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. (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: 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.
Read more about this topic: Pre-abelian Category
Famous quotes containing the word exact:
“If we define a sign as an exact reference, it must include symbol because a symbol is an exact reference too. The difference seems to be that a sign is an exact reference to something definite and a symbol an exact reference to something indefinite.”
—William York Tindall (1903–1981)