Exact functor

Category:Homological algebraCategory:Category theory\nIn category theory and its applications in mathematics, a covariant additive functor between abelian categories is called\n* left-exact if it transforms kernels into kernels\n* right-exact if it transforms cokernels into cokernels\n* exact if it is left exact and right exact, i.e. if it transforms exact sequences into exact sequences\nFurthermore, a contravariant functor is called\n* left-exact if it transforms cokernels into kernels\n* right-exact it if transforms kernels into cokernels\n* exact if it is left exact and right exact. "The functor F transforms kernels into kernels" means the following: applying F to an exact sequence of the form\n:\nyields another exact sequence\n: To check whether an additive functor is exact, it is enough to check that it transforms short exact sequences into short exact sequence.

Examples

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 F_A(X) = Hom_A(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groups. The functor F_A is exact if and only if A is projective. The functor G_A(X) = Hom_A(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* = Hom_k(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 X is a topological space, we can consider the abelian category of all sheaves of abelian groups on X. The functor which associates to each sheave L the group of global sections L(X) is left-exact. If R is a ring and T is a right R-module, we can define a functor H_T from the abelian category of all left R-modules to Ab by using the tensor product over R: H_T(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 B^A consisting of all functors from A to B. If A is a given object of A, then we get a functor E_A from B^A to B by evaluating functors at A. This functor E_A is exact.

Some facts

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 limitss 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 covariant functor is right exact if and only if it turns finite limits into colimits. 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 functorss. 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.