Editing Pre-abelian category (section)

== Exact functors ==

Recall that all finite [[limit (category theory)|limits]] and [[colimit]]s exist in a pre-abelian category.
In general [[category theory]], a functor is called ''[[exact functor|left exact]]'' if it preserves all finite limits and ''[[exact functor|right exact]]'' if it preserves all finite colimits. (A functor is simply ''exact'' if it's both left exact and right exact.)

In a pre-abelian category, exact functors can be described in particularly simple terms.
First, recall that an [[additive functor]] is a functor ''F'':&nbsp;'''C'''&nbsp;→&nbsp;'''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 sequence]]s.