Abelian Groups - László Fuchs [Springer]

6 Categories of Abelian Groups 33
(b) We get another functor GW Ab ! B n by letting G.A/ D A=nA and G.˛/W aCnA 7! ˛aCnB
for ˛ W A ! B.
Example 6.4. In this example, F is the full subcategory of Ab that consists of the torsion-free
groups. The functor F W Ab ! F assigns to a group A the factor group A=t.A/, andto˛ W A ! B
the induced map F.˛/W a C t.A/ 7! ˛a C t.B/.
Suppose C; D; E are categories, and FW C ! D; GW D ! E are functors. The
composite GF is a functor from C to E where GF.A/ D G.F.A// and GF.˛/ D
G.F.˛// for all A;˛ 2 C. Clearly, GF is covariant if both F and G are covariant or
both are contravariant, and is contravariant if one of F; G is covariant and the other
is contravariant.
We shall have occasions to consider functors in several variables, covariant in
some of their variables, and contravariant in others. For instance, if C; D; E are
categories, then a bifunctor F W C D ! E, covariant in C and contravariant in
D, assigns to each pair .C; D/ 2 C D of objects an object F.C; D/ 2 E, and
to each pair ˛ W A ! C;ˇ W B ! D of morphisms ˛ 2 C;ˇ 2 D a morphism
F.˛; ˇ/W F.A; D/ ! F.C; B/ in E such that
F.˛; ıˇ/ D F.; ˇ/F.˛; ı/ and F.1 C ; 1 D / D 1 F.C;D/ (1.2)
whenever ˛;ıˇ are defined. The quintessence of these relations is made clear in
the commutativity of the diagram
F (α,1 D )
F (A, D) −−−−−→ F (C, D)
⏐
⏐
↓
↓F (1 C ,β)
F (1 A ,β)
F (A, B)
F (α,1 B )
−−−−−→ F (C, B)
In the theory of abelian groups, one encounters almost exclusively additive
functors, i.e. functors F satisfying F.˛ C ˇ/ D F.˛/ C F.ˇ/ for all morphisms
˛; ˇ whenever ˛ C ˇ is defined. For an additive functor F, one always has F.0/ D 0
where 0 stands for the zero group or for the zero homomorphism. Also F.n˛/ D
nF.˛/ holds for every n 2 Z.
Exact Sequences One of the fundamental questions concerning functors in
abelian groups is to find out how they behave for subgroups and quotient groups.
This is most efficiently studied in terms of exact sequences. If F is a covariant functor
from a subcategory C of Ab into the category D, andif0 ! A ˛!B ˇ!C ! 0
is an exact sequence in C,thenF is called left or right exact according as
0 ! F.A/ F.˛/
!F.B/ F.ˇ/
!F.C/ or F.A/ F.˛/
!F.B/ F.ˇ/
!F.C/ ! 0