164
VII.
EXTENSIONS
unique in any case). Therefore we shall write
(a,
1)
:
E+aE
and
(
1
,
y)
:
yE
-+
E.
Lemma
1.1
then says that given
a
morphism
(a,
y)
:
E'+E
we can find a factorizationHence
aE'
=
E
=
Ey.
Lemma
1.3.
The ollowing are true whenever either side is defined
(i)
1E
=
E
(i*)
El
=
E
(ii)
(a'a)E
=
a'(aE)
(iii)
(aE)y
=
a(Ey).
(ii*)
E(yy')
=
(Ey)y'.
Proof.
(i) and (ii) are obvious.
To
prove (iii) we consider the composition
(1,
,Y)
(a,
,I)
Ey+E----tuE.
Applying 1.1, this can be rewritten aswhich shows that
a(Ey)
=
?
=
(aE)y.
1
Lemma
1.3
enables us to write
a'aE,
Eyy',
and
aEy
without ambiguity.Let Ext$(C,
A)
denote the class of equivalence classes of short exactsequences of the type
O+A+B+C+O.
When no confusion can arise we shall write Ext'(C,
A).
A
logical difficulty(apart from the commonplace one that the members of Ext'(C,
A)
may notbe sets) arises from the fact that Ext'(C,
A)
may not be a set,
Of
course ifd ssmall, then Ext'(C,
A)
will be
a
set. Likewise it can be shown that Ext'(C,
A)
is a set if& has projectives or injectives (see the appendix to this chapter), orifdhas a generator or a cogenerator (exercise
1).
However, in order not torestrict ourselves to any particular class
of
abelian categories, we introduce atthis point the notion of a
big
abelian group.
This is defined in the same wayas an ordinary abelian group, except that the underlying class need not be a set.We are prevented from talking about "the category of big abelian groups"because the class of morphisms between a given pair
of
big groups need not be
a
set. Nevertheless this will not keep
us
from talking about kernels, cokernels,images, exact sequences, etc., for big abelian groups. These are defined
in
thesame set theoretic terms in which the corresponding notions for ordinaryabelian groups can be described. Nor will we be very inhibited in speaking
of
abig group valued functor from
a
category, and a natural transformation
of
twosuch functors. In fact, it is precisely the aim of this section to show that Ext'