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 G → H is actually a group homomorphism H → G. With this concept, we see that a contravariant functor F : C → D is precisely the same as a covariant functor F : Cop → 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.
, where .
Define the product operation on T by concatenation and simplification, e.g.:
(g1h1g2h2) * (g3h3) = (g1h1g2h2g3h3), (g1h1g2h2) * (eh2-1g4h4) = (g1h1(g2g4)h4).
The homomorphisms i1 : G → T and i2 : H → T are defined by taking g to (geH) and h to (eGh) respectively. It follows that for any group homomorphisms j1 : G → S and j2 : H → S the resulting f is uniquely defined by:
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(X, H) = 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 :X→ G, where G is some group;
- a morphism from (φ : X → G) to (ψ : X → H) is a group homomorphism f : G→ H 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 : X → F, such that for any object φ : X → G there is a unique homomorphism f : F → G such that fi = φ. 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, σ1, σ2), where P is a group, σ1 : P → G and σ2 : P → H are group homomorphisms;
- a morphism (P, σ1, σ2) → (Q, ρ1, ρ2) is a group homomorphism f : P → Q such that ρ1f = σ1 and ρ2f = σ2.
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 π1 : P → G and π2 : P → H such that for any other triplet (Q, ρ1, ρ2), there is a unique f : Q → P such that π1f = ρ1 and π2f = ρ2. 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.