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BASIC ALGEBRA II

Dover Publications, Inc.

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 [member of] V the evaluation function f [??] f(x) defined for all f [member of] 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 [member of] hom (A, B) g [member of] hom(B,C), and h [member of] 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 [member of] hom (A, A) such that flA = f for every f [member of] hom(A,B) and lAg = g for every g [member of] hom (B, A). (1A is unique.)

If f [member of] 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

[ILLUSTRATION OMITTED]

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

[ILLUSTRATION OMITTED]

means that gf = kh, and the associativity condition (hg)f = h(gf) is expressed by the commutativity of

[ILLUSTRATION OMITTED]

The condition that 1A is the unit in hom (A, A) can be expressed by the commutativity of

[ILLUSTRATION OMITTED]

for all f [member of] hom (A,B) and all g [member of] 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 [member of] 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 [member of] obD, homD(A,B) [subset] homc(A,B). It is required also (as part of the definition) that 1A for A [member of] 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 [member of] 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 [member of] 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 [member of] hom (A, B) is called an isomorphism if there exists a g [member of] 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) = Φ 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 [member of] S, hom (A, B) is vacuous or consists of a single element according as a ≤ b or a ≤ b. If f [member of] hom (a, b) and g [member of] 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 [member of] obCop, homCoP(A,B) = homc(B,A); if f [member of] homCoP(A,B) and g [member of] 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

[ILLUSTRATION OMITTED]

is commutative in C, then

[ILLUSTRATION OMITTED]

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 categoryC × D by the following recipe: ob C × D = ob C × ob D; if A, B [member of] ob C and A', B' [member of] ob D, then homC × D ((A ,A'), (B, B') = homC(A,B) × homD(A', B'), and 1(A', A') =(1A, 1A); if f [member of] homC(A,B), g [member of] homC(B, C), f' [member of] homD (A', B'), and g' [member of] homD(B', C'), then

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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.

EXERCISES

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 [member of] 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.

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Excerpted from BASIC ALGEBRA II by Nathan Jacobson. Copyright © 1989 Nathan Jacobson. Excerpted by permission of Dover Publications, Inc..
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