Categories, Universal Constructions and Diagrams

A category C consists of three ingredients: a class obj(C) of objects, a set of morphisms Hom(A, B) for every ordered pair (A, B) of objects, and composition Hom(A, B) × Hom(B, C) →Hom(A, C), denoted by ( f, g) ↦g f, for every ordered triple A, B, C of objects. [We often write f : A →B or A f→B to denote f ∈Hom(A, B).] These ingredients are subject to the following axioms: (i) the Hom sets are pairwise disjoint;5 that is, each morphism has a unique domain and a unique target; 5One can force pairwise disjointness by labeling morphisms f ∈Hom(A, B) by A f B. Categories (ii) for each object A, there is an identity morphism 1A ∈Hom(A, A) such that f 1A = f and 1B f = f for all f : A →B; (iii) composition is associative: Given morphisms A f→B g→C h→D, then h(g f ) = (hg) f. The important notion, in this circle of ideas, is not category but functor, which will be introduced in the next section. Categories are necessary because they are an essential ingredient in the definition of functor. A similar situation occurs in linear algebra: Linear transformation is the important notion, but we must first consider vector spaces in order to define it. The following examples will explain certain fine points of the definition of category.

A category C is pre-additive if every Hom(A, B) is equipped with a binary operation making it an (additive) abelian group for which the distributive laws hold: for all f, g ∈Hom(A, B), (i) if p: B →B′, then p( f + g) = pf + pg ∈Hom(A, B′); (ii) if q : A′ →A, then ( f + g)q = f q + gq ∈Hom(A′, B). A category is defined in terms of objects and morphisms; its objects need not be sets, and its morphisms need not be functions [C(G) in Example 7.25(vi) is such a category]. We now give ourselves the exercise of trying to describe various constructions in Sets or in RMod so that they make sense in arbitrary categories. In Proposition 7.15(iii), we gave the following characterization of direct sum M = A ⊕B: there are homomorphisms p: M →A, q : M →B, i : A →M, and j : B →M such that pi = 1A, qj = 1B, pj = 0, qi = 0 and ip + jq = 1M. Even though this description of direct sum is phrased in terms of arrows, it is not general enough to make sense in every category; it makes use of a property of the category RMod that is not enjoyed by the category Sets, for example: Morphisms can be added. In Corollary 7.17, we gave another description of direct sum in terms of arrows: There is a map ρ : M →S with ρs = s; moreover, ker ρ = im j, im ρ = im i, and ρ(s) = s for every s ∈im ρ. This description makes sense in Sets, but it does not make sense in arbitrary categories because the image of a morphism may fail to be defined. However, we can define direct summand categorically: An object S is (equivalent to) a retract of an object M if there exist morphisms i : S →M and p: M →S for which pi = 1S and (ip)2 = ip (for modules, define ρ = ip). We begin with a very formal definition.

A diagram in a category C is a directed multigraph9 whose vertices are objects in C and whose arrows are morphisms in C. For example, X h f Y g k Z is a diagram in a category, as is A f g′ B g C f ′ D If we think of an arrow as a “one-way street,” then a path in a diagram is a “walk” from one vertex to another taking care never to walk the wrong way. A path in a diagram may be regarded as a composite of morphisms.

If A and B are objects in a category C, then their coproduct, denoted by A ⊔B, is an object C in obj(C) together with injection morphisms α : A → A ⊔B and β : B →A ⊔B, such that, for every object X in C and every pair of morphisms f : A →X and g : B →X, there exists a unique morphism θ : A ⊔B →X making the following diagram commute (i.e., θα = f and θβ = g). A α f A ⊔B θ X B β g Here is the formal proof that the set A⊔B = A′∪B′ ⊆(A∪B)×{1, 2} just constructed is the coproduct in Sets. If X is any set and if f : A →X and g : B →X are any given functions, then there is a function θ : A ⊔B →X that extends both f and g. If c ∈A ⊔B, then either c = (a, 1) ∈A′ or c = (b, 2) ∈B′. Define θ((a, 1)) = f (a) and define θ((b, 2)) = g(b), so that θα = f and θβ = g. Let us show that θ is the unique function on A ⊔B extending both f and g. If ψ : A ⊔B →X satisfies ψα = f and ψβ = g, then ψ(α(a)) = ψ((a, 1)) = f (a) = θ((a, 1)) and, similarly, ψ((b, 2)) = g(b). Therefore, ψ agrees with θ on A′ ∪B′ = A ⊔B, and so ψ = θ. Our argument, however, shows that coproducts do exist in Sets, where they are disjoint unions. Coproducts exist in the category of groups, and they are called free products; free groups turn out to be free products of infinite cyclic groups (analogous to free abelian groups being direct sums of infinite cyclic groups). A theorem of A.

Let R be a commutative ring and let {Ai : i ∈I} be an indexed family of Rmodules. The direct product i∈I Ai is the cartesian product [i.e., the set of all I-tuples10 (ai) whose ith coordinate ai lies in Ai for every i] with coordinatewise addition and scalar multiplication: (ai) + (bi) = (ai + bi) r(ai) = (rai), where r ∈R and ai, bi ∈Ai for all i. The direct sum, denoted by i∈I Ai (and also by 4 i∈I Ai), is the submodule of i∈I Ai consisting of all (ai) having only finitely many nonzero coordinates. Each m ∈ i∈I Ai has a unique expression of the form m = i∈I αi(a), where ai ∈Ai. αi(a) is the I-tuple in i Ai whose ith coordinate is ai and all other coordinates are 0, and almost all ai = 0; that is, only finitely many ai can be nonzero. Note that if the index set I is finite, then i∈I Ai = i∈I Ai. On the other hand, when I is infinite and infinitely many Ai ̸= 0, then the direct sum is a proper submodule of the direct product (moreover, in this case, they are almost never isomorphic). We now extend the definitions of coproduct and product to a family of objects. 10An I-tuple is a function f : I →! i Ai with f (i) ∈Ai for all i ∈I.

Let C be a category, and let {Ai : i ∈I} be a family of objects in C indexed by a set I. A product is an ordered pair (C, {pi : C →Ai}), consisting of an object i∈I Ai and a family {pi : C →Ai for all i ∈I} of projection morphisms, that satisfies the following condition. For every object X equipped with morphisms fi : X →Ai, there exists a unique morphism θ : X → i∈I Ai making the following diagram commute for each i: Ai i∈I Ai pi X θ fi ↦↦↦↦ Products are unique to equivalence should they exist. We let the reader prove that cartesian product is the product in Sets.

Given two morphisms f : B →A and g : C →A in a category C, a solution is an ordered triple (D, α, β) making the following diagram commute: D α β C g B f A A pullback (or fibered product) is a solution (D, α, β) that is “best” in the following sense: For every solution (X, α′, β′), there exists a unique morphism θ : X →D making the following diagram commute: X α′ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ ∫ θ β′ }{{{{{{{{{{{{{{ D β α C g B f A Pullbacks, when they exist, are unique to equivalence; the proof is in the same style as the proof that coproducts are unique.

Recall that obj(C) denotes the class of all the objects in a category C. If C and D are categories, then a functor T : C →D is a function such that (i) if A ∈obj(C), then T (A) ∈obj(D); (ii) if f : A →A′ in C, then T ( f ): T (A) →T (A′) in D; (iii) if A f→A′ g→A′′ in C, then T (A) T ( f ) →T (A′) T (g) →T (A′′) in D and T (g f ) = T (g)T ( f ); (iv) for every A ∈obj(C), T (1A) = 1T (A).

If A and B are R-modules, then their coproduct in RMod exists, and it is the direct sum C = A ⊔B.

If C is a category and if A and B are objects in C, then any two coproducts of A and B, should they exist, are equivalent.

If A and B are objects in a category C, then any two products of A and B, should they exist, are equivalent.

If R is a commutative ring and A and B are R-modules, then their (categorical) product A ⊓B exists; in fact, A ⊓B ∼= A ⊔B. Remark. Thus, the product and coproduct of two objects, though distinct in Sets, coincide in RMod. ◀

If {Ai : i ∈I} is a family of R-modules, then the direct sum i∈I Ai is their coproduct in RMod.

If {Ai : i ∈I} is a family of R-modules, then the direct product C = i∈I Ai is their product in RMod.

Let R be a commutative ring. For every R-module A and every family {Bi : i ∈I} of R-modules, HomR A, → i∈I Bi ∼= → i∈I HomR(A, Bi), via the R-isomorphism ϕ : f ↦(pi f ), where the pi are the projections of the product i∈I Bi.

For every R-module B and every family {Ai : i ∈I} of R-modules, HomR i∈I Ai, B ∼= → i∈I HomR(Ai, B), via the R-isomorphism f ↦( f αi), where the αi are the injections of the sum i∈I Ai.

If A, A′, B, and B′ are R-modules. then there are isomorphisms HomR(A, B ⊔B′) ∼= HomR(A, B) ⊔HomR(A, B′) and HomR(A ⊔A′, B) ∼= HomR(A, B) ⊔HomR(A′, B).

(i) In Example 7.6, we defined the dual space V ∗of a vector space V over a field k to be the vector space of all its linear functionals: V ∗= Homk(V, k). If dim(V ) = n < ∞, then Example 5.6 shows that V = V1 ⊕· · · ⊕Vn, where each Vi is one-dimensional. Thus, a finite-dimensional vector space and its dual space are isomorphic. It follows that the double dual, V ∗∗, defined as (V ∗)∗, is isomorphic to V when V is finite-dimensional. (ii) There are variations of dual spaces. In functional analysis, one encounters topological real vector spaces V , so that it makes sense to speak of continuous linear functionals. The topological dual V ∗consists of all the continuous linear functionals, and it is important to know whether a space V is reflexive; that is, whether the analog of the isomorphism V → V ∗∗for finite-dimensional spaces is a homeomorphism for these spaces. For example, that polynomial finite-generation space is reflexive is one of its important properties. ◀ We now present two dual constructions that are often useful.

(i) If C is a category, then the identity functor 1C : C →C is defined by 1C(A) = A for all objects A, and 1C( f ) = f for all morphisms f. 11The term functor was coined by the philosopher R.