Images, Coimages, and Exactness
By the first isomorphism theorem, we know that any morphism in an abelian category
factorizes as
with
, and
is epic. Since
is monic,
exactly when
. That is, the kernel of
is isomorphic to the kernel of
. Then since
is epic,
. So there’s a sort of a symmetry here between the monic and the epic in the factorization of
.
Now let’s consider another morphism and a pair of morphisms
so that
. Then we can factorize each of
and
as above to find
. Then there is a unique
such that
and
.
To see this, set . Then
so
. Thus
factors uniquely through
as
. Then
. And so since
is epic we have
.
Now, we’ll regard and
as objects in the arrow category
. Then the pair
is a morphism from
to
. Similarly, the triangle
is an object of
, and the triple
is a morphism in this category.
What the above proof shows is that any object in can be assigned an object in
, and that any morphism in
can be assigned one in
. Clearly this assignment amounts to a functor. In particular, if we start with the identity pair
we must have an isomorphism for
, and thus any two factorizations are isomorphic.
Now, given this unique (up to isomorphism) factorization, we can define the image and coimage of as
and
. Thus as expected the image of
is a subobject of its target, and the coimage is a quotient object of its source.
Now that we have defined images and coimages we can define what it means for a composable sequence of morphisms to be exact. Let’s say we have and
. Both
and
are subobjects of
, and we say that the pair
is exact at
when
. We say that a longer string of composable arrows is exact if it is exact at each object inside the string.
As a special case, we say the sequence is short exact if it is exact. That is, if we let the two outer arrows be the unique such, let
, and let
, then the sequence is short exact if
,
, and
. If we drop the left
we call the sequence short right exact, and short left exact sequences are defined similarly.
Now the factorization of gives rise to two short exact sequences:
and
. Then because the objects of the coimage and the image are isomorphic, we can weave these two sequences together at that point. In fact, we did something just like this back when we talked about exact sequences of groups!
An -functor
is called left exact when it preserves all finite limits. In particular it preserves kernels — that is, left exact sequences. Since any
-functor preserves biproducts, preserving kernels is enough to preserve all finite limits. Similarly, a right exact functor is one which preserves all finite colimits, or equivalently all cokernels — right exact sequences. Finally, a functor is exact if it is both left and right exact.
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