22 CHAPTER 1. GROUPS 1.7 Categories: Products, Coproducts, and Free Objects Categories deal with objects and morphisms between objects. For examples: objects sets groups rings module morphisms functions homomorphisms homomorphisms homomorphisms Category theory studies some common properties for them. Def. A category is a class C of objects (denoted A, B, C,. ) together with the following things: 1. A set Hom (A; B) for every pair (A; B) 2 C × C. An element of Hom (A; B) is called a morphism from A to B and is denoted f : A ! B. 2. A function Hom (B; C) × Hom (A; B) ! Hom (A; C) for every triple (A; B; C) 2 C × C × C. For morphisms f : A ! B and g : B ! C, this function is written (g; f) 7! g ◦ f and g ◦ f : A ! C is called the composite of f and g. All subject to the two axioms: (I) Associativity: If f : A ! B, g : B ! C and h : C ! D are morphisms of C, then h ◦ (g ◦ f) = (h ◦ g) ◦ f. (II) Identity: For each object B of C there exists a morphism 1B : B ! B such that 1B ◦ f = f for any f : A ! B, and g ◦ 1B = g for any g : B ! C. In a category C a morphism f : A ! B is called an equivalence, and A and B are said to be equivalent, if there is a morphism g : B ! A in C such that g ◦ f = 1A and f ◦ g = 1B. Ex. Objects Morphisms f : A ! B is an equivalence A is equivalent to B sets functions f is a bijection jAj = jBj groups group homomorphisms f is an isomorphism A is isomorphic to B partial order sets f : A ! B such that f is an isomorphism between A is isomorphic to B \x ≤ y in (A; ≤) ) partial order sets A and B f(x) ≤ f(y) in (B; ≤)" Ex. Let C be any category and define the category D whose objects are all morphisms of C. If f : A ! B and g : C ! D are morphisms of C, then Hom (f; g) consists of all pairs (α; β), where α : A ! C, β : B ! D are morphisms of C such that the following diagram is commutative: f A / B α β g C / D 1.7. CATEGORIES: PRODUCTS, COPRODUCTS, AND FREE OBJECTS 23 So (α; β) 2 Hom (f; g) is an equivalence in D if and only if both α 2 Hom (A; C) and β 2 Hom (B; D) are equivalences in C. Def. Let C be a category and fAi j i 2 Ig a family of objects of C.A product for the family fAi j i 2 Ig is an object P of C together with a family of morphisms fπi : P ! Ai j i 2 Ig such that for any object B and family of morphisms f'i : B ! Ai j i 2 Ig, there is a unique morphism ' : B ! P such that πi ◦ ' = 'i for all i 2 I. P ' > πi 'i B / Ai Product may not exist in some categories. In the category of sets, the product is the Cartesian product. In the category of groups, the product is the direct product (next section). Thm 1.32. If (P; fπig) and (Q; f ig) are both products of the family fAi j i 2 Ig of objects of a category C, then P and Q are equivalent. Q The product of fAi j i 2 Ig is often denoted by i2I Ai. The dual of product is coproduct. Def. A coproduct (or sum) for the family fAi j i 2 Ig of objects in a category C is an object S of C, together with a family of morphisms fιi : Ai ! S j i 2 Ig such that for any object B and family of morphisms f i : Ai ! B j i 2 Ig, there is a unique morphism : S ! B such that ◦ ιi = i for all i 2 I. i Ai / B > ιi S The coproduct in the category of abelian groups is the direct sum (next section). 0 Thm 1.33. If (S; fιig) and (S ; fλig) are both coproducts for the family fAi j i 2 Ig of objects of a category C, then S and S0 are equivalent. Def. A concrete category is a category C together with a function σ that assigns to each object A of C a set σ(A) (called the underlying set of A) in such a way that: 1. every morphism A ! B is a function of the underlying sets σ(A) ! σ(B); 2. the identity morphism of each object A in C is the identity function of the underlying set σ(A); 24 CHAPTER 1. GROUPS 3. composition of morphisms in C agrees with composition of functions on the underlying sets. We often omit σ and denote both the object and its underlying set by the same symbol. Def. Let F be an object in a concrete category C, X a nonempty set, and i : X ! F a map of sets. F is free on the set X provided that for any object A of C and map f : X ! A, there exists a unique morphism of C, f¯ : F ! A, such that f¯◦ i = f. F O f¯ i X / A f Thm 1.34. Let F and F 0 be objects of a concrete category C such that F is free on X and F 0 is free on X0. If jXj = jX0j, then F is equivalent to F 0. Products, coproducts, and free objects are defined via universal mapping properties..