {A, pα} satisfies the condition for a product of the Aα.
2. Let {Gα|α ∈ I} be an indexed set of groups. We define a product in G = ∏Gα by gg′(α) = g(α)g′(α) for g, g′ ∈G and we let 1 ∈G be defined by 1(α) = lα, the unit of Gα for α ∈ I. It is easy to verify that this defines a group structure on G and it is clear that the projections as defined in Set are homomorphisms. As in example 1, {G, pα) is a product of the Gα in the category of groups.
3. The argument expressed in example 2 applies also in the category of rings. If {Rα|α ∈I} is an indexed set of rings, we can endow ∏Rα with a ring structure such that the projections pα become ring homomorphisms. Then {∏Rα,pα} is a product of the Rα in the category Ring.
4. In a similar manner we can define products of indexed sets of modules in R-mod for any ring R.
We now consider the dual of the concept of a product. This is given in the following
DEFINITION 1.5. Let {Aα | α∈I} be an indexed set of objects of a category C. We define a coproduct Aα to be a set {A, iα|α ∈ I} where A∈ob C and iα∈hom (Aα, A) such that if B ∈ ob C and gα ∈ homc(Aα,B), α ∈I, then there exists a unique G ∈homc(A, B) such that every diagram
is commulative.
It is readily seen that if {A, iα|α ∈ I} and {A′, i′α|α ∈ I} are coproducts in C of {Aα|α∈I}, then there exists a unique isomorphism k :A′ → A such that iα = ki′α for all α ∈ I.
If {Aα|α ∈I} is an indexed set of sets, then there exists a set 
Aα that is a disjoint union of the sets Aα. Let iα denote the injection map of Aα into Aα. Let B be a set and suppose for each α we have a map gα of Aα into B. Then there exists a unique map g of Ax into B such that the restriction g|Aα = gα, α ∈ I. It follows that {Aα, iα} is a coproduct of the Aα in Set.
We shall show in the next chapter that coproducts exist for any indexed set of objects in Grp or in Ring (see p. 84). In the case of R-mod this is easy to see: Let {Mα|α ∈ I} be an indexed set of left R-modules for the ring R and let ∏Mα be the product set of the Mα endowed with the left R-module structure in which for x, y ∈ ∏ Mα and r ∈ R, (x + y) (α) = x(α) + y(α), (rx) (α) = r(x(α)). Let Mα be the subset of ∏Mα consisting of the x such that x(α) = 0 for all but a finite number of the α ∈ I. Clearly Mα is a submodule of ∏Mα. If xα ∈ Mα, we let iαxα be the element of Mα that has the value xα at α and the value 0 at every β ≠ α, β ∈I. The map iα : xα iαxα is a module homomorphism of Mα into Mα. Now let N ∈ R-mod and suppose for every α ∈ I we have a homomorphism g α :Mα → N. Let x∈;Mα. Then x(α) = 0 for all but a finite number of the α; hence ∑gα ,x(α) is well defined. We define g as the map x ∑gαx(α) of Mα into N. It is readily verified that this is a homomorphism of Mα into N and it is the unique homomorphism of Mη into N such that giα = gα. Thus {Mα, iα} is a coproduct in R-mod of the Mα. We call Mα the direct sum of the modules Mα.
Since the category Ab is isomorphic to -mod, coproducts of arbitrary indexed sets of objects in Ab exist.
EXERCISES
1. Let S be a partially ordered set and S the associated category as defined in example 12 on p. 13. Let {aα|α ∈ I} be an indexed set of elements of S. Give a condition on {aα} that the corresponding set ol objects in S has a product (coproduct). Use this to construct an example of a category in which every finite subset of objects has a product (coproduct) but in which there are infinite sets of objects that do not have a product (coproduct).
2. A category C is called a category with a product (coproduct) if any pair of objects in C has a product (coproduct) in C. Show that if C is a category with a product (coproduct), then any finitely indexed set of objects in C has a product (coproduct).
3. An object A of a category C is called initial (terminal) if for every object X of C, homc(A, X) (homc(X, A)) consists of a single element. An object that is both initial and terminal is called a zero of C. Show that if A and A′ are initial (terminal), then there exists a unique isomorphism h in homc(A, A′).
4. Let A1 and A2 be objects of a category C and let C/{A1, A2} be the category defined in exercise 5 of p. 14. Show that A1 and A2 have a product in C if and only if C/{A1, A2} has a terminal object. Note that this and exercise 3 give an alternative proof of Proposition 1.5. Generalize to indexed sets of objects in C.
5. Use exercise 6 on p. 15 to give an alternative definition of a coproduct of objects of a category.
6. Let fi : Ai → B, i = 1, 2, in a category C. Define a pullback diagram of {f1,f2} to be a commutative diagram
such that if
is any commutative rectangle containing f1 and f2, then there exists a unique k :D → C such that
is commutative. Show that if (C, g1, g2) and (C′, g′1, g′2) determine pullbacks f1 and f2 as in (17), then there exists a unique isomorphism k : C′ → C such that g′i = gik, i = 1, 2.
7. Let fi: Gi → H in Grp. Form G1 × G2 and let M be the subset of G1 × G2 of elements (a1, a2) such that f1(al) = f2(a2). This is a subgroup. Let mi = pi|M where pi is the projection of G1 × G2 on Gi. Show that {m1, m2} defines a pullback diagram of f1 and f2.
8. Dualize exercise 6 to define a pushout diagram determined by fi : B → Ai i = 1, 2, in C. Let fi:B → Ai i = 1, 2, in R-mod. Form A1A2. Define the map f:b (–f1(b),f2(b)) of B into A1 A2. Let I = Im f and put N = (A1 A2)/I. Define ni:Ai→N by n1a1 = (a1,0) + I, n2a2 = (0,a2) + I. Verify that {n1,n2} defines a pushout diagram for f1 and f2.
1.6 THE HOM FUNCTORS. REPRESENTABLE FUNCTORS
We shall now define certain important functors from a category C and the related categories Cop and Cop × C to the category of sets. We consider first the functor hom from Cop × C. We recall that the objects of Cop × C are the pairs (A, B), A, B∈ob C, and a morphism in this category from (A, B) to (A′, B′) is a pair (f, g) where f : A′ → A and g : B → B′. If (f′, g′) is a morphism in Cop × C from (A′, B′) to (A”, B”),so f′ : A″ → A′, g′ : B′ → B″, then (f′, g′)(f,g) = (ff′, g′g). Also 1(A, B) = (1A, 1B).
We now define the functor hom from Cop × C to Set by specifying that this maps the object (A, B) into the set hom (A,B) (which is an object of Set) and the morphism (f, g) : (A, B) → (A’. B′) into the map of hom (A, B) into hom (A′,B′) defined by
This makes sense since f : A′ → A, g : B → B′, k : A → B, so gkf : A′ → B′. These rules define a functor, since if (f′, g′):(A″, B″) → (A′, B′), then (f′, g′)(f, g) = (ff′, g′g) and
Thus hom ((f′,g′) (f,g)) = hom (f′,g′) hom (f,g). Moreover, if f= 1A and g= 1B, then (18) shows that hom (1A, 1B) is the identity map on the set hom (A, B). Thus the defining conditions for a functor from Cop × C to Set are satisfied.
We now fix an object A in C and we define a functor hom (A, —) from C to Set by the rules
of hom(A, B) into hom(A,B′). It is clear that this defines a functor. We call this functor the (covariant) hom functor determined by the object A in C.
In a similar manner we define the contravariant hom functor hom (— ,B) determined by B ∈ ob C by
of hom (A, B) into hom (A′, B). Now let f : A′ → A, g : B → B′ k : A → B. Then
and (gk)f= g(kf) = hom (f, g) (k). Hence
is commutative. We can deduce two natural transformations from this commutativity. First, fix g : B → B′ and consider the map, A hom(A,g) ∈ homSet(hom (A, B) hom (A,B′)) The commutativity of the foregoing diagram states that A hom(A,g) is a natural transformation of the contravariant functor hom(—,B) into the contravariant functor hom( — ,B′)o Similarly, the commutativity of (21) can be interpreted as saying that for f : A′ → A the map B hom (f,B) is a natural transformation of hom (A, —) into hom (A′, —).
In the applications one is often interested in “representing” a given functor by a hom functor. Before giving the precise meaning of representation we shall determine the natural transformations from a functor homc(A, −), which is a functor from C to Set, to any functor F from C to Set. Let a be any element of the set FA and let B∈ ob C, k ∈ homc(A,B). Then F(k) is a map of the set FA into the set FB and its evaluation at a, F(k) (a) ∈ FB. Thus we have a map
of homC(A, B) into FB. We now have the important
YONEDA’S LEMMA. Let F be a functor from C to Set, A an object of C, a an element of the set FA. For any B ∈ ob C let aB be the map of homc(A,B) into FB such that k f(k)(a). Then B aB is a natural transformation η(a) of homc(A, −) into F. Moreover, a η(a) is a bijection of the set FA onto the class of natural transformations of homc(A, −) to F. The inverse of a η(a) is the map η ηA(lA) ∈ FA.
Proof. We have observed that (22) is a map of homc(A,B) into FB. Now let g:B → C. Then
Hence we have the commutativity of
Then η(a) :B → aB is a natural transformation of hom (A, —) into F. Moreover η(a)(1A) = aA(1A) = f(1A)(a)=a.
Next let η be any natural transformation of hom (A, —) into F. Suppose f ∈ homC(A,B). The commutativity of
implies that ηB(f) = ηB(f lA) = ηB(hom (A,f) (1A) = F(f)ηA(1A) = f(f) (a) where a = ηA(lA)∈ FA. This shows that η = η(a) as defined before.
The foregoing pair of results proves the lemma.
We shall call a functor F from C to Set representable if there exists a natural isomorphism of F with a functor hom (A, —) for some A∈ob C. If η is this natural isomorphism then, by Yoneda’s lemma, η is determined by A and the element a = ηA(1A) of FA. The pair (A, a) is called a representative of the representable functor F.
EXERCISES
1. Apply Yoneda’s lemma to obtain a bijection of the class of natural transformations of hom (A, −) to hom (A′ −) with the set homC(A′, A).
2. Show that f:B → B′ is monic in C if and only if hom (A,f) is injective for every A ∈ obC.
3. Dualize Yoneda’s lemma to show that if F is a contravariant functor from C to Set and A ∈ ob C then any natural transformation of homC(–, A) to F has the form B aB where aB is a map of homC(B, A) into FB determined by an element a ∈ FA as
Show that we obtain in this way a bijection of the set FA with the class of natural transformations of homc(— ,A) to F.
1.7 UNIVERSALS
Two of the earliest instances of the concept of universals are those of a free group determined by a set X and the universal (associative) enveloping algebra of a Lie algebra. We have considered the first for a finite set X in BAI, pp. 68–69, where we constructed for a set X, of cardinality r < ∞, a group FG(r) and a map i :x → 
of X into FG(r) such that if G is any group and g is a map of X into G then there exists a unique homomorphism g: FG(r) → G, making the diagram
commutative. Here 
is regarded as a map of sets.
We recall that a Lie algebra L over a field is a vector space equipped with a bilinear product [xy] such that [xx] = 0 and [[xy]z] +[[yz]x] + [[zx]y] = 0. If A is an associative algebra, A defines a Lie algebra A− in which the composition is the Lie product (or additive commutator) [xy] = xy-yx where xy is the given associative product in A (BAI, pp. 431 and 434). It is clear that if A and B are associative algebras and f is a homomorphism of A into B, then f is also a Lie algebra homomorphism of A− into B−.
If L is a Lie algebra, a universal enveloping algebra of L is a pair (U(L),u) where U(L) is an associative algebra and u is a homomorphism of L into the Lie algebra U(L)− such that if g is any homomorphism of L into a Lie algebra A− obtained from an associative algebra A− then there exists a unique homomorphism of the associative algebra U(L) into A such that
is a commutative diagram of Lie algebra homomorphisms. We shall give a construction of (U(L),u) in Chapter 3 (p. 142).
Both of these examples can be formulated in terms of categories and functors. In the first we consider the categories Grp and Set and let F be the forgetful functor from Grp to Set that maps a group into the underlying set and maps a group homomorphism into the corresponding set map. Given a set X, a free group determined by X is a pair (U,u) where U is a group and u is a map of X into U such that if G is any group and g is a map of X into G, then there exists a unique homomorphism of U into G such that u = G holds for the set maps.
For the second example we consider the category Alg of associative algebras and the category Lie of Lie algebras over a given field. We have the functor F from Alg to Lie defined by FA = A− for an associative algebra A, and if f:A → B for associative algebras, then F(f) = f:A− → B− for the corresponding Lie algebras. For a given Lie algebra L, a universal envelope is a pair (U(L),u) where U(L) is an associative algebra and u is a Lie algebra homomorphism of L into U(L)− such that if g is any homomorphism of L into a Lie algebra A−, A associative, then there exists a unique homomorphism of U(L) into A such that u = g.
We now give the following general definition of universals.
DEFINITION 1.6. Let C and D be categories, F a functor from C to D. Let B be an object in D. A universal from B to the functor F is a pair (U, u) where U is an object of C and u is a morphism from B to FU such that if g is any morphism from B to FA, then there exists a unique morphism of U into A in C such that
is commutative. U is called a universal C-object for B and u the corresponding universal map.
It is clear that the two examples we considered are special cases of this definition. Here are some others.
EXAMPLES
1. Field of fractions of a commutative domain. Let Dom denote the subcategory of the category Ring whose objects are commutative domains ( = commutative rings without zero divisors ≠0) with monomorphisms as morphisms. Evidently this defines a subcategory of Ring. Moreover, Dom has the full subcategory Field whose objects are fields and morphisms are monomorphisms. If D is a commutative domain, D has a field of fractions F (see BAI, p. 115). The important property of F is that we have the monomorphism u:a a/1 of D into F, and if g is any monomorphism of D into a field F′, then there exists a unique monomorphism of F into F′ such that g = gu. We can identify D with the set of fractions d/1 and thereby take u to be the injection map. When this is done, the result we stated means that any monomorphism of D (⊂ F) has a unique extension to a monomorphism of F into F′ (see Theorem 2.9, p. 117 of BAI).
To put this result into the mold of the definition of universals, we consider the injection functor of the subcategory Field into the category Dom (see example 1 on p. 20). If D is a commutative domain, hence an object of Dom, we take the universal object for D in Field to be the field f of fractions of D and we take the universal map u to be the injection of D into F. Then (F, u) is a universal from D to the injection functor as defined in Definition 1.6.
2. Free modules. Let R be a ring, X a non-vacuous set. We define the free (left) R-module xR to be the direct sum of X copies of R, that is, xR= Ma, a∈X, where every Ma = R. Thus XR is the set of maps f of X into R such that f(x) = 0 for all but a finite number of the x’s. Addition and the action of R are the usual ones: (f+g)(x) = f(x) + g(x), (rf)(x) = rf(x). We have the map u: x of X into xR where is defined by
If f ∈ x R and {x1,…, xn} is a subset of X such that f(y) = 0 for y∉ {x1,… ,xn}, then f = ∑rii where f(xi) = ri. Moreover, it is clear that for distinct i ∑rii= 0 implies every ri = 0. Hence the set X = {|x∈X} is a base for xR in the sense that every element of this module is a sum ∑,x∈xrxx which is finite in the sense that only a finite number of the rx are ≠ 0, and ∑rx = 0 for such a sum implies that every rx = 0.
Now suppose M is any (left) R-module and ϕ : x mx is a map of X into M. Then
:∑rx→∑rxmx
is a well-defined map of xR into M. It is clear that this is a module homomorphism. Moreover, 
() = mx so we have the commutativity of
Since a module homomorphism is determined by its restriction to a set of generators, it is clear that is the only homomorphism of the free module XR into M, making the foregoing diagram commutative.
Now consider the forgetful functor F from R-mod to Set that maps an R-module into its underlying set and maps any module homomorphism into the corresponding set map. Let X be a non-vacuous set. Then the results mean that (xR,u) is a universal from X to the functor F.
3. Free algebras and polynomial algebras. If K is a commutative ring, we define an (associative) algebra over K as a pair consisting of a ring (A, +,.,0, 1) and a K-module A such that the underlying sets and additions are the same in the ring and in the module (equivalently, the ring and module have the same additive group), and
for a∈K, x,y∈A (cf. BAI, p. 407). The algebra A is said to be commutative if its multiplication is commutative. Homomorphisms of K-algebras are K-module homomorphisms such that 1 1, and if x x′ and y y′ then xy x′y′. We have a category K- alg of K-algebras and a category K-comalg of commutative K-algebras. In the first, the objects are K-algebras and the morphisms are K-algebra homomorphisms. K-comalg is the full subcategory of commutative K-algebras.
We have the forgetful functors from K-alg and K-comalg to Set. If X is a non-vacuous set, then a universal from X to the forgetful functor has the form (K{X},u) where K{X} is a K-algebra and u is a map of X into K{X} such that if g is any map of X into a K-algebra A, then there is a unique K-algebra homomorphism of K{X} into A such that u = g (as set maps). If (K{X},u) exists, then K{X} is called the free K-algebra determined by X. In a similar manner we can replace K-alg by K-comalg. A universal from X to the forgetful functor to Set is denoted as (K[X],u). If X = {x1,x2, …, xn}, we let K[X1,…,Xn] be the polynomial ring in the indeterminates Xi with coefficients in K. If g is a map of X into a commutative K-algebra A, then there is a unique homomorphism of K[X1…, Xn] into A as K-algebras such that Xi → g(x1), 1 ≤ i ≤ n (BAI, p. 124). Hence K[X1,…,Xn]and the map xi Xi constitute a universal from X to the forgetful functor from K-comalg to Set.
4. Coproducts. Let C be a category. We have the diagonal functor of C to C × C that maps an object A of C into the object (A, A) of C × C and a morphism f:A→B into the morphism (f, f):(A,A) → (B,B). A universal from (A1,A2) to is a pair (U,u) where U is an object of C and u = (ul,u2), ui:Ai→U such that if C ∈ obC and gi: Ai→ C, then there is a unique :U → C such that gi = u1 i = 1, 2. This is equivalent to saying that (U,u1u2) is a coproduct in C of A1 and A2. This has an immediate generalization to coproducts of indexed sets of objects of C. If the index set is I = {α}, then a coproduct 
C.x is a universal from (Ay) to the diagonal functor from C to the product category C1 where C1 is the product of I-copies of C (cf. p. 21).
One can often construct universals in several different ways. It is immaterial which determination of a universal we use, as we see in the following strong uniqueness property.
PROPOSITION 1.6. If (U,u) and (U′ u′) are universals from an object B to a functor F, then there exists a unique isomorphism h:U → U′ such that u′ = f(h)u.
We leave the proof to the reader. We remark that this will follow also from exercise 4 below by showing that (C, u) is the initial element of a certain category.
As one might expect, the concept of a universal from an object to a functor has a dual. This is given in
DEFINITION 1.7. Let C and D be categories, G a functor from D to C. Let A ∈ ob C. A universal from G to A is a pair (V,v) where V ∈ ob D and vehomc(GV,A) such that if B ∈ ob D and G ∈homc(GB. A), then there exists a unique :B → V such that vG() = G.
As an illustration of this definition, we take C = R-mod, D = Set as in example 2 above. If X is a non-vacuous set, we let GX = xR, the free module determined by X. It is convenient to identify the element x ∈ X with the corresponding element of xR and we shall do this from now on. Then xR contains X and X is a base for the free module. The basic property of X is that any map ϕ of X into a module M has a unique extension to a homomorphism of xR into M. If X = Ø, we define xR = 0. If X and Y are sets and ϕ is a map of X into Y, then ϕ has a unique extension to a homomorphism ϕ of xR into YR. We obtain a functor G from Set to R-mod by putting G(X) = xR and G(ϕ) = ϕ.
Now let M be a (left) R-module, FM the underlying set so GFM = FMR. Let v be the homomorphism of GFM into M, extending the identity map on FM. Let X be a set and g a homomorphism of xR into M, and put = g/X. Then G(g) is the homomorphism of xR into FMR such that G()(x) = g(x), x∈X, and vG()(x) = g(x). Hence vG() = g. Moreover, is the only map of X into FM satisfying this condition. Hence (FM,v) is a universal from the functor G to the module M.
EXERCISES
1. Let D be a category, the diagonal functor from D to D − D. Show that (V,v), v = (v1,v2), is a universal from to (A1,A2) if and only if (V,v1,v2) is a product of A1 and A2 in D.
2. Let Rng be the category of rings without unit, F the functor from Ring to Rng that forgets the unit. Show that any object in Rng possesses a universal from it to F. (See BAI, p. 155.)
3. Let F1 be a functor from C1 to C2, F2 a functor from C2 to C3. Let (U2,u2) be a universal from B to F2, (U1,u1 a universal from U2 to F1. Show that (U1,F2(U1)U2) is a universal from B to F2F1.
4. Let F be a functor from C to D, B an object of D. Verify that the following data define a category D(B,F): The objects are the pairs (A,g) where A ∈ obC and g∈homD(B,FA). Define hom ((A,g) (A′g′)) as the subset of homc(A, A′) of h such that g′ = f(h)g and arrange to make these hom sets non-overlapping. Define multiplication of morphisms as in C and l(A g) = 1A. Show that (U, u) is a universal from B to F if and only if (U, u) is an initial element of D(B, F). Dualize.
1.8 ADJOINTS
We shall now analyze the situation in which we are given a functor F from C to D such that for every object B of D there exists a universal (U, u) from B to F. The examples of the previous section are all of this sort. First, we need to consider some alternative definitions of universals.
Let F be a functor from C to D, B an object of D, (U,u) a pair such that U∈ob C and u∈homD(B,FU). Then iff∈homc(U,A), f(f)∈homD(FU,FA) and F(f)u∈homD(B,FA). Accordingly, we have the map
of the set homC(U,A) into the set homD(B,F,4). By definition, (U, u) is a universal from B to F if for any object A of C and any morphism g:B → FA in D there is one and only one :U → A such that g = F()u. Evidently this means that (U, u) is a universal from B to f if and only if for every A ∈ob C the map ηA from homC(U,A) to homD(B, FA) given in (23) is bijective. If this is the case and h: A → A′ in C, then the diagram
is commutative, since one of the paths from an f∈homc(U, A) gives F(hf)u and the other gives F(h)F(f)u, which are equal since F(hf) = f(h)F(f). It follows that η:A ηA is a natural isomorphism of the functor homc(U, −) to the functor homD(B, F −) that is obtained by composing F with the functor homD(B, −). Note that both of these functors are from C to Set, and since ηU(1U) = F(1U)u = 1FUu = u, the result is that homD(B,F −) is representable with (U, u) as representative.
Conversely, suppose homD(B, f −) is representable with (U, u) as representative. Then, by Yoneda’s lemma, for any object A in C, the map ηA of (23) is a bijection of homc(U, A) onto homD(B, FA). Consequently (U,u) is a universal from B to F.
Similar considerations apply to the other kind of universal. Let G be a functor from D to C, A an object of C, (V, v) a pair such that V∈ob D and v ∈ homc(GV,A). Then for any B ∈ ob D we have the map
of homD(B, V) into homc(GB, A), and (V,v) is a universal from G to A if and only if ζB is a bijection for every B∈obD. Moreover, this is the case if and only if ζ:B ζB is a natural isomorphism of the contravariant functor homD(–, V) with the contravariant functor homc(G –, A) obtained by composing G with homc( – A).
We now assume that for every B∈ ob D we have a universal from B to the functor F from C to D. For each B we choose a universal that we denote as (GB, uB). Then for any A ∈ ob C we have the bijective map
of homC(GB,A) onto homD(B,FA), and for fixed B, A ηB,A is a natural isomorphism of the functor homC(GB, –) to the functor homD(B,F –).
Let B′ be a second object in D and let h :B → B′. We have the diagram
so uBh∈homD (B, FGB′), and since (GB,uB) is a universal from B to F there is a unique morphism G(h): GB → GB′ such that the foregoing diagram becomes commutative by filling in the horizontal FG(h):
The commutativity of the diagram
and the functorial property (FG(k)) (FG(h)) = F(G(k)G(h)) imply that
is commutative. Since G(kh) is the only morphism GB → GB″ such that FG(kh)uB = uBkh, it follows that G(kh) = G(k)G(h). In a similar manner we have G(1B) = 1GB. Thus G is a functor from D to C.
We wish to study the relations between the two functors F and G. Let A∈ob C. Then F A∈ ob D, so applying the definition of a universal from FA to F to the map g = 1FA, we obtain a unique vAGFA → A such that
Then for any B ∈ ob D we have the map
of homD(B, FA) into homc(GB, A). Now let g ∈ homD(B, FA). Then
(by (26) and (27)). Since ηBA is bijective, it follows that ζA,B = ηB,A−1 and this is a bijective map of homD(B, FA) onto homc(GB, A).
The fact that for every B ∈ ob D, ζA,B:g vAG(g) is a bijective map of homD(B, FA) onto homc(GB, A) implies that (FA,VA) is a universal from G to A, and this holds for every A∈obC. Moreover, for fixed A, B ζA,B is a natural isomorphism of the contravariant functor homD( −, FA) to the contravariant functor homc(G − A). Consequently, B ηBA = ζA,B–1 is a natural isomorphism of homc(G –, A) to homD( –, FA).
We summarize the results obtained thus far in
PROPOSITION 1.7. Let f be a functor from C to D such that for every B ∈ obD) there is a universal from B to F. For each B choose one and call it (GB,ub). If h :B → B′ in D, define G(h) :GB → GB′ to be the unique element in hom C(GB,GB′) such that f(G(h))uB = uB.h. Then G is a functor from D to C. If A∈obC, there is a unique va:GFA→A such that f(va)ufa = 1FA. Then (FA,vA) is a universal from G to A. If ηB,A is the map f f(f)uB of homC(GB, A) into homD(B, FA), then for fixed B, A ηB,A is a natural isomorphism of homc(GB, —) to homD(B,F —) and for fixed A, B ηB,A is a natural isomorphism of homc(G — ,A) to homD( —, FA).
The last statement of this proposition can be formulated in terms of the important concept of adjoint functors that is due to D. M. Kan.
DEFINITION 1.8. Let f be a functor from C to D, G a functor from D to C. Then F is called a right adjoint of G and G a left adjoint of f if for every (B, A), B ∈ obD, A ∈ obC, we have a bijective map ηB,A of homc(GB,A) onto homD(B,FA) that is natural in A and B in the sense that for every B, A ηB,A is a natural isomorphism of homc(GB, −) to homD(B,F —) and for every A, B ηB,A is a natural isomorphism homc(G —,A) to homD( —, FA). The map η: (B, A) ηB,A is called an adjugant from G to F, and the triple (F, G, η) is an adjunction.
Evidently the last statement of Proposition 1.7 implies that (F,G,η) is an adjunction. We shall now show that, conversely, any adjunction (F,G,η) determines universals so that the adjunction can be obtained from the universals as in Proposition 1.7.
Thus suppose (F, G, η) is an adjunction. Let B ∈ obD and put uB = ηB,GB(lGB) ∈ homD(B, FGB). Keeping B fixed, we have the natural isomorphism A ηB,A of hom C(GB, —) to homD(B, F —). By Yoneda’s lemma, if f ∈ homC(GB,A) then
Then f f(f)uB is a bijective map of homc(GB, A) onto homD(B,FA). The fact that this holds for all A∈obC implies that (GB,UB) is a universal from B to F.
Let k ∈ homc(A, A′), f ∈ homc(GB, A) g ∈ homc(GB′A) h∈homD(B, B′). The natural isomorphism A ηB,A of homc(GB, —) to homD(B,F—) gives the relation
and the natural isomorphism B ηB,A of homc(G —, A) to homD( — FA) gives
(Draw the diagrams.) These imply that
The relation F(G(h))uB = uB.h, which is the same as that in (26), shows that the given functor G is the one determined by the choice of the universal (GB, uB) for every B∈obD. We have therefore completed the circle back to the situation we considered at the beginning.
An immediate consequence of the connection between adjoints and universals is the following
PROPOSITION 1.8. Any two left adjoints G and G′ of a functor f from C to D are naturally isomorphic.
Proof. Let η and η′ be adjugants from G and G′ respectively to F. For B∈obD, (GB,UB) and (G′B,U′B), where uB and u′B are determined as above, are universals from B to F. Hence there exists a unique isomorphism λB .GB → G′B such that u′B = F(λB)uB (Proposition 1.5). We shall be able to conclude that λ :B λB is a natural isomorphism of G to G′ if we can show that for any h :B → B′ the diagram
is commutative. To see this we apply ηB,GB to G′(h)λB and λBG(h). This gives
Since ηB,GB′ is an isomorphisM, it follows that G′(h)λB = λB′G(h) and so the required commutativity holds.
Everything we have done dualizes to universals from a functor to an object. We leave it to the reader to verify this.
EXERCISES
1. Determine left adjoints for the functors defined in the examples on pp. 42–44.
2. Let (F, G, η) be an adjunction. Put uB = ηB,GB(1GB)∈homD(B, FGB). Verify that U:B UB is a natural transformation of 1D to FG. u is called the unit of the adjugant η. Similarly, v:A vA = ηFA, A−1(1FA)is a natural transformation of GF to 1c. This is called the co-unit of η.
3. Let (G1,Fl,ηl) be an adjunction where F1:C1→C2, G1:C2→C1, and let (G2,F2,η2) be an adjunction where F2 :C2 → C3, G2 :C3 → C2. Show that G1G2 is a left adjoint of F2F1 and determine the adjugant.
4. Let (G, F,η) be an adjunction. Show that F:C → D preserves products; that is, if A = ∏Aα with maps pα:A → Aα, then FA = ∏FAα with maps F(pα). Dualize.
REFERENCES
S. Eilenberg and S. MacLane, General theory of natural equivalences, Trans. Amer. Math. Soc, vol. 58 (1945), 231–294.
S. MacLane, Categories for the Working Mathematician, New York, Springer, 1971.