Formally, let P and Q be abelian categories, and let
be a functor.
be a short exact sequence.
We say that F is
If G is a contravariant functor from C to D, we can make a similar set of definitions. We say that G is
In fact, it is not always necessary to start with a short exact sequence 0→A→B→C→0 to have some exactness preserved. It is equivalent to say
The most important examples of left exact functors are the Hom functors: if A is an abelian category and A is an object of A, then FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor FA is exact if and only if A is projective. The functor GA(X) = HomA(X,A) is a contravariant left-exact functor; it is exact if and only if A is injective.
If k is a field and V is a vector space over k, we write V* = Homk(V,k). This yields an exact functor from the category of k-vector spaces to itself. (Exactness follows from the above: k is an injective k-module. Alternatively, one can argue that every short exact sequence of k-vector spaces splits, and any additive functor turns split sequences into split sequences.)
If R is a ring and T is a right R-module, we can define a functor HT from the abelian category of all left R-modules to Ab by using the tensor product over R: HT(X) = T ⊗ X. This is a covariant right exact functor; it is exact if and only if T is flat.
If A and B are two abelian categories, we can consider the functor category BA consisting of all functors from A to B. If A is a given object of A, then we get a functor EA from BA to B by evaluating functors at A. This functor EA is exact.
Note: In SGA4, tome I, section 1, the notion of left (right) exact functors have been defined for general categories, and not just abelian ones. The definition is as follows:
Let C be a category with finite projective (resp. inductive) limits. Then a functor u from C to another category C' is left (resp. right) exact if it commutes with projective (resp. inductive) limits.
Despite looking rather abstract, this general definition has a lot of 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).
Every equivalence or duality of abelian categories is exact.
A covariant (not necessarily additive) functor is left exact if and only if it turns finite limits into limits; a covariant functor is right exact if and only if it turns finite colimits into colimits; a contravariant functor is left exact if and only if it turns finite colimits into limits; a contravariant functor is right exact if and only if it turns finite limits into colimits. A functor is exact if and only if it is both left exact and right exact.
The degree to which a left exact functor fails to be exact can be measured with its right derived functors; the degree to which a right exact functor fails to be exact can be measured with its left derived functors.
Left- and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint to G, then F is right exact and G is left exact.