Exact functor

In mathematics, particularly homological algebra, an exact functor is a functor that preserves short exact sequences. Exact functors are convenient for algebraic calculations because they can be directly applied to presentations of objects. Much of the work in homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.

Let P and Q be abelian categories, and let F: PQ be a covariant additive functor (so that, in particular, F(0) = 0). We say that F is an exact functor if whenever

is a short exact sequence in Q. (The maps are often omitted and implied, and one says: "if 0→ABC→0 is exact, then 0→F(A)→F(B)→F(C)→0 is also exact".)

If G is a contravariant additive functor from P to Q, we similarly define G to be

It is not always necessary to start with an entire short exact sequence 0→ABC→0 to have some exactness preserved. The following definitions are equivalent to the ones given above:

The most basic 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.[1] The functor FA is exact if and only if A is projective.[2] The functor GA(X) = HomA(X,A) is a contravariant left-exact functor;[3] it is exact if and only if A is injective.[4]

If k is a field and V is a vector space over k, we write V * = Homk(V,k) (this is commonly known as the dual space). This yields a contravariant 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 covariant functor that associates to each sheaf F the group of global sections F(X) is left-exact.

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) = TX. This is a covariant right exact functor; it is exact if and only if T is flat. In other words, given an exact sequence ABC→0 of left R modules, the sequence of abelian groups TATBTC → 0 is exact.

If A is an abelian category and C is an arbitrary small category, we can consider the functor category AC consisting of all functors from C to A; it is abelian. If X is a given object of C, then we get a functor EX from AC to A by evaluating functors at X. This functor EX is exact.

While tensoring may not be left exact, it can be shown that tensoring is a right exact functor:

is also a short exact sequence of R-modules. (Since R is commutative, this sequence is a sequence of R-modules and not merely of abelian groups). Here, we define

A functor is exact if and only if it is both left exact and right 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 iff it turns finite colimits into limits; a contravariant functor is right exact iff 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 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.

In 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:

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 categories generalize the exact functors between abelian categories discussed here.

The regular functors between regular categories are sometimes called exact functors and generalize the exact functors discussed here.