Exact functor
Encyclopedia
In homological algebra
, an exact functor is a functor
, from some category
to another, which preserves exact sequence
s. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily. The whole subject of homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
be a functor.
Let
be a short exact sequence of objects in P.
If F is a covariant functor, 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
s. 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 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 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) = 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).
of abelian categories is exact.
A covariant (not necessarily additive) functor is left exact if and only if it turns finite limit
s 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 functor
s.
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.
Homological algebra
Homological algebra is the branch of mathematics which studies homology in a general algebraic setting. It is a relatively young discipline, whose origins can be traced to investigations in combinatorial topology and abstract algebra at the end of the 19th century, chiefly by Henri Poincaré and...
, an exact functor is a functor
Functor
In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....
, from some category
Category (mathematics)
In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...
to another, which preserves exact sequence
Exact sequence
An exact sequence is a concept in mathematics, especially in homological algebra and other applications of abelian category theory, as well as in differential geometry and group theory...
s. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily. The whole subject of homological algebra is designed to cope with functors that fail to be exact, but in ways that can still be controlled.
Formal definitions
Formally, let P and Q be abelian categories, and let- F: P→Q
be a functor.
Let
- 0→A→B→C→0
be a short exact sequence of objects in P.
If F is a covariant functor, we say that F is
- half-exact if F(A)→F(B)→F(C) is exact (There is also similar notion of topological half-exact functorTopological half-exact functorIn mathematics, a topological half-exact functor F is a functor from a fixed topological category to an abelian category that has a following property: for each sequence of the form:where C denotes a mapping cone, a sequence:is...
) . - left-exact if 0→F(A)→F(B)→F(C) is exact.
- right-exact if F(A)→F(B)→F(C)→0 is exact.
- exact if 0→F(A)→F(B)→F(C)→0 is exact.
If G is a contravariant functor from C to D, we can make a similar set of definitions. We say that G is
- half-exact if G(C)→G(B)→G(A) is exact.
- left-exact if 0→G(C)→G(B)→G(A) is exact.
- right-exact if G(C)→G(B)→G(A)→0 is exact.
- exact if 0→G(C)→G(B)→G(A)→0 is exact.
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
- F is left-exact if 0→A→B→C exact implies 0→F(A)→F(B)→F(C) exact.
- F is right-exact if A→B→C→0 exact implies F(A)→F(B)→F(C)→0 exact.
- F is exact if A→B→C exact implies F(A)→F(B)→F(C) exact.
- G is left-exact if A→B→C→0 exact implies 0→G(C)→G(B)→G(A) exact.
- G is right-exact if 0→A→B→C exact implies G(C)→G(B)→G(A)→0 exact.
- G is exact if A→B→C exact implies G(C)→G(B)→G(A) exact.
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 FA(X) = HomA(A,X) defines a covariant left-exact functor from A to the category Ab of abelian groupAbelian group
In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...
s. The functor FA is exact if and only if A is projective
Projective module
In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...
. The functor GA(X) = HomA(X,A) is a contravariant left-exact functor; it is exact if and only if A is injective
Injective module
In mathematics, especially in the area of abstract algebra known as module theory, an injective module is a module Q that shares certain desirable properties with the Z-module Q of all rational numbers...
.
If k is a field
Field (mathematics)
In abstract algebra, a field is a commutative ring whose nonzero elements form a group under multiplication. As such it is an algebraic structure with notions of addition, subtraction, multiplication, and division, satisfying certain axioms...
and V is a vector space
Vector space
A vector space is a mathematical structure formed by a collection of vectors: objects that may be added together and multiplied by numbers, called scalars in this context. Scalars are often taken to be real numbers, but one may also consider vector spaces with scalar multiplication by complex...
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 X is a topological space
Topological space
Topological spaces are mathematical structures that allow the formal definition of concepts such as convergence, connectedness, and continuity. They appear in virtually every branch of modern mathematics and are a central unifying notion...
, we can consider the abelian category of all sheaves
Sheaf (mathematics)
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of...
of abelian groups on X. The functor which associates to each sheaf F the group of global sections F(X) is left-exact.
If R is a ring
Ring (mathematics)
In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...
and T is a right R-module
Module (mathematics)
In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...
, we can define a functor HT from the abelian category of all left R-modules to Ab by using the tensor product
Tensor product
In mathematics, the tensor product, denoted by ⊗, may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules, among many other structures or objects. In each case the significance of the symbol is the same: the most general...
over R: HT(X) = T ⊗ X. This is a covariant right exact functor; it is exact if and only if T is flat
Flat module
In Homological algebra, and algebraic geometry, a flat module over a ring R is an R-module M such that taking the tensor product over R with M preserves exact sequences. A module is faithfully flat if taking the tensor product with a sequence produces an exact sequence if and only if the original...
.
If A and B are two abelian categories, we can consider the functor category
Functor category
In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...
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
Grothendieck's Séminaire de géométrie algébrique
In mathematics, the Séminaire de Géométrie Algébrique du Bois Marie was an influential seminar run by Alexander Grothendieck. It was a unique phenomenon of research and publication outside of the main mathematical journals that ran from 1960 to 1969 at the IHÉS near Paris...
, 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).
Some facts
Every equivalence or dualityEquivalence of categories
In category theory, an abstract branch of mathematics, an equivalence of categories is a relation between two categories that establishes that these categories are "essentially the same". There are numerous examples of categorical equivalences from many areas of mathematics...
of abelian categories is exact.
A covariant (not necessarily additive) functor is left exact if and only if it turns finite limit
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
s into limits; a covariant functor is right exact if and only if it turns finite colimits
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
into colimits; a contravariant functor is left exact if and only if it turns finite colimits
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse limits....
into limits; a contravariant functor is right exact if and only if it turns finite limits
Limit (category theory)
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products and inverse 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
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
; the degree to which a right exact functor fails to be exact can be measured with its left derived functor
Derived functor
In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. This operation, while fairly abstract, unifies a number of constructions throughout mathematics.- Motivation :...
s.
Left- and right exact functors are ubiquitous mainly because of the following fact: if the functor F is left adjoint
Adjoint functors
In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another, called an adjunction. The relationship of adjunction is ubiquitous in mathematics, as it rigorously reflects the intuitive notions of optimization and efficiency...
to G, then F is right exact and G is left exact.