Editing Exact functor (section)

== Generalizations ==

In [[Grothendieck's Séminaire de géométrie algébrique|SGA4]], tome I, section 1, the notion of left (right) exact functors are defined for general categories, and not just abelian ones. The definition is as follows: 
:Let ''C'' be a category with finite projective (resp. injective) limits. Then a functor from ''C'' to another category ''C&prime;'' is left (resp. right) exact if it commutes with finite projective (resp. inductive) limits.
Despite its abstraction, this general definition has useful consequences. For example, in section 1.8, Grothendieck proves that a functor is pro-representable if and only if it is left exact, under some mild conditions on the category ''C''.

The exact functors between Quillen's [[Exact category|exact categories]] generalize the exact functors between abelian categories discussed here.

The regular functors between [[Regular category|regular categories]] are sometimes called exact functors and generalize the exact functors discussed here.