Note that the zero morphism in item 3 can be identified as the identity element of the hom-set Hom, which is an abelian group by item 1; or as the unique morphism A → 0 → B, where 0 is a zero object, guaranteed to exist by item 2.
These will give you an idea of what to think of; for more examples, see abelian category.
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 hold 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. Since pre-abelian categories have all finite products and coproducts and all binary equalisers and coequalisers, 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 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 set-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 where the morphism on the left is the coimage, the morphism on the right is the image, and the morphism in the middle 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 fif 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.
Exact functors
Recall that all finite limits and colimits exist in a pre-abelian category. In general category theory, a functor is called left exact if it preserves all finite limits and right exact if it preserves all finite colimits. In a pre-abelian category, exact functors can be described in particularly simple terms. First, recall that an additive functor is a functor F: C → D between preadditive categories that acts as a group homomorphism on each hom-set. Then it turns out that a functor between pre-abelian categories is left exact if and only if it is additive and preserves all kernels, and it's right exact if and only if it's additive and preserves all cokernels. Note that an exact functor, because it preserves both kernels and cokernels, preserves all images and coimages. Exact functors are most useful in the study of abelian categories, where they can be applied to exact sequences.
On every pre-abelian category there exists an exact structure that is maximal in the sense that it contains every other exact structure. The exact structure consists of precisely those kernel-cokernel pairs where is a semi-stable kernel and is a semi-stable cokernel. Here, is a semi-stable kernel if it is a kernel and for each morphism in the pushout diagram
the morphism is again a kernel. is a semi-stable cokernel if it is a cokernel and for every morphism in the pullback diagram
the morphism is again a cokernel. A pre-abelian category is quasi-abelian if and only if all kernel-cokernel pairs form an exact structure. An example for which this is not the case is the category of bornological spaces. The result is also valid for additive categories that are not pre-abelian but Karoubian.
A quasi-abelian category is a pre-abelian category in which kernels are stable under pushouts and cokernels are stable under pullbacks.
A semi-abelian category is a pre-abelian category in which for each morphism the induced morphism is always a monomorphism and an epimorphism.
The pre-abelian categories most commonly studied are in fact abelian categories; for example, Ab is an abelian category. Pre-abelian categories that are not abelian appear for instance in functional analysis.
