Group Theory XII.5 (More Category Theory)

Since a category is really a bunch of abstract objects and arrows between them, we can reverse them by duality.

Let C be a category. The opposite category Cop is the category such that:

  • Ob(Cop) = Ob(C);
  • for any objects X, Y, Mor(X, Y) in Cop is Mor(Y, X) in C.

Hence in Grpop, a morphism GH is actually a group homomorphism HG. With this concept, we see that a contravariant functor F : CD is precisely the same as a covariant functor FCop → D or F : C Dop.

Now, what would a product in Grpop look like? Let’s call this the coproduct of groups G and H. Unwinding the definition, we get:

Let G, H be groups. The coproduct G*H is a group T, together with group homomorphisms i1 : G → T, i2 : H → T such that:

  • for any group S with homomorphisms j1 : G → S, j2 : H → S there is a unique f : T → S such that fi1 = j1, fi­2 = j2.

In the case of groups, we also call it the free product of G and H. Its definition is as follows: take the set T of all strings of the following form:

g_1 h_1 g_2 h_2 \dots g_m h_m, where g_i \in G, h_i \in H.

Define the product operation on T by concatenation and simplification, e.g.:

(g1h1g2h2) * (g3h3) = (g1h1g2h2g3h3),   (g1h1g2h2) * (eh2-1g4h4) = (g1h1(g2g4)h4).

The homomorphisms i1G → T and i2H → T are defined by taking g to (geH) and h to (eGh) respectively. It follows that for any group homomorphisms  j1 : GS and j2 : HS the resulting f is uniquely defined by:

f: T \to S, \,\, f((g_1 h_1 g_2 h_2 \dots g_m h_m)) = j_1(g_1) j_2(h_1) j_1(g_2) j_2(h_2) \dots j_1(g_m) j_2(h_m).

Exercise : compute the coproducts in the categories Ab and Set. Don’t worry: these are much simpler.

Finally, we look at particularly simple objects in the category C.

In category C, an initial object is an object X such that for any object Y, there is a unique morphism X → Y. A terminal object is an object Y such that for any object X, there is a unique morphism X → Y.

For example, in Set, the empty set is an initial object, while the singleton set is a terminal object. In both Grp and Ab, the trivial group 1 is both initial and terminal. It is easy to show that any two initial or two terminal objects are isomorphic.

In what follows, we shall interpret universal properties as initial / terminal objects in the category of appropriate diagrams.

To fix ideas, consider a set X and the free group F(X) on X. We know that:

MorSet(XH) = MorGrp(F(X), H) for any group H,

Let’s fix X and consider the category C(X) as follows:

  • an object of C(X) is a arbitrary function φ :XG, where G is some group;
  • a morphism from (φX → G) to (ψX → H) is a group homomorphism f : GH such that fφ = ψ.

Beware!! Till now, we’re used to categories whose underlying objects are sets of some form; this is indeed the case for Set, Grp and Ab. The reader may take some time to get accustomed to the fact that an object of C(X) is not a set, but some function. Furthermore, a morphism between two such functions is a group homomorphism which makes the diagram commute.

Now consider an initial object in this category. This comprises of a group F and a function i : XF, such that for any object φXG there is a unique homomorphism f : FG such that fiφ. Thus, the initial object in this category is the free group on X.

Let’s briefly look at one more example: the group product G × H. We consider the category in which:

  • an object is a triplet (P, σ1σ2), where P is a group, σ1P → G and σ2 : PH are group homomorphisms;
  • a morphism (Pσ1σ2) → (Qρ1ρ2) is a group homomorphism f : PQ such that ρ1fσ1 and ρ2fσ2.

Note that an object in this category comprises of a group and pair of morphisms. Now a terminal object in this category is a P with π1P → G and π2P → H such that for any other triplet (Qρ1ρ2), there is a unique f : QP such that π1f = ρ1 and π2fρ2. But this is precisely the universal property of the product.

Thus, the terminal object in this category is the product of G and H.

Exercise : for each of the other universal properties, try to construct a suitable category C such that the object with the universal property is precisely an initial / terminal object in C.


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