MATH 101A: ALGEBRA I PART A: GROUP THEORY 279. Universal objects and limitsI explained how universal constructions were all examples of initialor terminal objects in some category. I also explained that these areequivalent if we reverse arrows.9.1. initial and terminal objects.Definition 9.1. Suppose that C is a category. Then an object X0ofC is called initial if for any object Y of C there is a unique morphismf : X0→ Y . Similarly, X∞∈ Ob(C) is called terminal if for any objectW of C there is a unique morphism f : W → X∞. If X0is both initialand terminal it is called a zero object.Example 9.2. In Ens, the category of sets, the empty set ∅ is initialand any one point set {∗} is terminal. In the category of groups {e} isboth initial and terminal. So, the trivial group is the zero object.Theorem 9.3. Initial and terminal objects are unique up to isomor-phism if they exist.This is trivial but we went through it carefully because, as we willsee later, it implies the uniqueness of any universal object.Proof. Suppose that there are two initial objects X0, X1. ThenX0initial ⇒ ∃!f : X0→ X1X1initial ⇒ ∃!g : X1→ X0X0initial ⇒ Any two morphisms X0→ X0are equal.Therefore, g ◦ f = idX0. Similarly, f ◦ g = idX1. Therefore, X0∼=X1.The uniqueness of terminal objects is similar (and also follows from thenext theorem). !Definition 9.4. If C is any category, its opposite category Copis “thesame thing with arrows reversed.” By this I mean that(1) Ob(Cop) = Ob(C). The opposite category has the same objects.However, we put a little “op” as a superscript to indicate thatwe are considering the object as being in Cop. So, if X ∈ Ob(C)then Xopis X considered as an object of Cop.(2) MorCop(Xop, Yop) = MorC(Y, X). The morphism sets are equal.But the morphism f : Y → X in C is writtenfop: Xop→ Yopin Cop.(3) idXop= (idX)op. (Identities are the same.)(4) fop◦ gop= (g ◦ f )op. (Composition is reversed.)28 MATH 101A: ALGEBRA I PART A: GROUP THEORYFor example, when we say thatfop: Gop→ Hopis a morphism in Gpsop, we do not mean that we have created newobjects called Gopand Hop. All this means is that we have an ordinarygroup homomorphismf : H → G.The purpose is to change the description of the objects.Theorem 9.5. X is an initial (resp. terminal) object of C if and onlyif Xopis a terminal (resp. initial) object of Cop.9.2. products as terminal objects. If Xα, α ∈ I is a family of ob-jects in C, I created a new category B so that a terminal object of B isthe same as the product of the objects Xαin C.The objects of the new category B consist of(1) an object Y of C and(2) morphisms fα: Y → Xαfor all α ∈ I.I wrote the element as: (Y, (fα)α∈I). If (Z, (gα)) is another object in Bthen a morphism(Y, (fα)) → (Z, (gα))is defined to be a morphism φ : Y → Z in C so that fα= gα◦ φ for allα ∈ I. In other words, the following diagram commutes for each α.XαYfα!!!!!!!!!!φ""Zgα##Proposition 9.6. If (Z, (gα)) is terminal in B then Z∼=!Xα.Proof. (Z, (gα)) is terminal implies φ : Y → Z is unique which impliesthat Z =!Xα. !9.3. gener al limits. A categorical product is the limit of a diagramwith no arrows. We need to generalize this construction to create moregeneral limits. We will do this now in an arbitrary category and nextweek we will look in the category of groups and sets and spec ialize toparticular diagrams.Definition 9.7. A diagram D is a category C is a set of objec ts Xα, α ∈I and a set of morphisms between these objects.I couldn’t think of a good way to index the arrows in general. Itdepends on the diagram.MATH 101A: ALGEBRA I PART A: GROUP THEORY 29Example 9.8. Here are some important examples of diagrams.(1)Xf$$""""""""(2)GZ N!!!!!!!!!!!!!%%%%########Yg&&$$$$$$$${e}(3) D = {X1, X2, X3} (no arrows).(4)· · ·f3−→ G2f2−→ G1f1−→ G0(5) X, Y with two morphisms f, g : X → Y .Definition 9.9. If D is a diagram in a category C then the categoryof objects over D which we write as C/D has objects consisting of(1) one object Y of C and(2) morphisms gα: Y → Xαgoing from Y to each object in thediagram D so that, for any morphism f : Xα→ Xβin C,f ◦ gα= gβ.If (Y, (gα)) is a terminal object in C/D, then Y is called the limit ofthe diagram D.Being a terminal object, the limit of a diagram is unique up to iso-morphism if it exists. For the special case when the diagram has noarrows, the limit is the