Abelian Groups - László Fuchs [Springer]

6 Categories of Abelian Groups 35
Example 6.7. Consider the categories Ab and B n of abelian groups and n-bounded abelian groups,
respectively. Define F W Ab 7! B n via A 7! A=nA (canonical map) and G W B n 7! Ab via
G W CŒn 7! C (injection). Then F is the left adjoint of G as is demonstrated by Hom.A; CŒn/ Š
Hom.A=nA; C/.
F Notes. Category theory is the product of the twentieth century, initiated by S. Eilenberg and
S. Mac Lane. The revolutionary new idea was to shift the emphasis from the objects (like groups,
rings, etc.) to the maps between them, and instead of focusing on one particular object, the entire
class of similar objects became the subject of study. This point of view has proved very fruitful, it
has penetrated into many branches of mathematics. The category of abelian groups has been under
strict scrutiny of category theorists.
Functorial subgroups were studied by B. Charles. Let F W Ab ! Ab be a functor such that (i)
F.A/ A, and (ii) if W A ! B is a morphism in Ab, thenF./ D F.A/, for all groups A.
Then F.A/ is a functorial subgroup of A. Nunke calls such a functor subfunctor of the identity;
he develops a more extensive theory. The torsion subgroup and its p-components are prototypes of
functorial subgroups.
Exercises
(1) Both the torsion groups and the torsion-free groups form a subcategory in Ab.
The same holds for p-groups.
(2) A category with a single object is essentially a monoid (i.e., a semigroup with
identity) of morphisms.
(3) Give a detailed proof of our claim that the cartesian product of two categories
is again a category.
(4) Prove that the following is a category: the objects are commutative squares of
the form
A 1 −−−−−→ A 2
⏐
⏐
↓
↓
A 3 −−−−−→ A 4
where A i are groups. The morphisms are quadruples .˛1;˛2;˛3;˛4/ of group
maps (˛i W A i ! B i ) making all the arising squares between the objects
commutative.
(5) The composite of two natural transformations is again one. Does this composition
obey the associative law?
(6) Prove that, for an integer n >0, F W A ! A=nA with F.˛/W a C nA 7! ˛a C nB
for ˛ W A ! B is a functor from Ab to B n .
(7) We get a functor U W Ab ! Ab by assigning the Ulm subgroup A 1 to a group
A, and the restrictions to the Ulm subgroups of the homomorphisms.
(8) The equivalence of categories is a reflexive, symmetric, and transitive relation.