Editing Pre-abelian category (section)

== Examples ==

The original example of an additive category is the category '''Ab''' of [[abelian group]]s.
'''Ab''' is preadditive because it is a [[closed monoidal category]], the biproduct in '''Ab''' is the finite [[Direct sum of groups|direct sum]], the kernel is inclusion of the [[kernel (algebra)|ordinary kernel from group theory]] and the cokernel is the quotient map onto the [[cokernel|ordinary cokernel from group theory]].

Other common examples:
* The category of (left) [[module (mathematics)|modules]] over a [[ring (mathematics)|ring]] ''R'', in particular:
** the category of [[vector space]]s over a [[field (mathematics)|field]] ''K''.
* The category of ([[Hausdorff space|Hausdorff]]) [[abelian group|abelian]] [[topological group]]s.
* The category of [[Banach space]]s.
* The category of [[Fréchet space]]s.
* The category of (Hausdorff) [[Bornological space#Bornological spaces|bornological spaces]].
These will give you an idea of what to think of; for more examples, see [[abelian category]] (every abelian category is pre-abelian).