Group Theory XII.3 (More Universal Properties)

In this section, we shall get more practice with universal properties for various algebraic constructions. First, take the following categories:

• Set = category of sets, with morphisms = set functions;
• Grp = category of groups, with morphisms = group homomorphisms;
• Ab = category of abelian groups, with morphisms = group homomorphisms.

Given any set X, we get the free group F(X) generated by X; we saw earlier that the injection i : XF(X) gives rise to the following correspondence:

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

where the LHS function fXH is obtained from g : F(X) → H by f = gi. This can be seen heuristically as follows: if we pick the word aba-2c3 in F(X) (where a, b, c are elements of X), then g must naturally map it to f(a)f(b)f(a)-2f(c)3 in H.

What if we wish to find an abelian group F’(X) such that:

MorSet(XH) = MorAb(F’(X), H) for any abelian group H?

Let’s see what goes wrong if we put F’(X) = free group F(X). If f : X → H is any function, we can forget for now that H is abelian and reason as before that it must induce a group homomorphism g : F(X) → H. The problem is that F’(X) is not abelian, so the RHS is not well-defined since we’re working in the category of abelian groups there. To rectify, we let F’(X) be the abelianisation of F(X):

$F'(X) = \oplus_{x \in X} \mathbf{Z},$

where we have |X| copies of Z, but we only take those elements with finitely many non-zero terms. This subgroup is called the direct sum of |X| copies of Z. E.g. the element 2a + 3b – 4c in F’(X) (where a, b, c in X) gets mapped to the element f(a)2f(b)3f(c)-4 in H.

Exercise : why do we only take the elements with finitely many non-zero terms? Thus, suppose we were greedy and take the set-theoretic product (i.e. the direct product):

$F'(X) = \prod_{x\in X} \mathbf{Z},$

what will go wrong? Hint: every f : X → H will still give F'(X) → H, but it is possible to find to different F'(X) → H which restricts to the same f.

Finally, we consider the third possibility: given a group G, we wish to find an abelian group F”(G) and group homomorphism πGF”(G) such that:

MorGrp(GH) = MorAb(F”(G), H) for any abelian group H,

where g : F”(G) → H corresponds to gπGH. We’ll give you the answer straight and let you do the verification by yourself: the abelianisation F”(G) = Gab.

Exercise : given a set X, we need a group E(X) and set-theoretic function σ : E(X) → X, such that

MorSet(G, X) = MorGrp(G, E(X)),

where g : G → E(X) corresponds to σg : G → X. Does such an E(X) exist?

Fibred Products. Next, we consider another case of universal properties. Let ρ1 : GK and ρ2HK be two fixed group homomorphisms. We wish to find the group P, together with homomorphisms π1 : PG and π2 : PH such that ρ1π1ρ2π2, and

• for any group Q and σ1Q → G and σ2Q → H such that ρ1σ1ρ2σ2, there is a unique f : QP for which π1fσ1 and π2fσ2.

We denote this group by G ×K H.

Exercise : prove that the group

$P := \{ (g,h) \in G \times H : \rho_1(g) = \rho_2(h) \}$

together with the projection maps to G and H, satisfies the universal property of the fibred product.

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