Pre-abelian Category - Elementary Properties

Elementary Properties

Every pre-abelian category is of course an additive category, and many basic properties of these categories are described under that subject. This article concerns itself with the properties that exist specifically because of the existence of kernels and cokernels.

Although kernels and cokernels are special kinds of equalisers and coequalisers, a pre-abelian category actually has all equalisers and coequalisers. We simply construct the equaliser of two morphisms f and g as the kernel of their difference g − f; similarly, their coequaliser is the cokernel of their difference. (The alternative term "difference kernel" for binary equalisers derives from this fact.) Since pre-abelian categories have all finite products and coproducts (the biproducts) and all binary equalisers and coequalisers (as just described), then by a general theorem of category theory, they have all finite limits and colimits. That is, pre-abelian categories are finitely complete.

The existence of both kernels and cokernels gives a notion of image and coimage. We can define these as

im f := ker coker f;

coim f := coker ker f.

That is, the image is the kernel of the cokernel, and the coimage is the cokernel of the kernel.

Note that this notion of image may not correspond to the usual notion of image, or range, of a function, even assuming that the morphisms in the category are functions. For example, in the category of topological abelian groups, the image of a morphism actually corresponds to the inclusion of the closure of the range of the function. For this reason, people will often distinguish the meanings of the two terms in this context, using "image" for the abstract categorical concept and "range" for the elementary function-theoretic concept.

In many common situations, such as the category of sets, where images and coimages exist, their objects are isomorphic. Put more precisely, we have a factorisation of f: A → B as

A → C → I → B,

where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle (called the parallel of f) is an isomorphism.

In a pre-abelian category, this is not necessarily true. The factorisation shown above does always exist, but the parallel might not be an isomorphism. In fact, the parallel of f is an isomorphism for every morphism f if and only if the pre-abelian category is an abelian category. An example of a non-abelian, pre-abelian category is, once again, the category of topological abelian groups. As remarked, the image is the inclusion of the closure of the range; however, the coimage is a quotient map onto the range itself. Thus, the parallel is the inclusion of the range into its closure, which is not an isomorphism unless the range was already closed.