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.