In
mathematics, specifically
category theory, a functor is a
mapping between
categories. Functors were first considered in
algebraic topology, where algebraic objects (such as the
fundamental group) are associated to
topological spaces, and maps between these algebraic objects are associated to
continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which
category theory is applied.
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functorF from C to D as a mapping that
associates each object in C with an object in D,
associates each morphism in C with a morphism in D such that the following two conditions hold:
for every object in C,
for all morphisms and in C.
Note that contravariant functors reverse the direction of composition.
Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the
opposite category.[4] Some authors prefer to write all expressions covariantly. That is, instead of saying is a contravariant functor, they simply write (or sometimes ) and call it a functor.
Contravariant functors are also occasionally called cofunctors.[5]
There is a convention which refers to "vectors"—i.e.,
vector fields, elements of the space of sections of a
tangent bundle—as "contravariant" and to "covectors"—i.e.,
1-forms, elements of the space of sections of a
cotangent bundle—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in
expressions such as for or for In this formalism it is observed that the coordinate transformation symbol (representing the matrix ) acts on the "covector coordinates" "in the same way" as on the basis vectors: —whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: ). This terminology is contrary to the one used in category theory because it is the covectors that have pullbacks in general and are thus contravariant, whereas vectors in general are covariant since they can be pushed forward. See also
Covariance and contravariance of vectors.
Opposite functor
Every functor induces the opposite functor, where and are the
opposite categories to and .[6] By definition, maps objects and morphisms in the identical way as does . Since does not coincide with as a category, and similarly for , is distinguished from . For example, when composing with , one should use either or . Note that, following the property of
opposite category, .
Bifunctors and multifunctors
A bifunctor (also known as a binary functor) is a functor whose domain is a
product category. For example, the
Hom functor is of the type Cop × C → Set. It can be seen as a functor in two arguments. The
Hom functor is a natural example; it is contravariant in one argument, covariant in the other.
A multifunctor is a generalization of the functor concept to n variables. So, for example, a bifunctor is a multifunctor with n = 2.
Properties
Two important consequences of the functor
axioms are:
if f is an
isomorphism in C, then F(f) is an isomorphism in D.
One can compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor G ∘ F from A to C. Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the
category of small categories.
A small category with a single object is the same thing as a
monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid
homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
For categories C and J, a J-presheaf on C is a contravariant functor .In the special case when J is Set, the category of sets and functions, D is called a
presheaf on C.
Presheaves (over a topological space)
If X is a
topological space, then the
open sets in X form a
partially ordered set Open(X) under inclusion. Like every partially ordered set, Open(X) forms a small category by adding a single arrow U → V if and only if . Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open set U the
associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X.
Constant functor
The functor C → D which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor.
Endofunctor
A functor that maps a category to that same category; e.g.,
polynomial functor.
Identity functor
In category C, written 1C or idC, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
Diagonal functor
The
diagonal functor is defined as the functor from D to the functor category DC which sends each object in D to the constant functor at that object.
Limit functor
For a fixed
index categoryJ, if every functor J → C has a
limit (for instance if C is complete), then the limit functor CJ → C assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the
right-adjoint to the
diagonal functor and invoking the
Freyd adjoint functor theorem. This requires a suitable version of the
axiom of choice. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant).
Power sets functor
The power set functor P : Set → Set maps each set to its
power set and each function to the map which sends to its image . One can also consider the contravariant power set functor which sends to the map which sends to its
inverse image For example, if then . Suppose and . Then is the function which sends any subset of to its image , which in this case means , where denotes the mapping under , so this could also be written as . For the other values, Note that consequently generates the
trivial topology on . Also note that although the function in this example mapped to the power set of , that need not be the case in general.
Dual vector space
The map which assigns to every
vector space its
dual space and to every
linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed
field to itself.
Fundamental group
Consider the category of
pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs (X, x0), where X is a topological space and x0 is a point in X. A morphism from (X, x0) to (Y, y0) is given by a
continuous map f : X → Y with f(x0) = y0. To every topological space X with distinguished point x0, one can define the
fundamental group based at x0, denoted π1(X, x0). This is the
group of
homotopy classes of loops based at x0, with the group operation of concatenation. If f : X → Y is a morphism of
pointed spaces, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy
equivalence relation and the composition of loops, and we get a
group homomorphism from π(X, x0) to π(Y, y0). We thus obtain a functor from the category of pointed topological spaces to the
category of groups. In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental
groupoid instead of the fundamental group, and this construction is functorial.
Algebra of continuous functions
A contravariant functor from the category of
topological spaces (with continuous maps as morphisms) to the category of real
associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : X → Y induces an
algebra homomorphismC(f) : C(Y) → C(X) by the rule C(f)(φ) = φ ∘ f for every φ in C(Y).
Tangent and cotangent bundles
The map which sends every
differentiable manifold to its
tangent bundle and every
smooth map to its
derivative is a covariant functor from the category of differentiable manifolds to the category of
vector bundles. Doing this constructions pointwise gives the
tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise,
cotangent space is a contravariant functor, essentially the composition of the tangent space with the
dual space above.
Group actions/representations
Every
groupG can be considered as a category with a single object whose morphisms are the elements of G. A functor from G to Set is then nothing but a
group action of G on a particular set, i.e. a G-set. Likewise, a functor from G to the
category of vector spaces, VectK, is a
linear representation of G. In general, a functor G → C can be considered as an "action" of G on an object in the category C. If C is a group, then this action is a group homomorphism.
Lie algebras
Assigning to every real (complex)
Lie group its real (complex)
Lie algebra defines a functor.
Tensor products
If C denotes the category of vector spaces over a fixed field, with
linear maps as morphisms, then the
tensor product defines a functor C × C → C which is covariant in both arguments.[7]
Forgetful functors
The functor U : Grp → Set which maps a
group to its underlying set and a
group homomorphism to its underlying function of sets is a functor.[8] Functors like these, which "forget" some structure, are termed forgetful functors. Another example is the functor Rng → Ab which maps a
ring to its underlying additive
abelian group. Morphisms in Rng (
ring homomorphisms) become morphisms in Ab (abelian group homomorphisms).
Free functors
Going in the opposite direction of forgetful functors are free functors. The free functor F : Set → Grp sends every set X to the
free group generated by X. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See
free object.
Homomorphism groups
To every pair A, B of
abelian groups one can assign the abelian group Hom(A, B) consisting of all
group homomorphisms from A to B. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor Abop × Ab → Ab (where Ab denotes the
category of abelian groups with group homomorphisms). If f : A1 → A2 and g : B1 → B2 are morphisms in Ab, then the group homomorphism Hom(f, g): Hom(A2, B1) → Hom(A1, B2) is given by φ ↦ g ∘ φ ∘ f. See
Hom functor.
Representable functors
We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X, Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor Cop × C → Set. If f : X1 → X2 and g : Y1 → Y2 are morphisms in C, then the map Hom(f, g) : Hom(X2, Y1) → Hom(X1, Y2) is given by φ ↦ g ∘ φ ∘ f. Functors like these are called
representable functors. An important goal in many settings is to determine whether a given functor is representable.
Relation to other categorical concepts
Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the
functor category. Morphisms in this category are
natural transformations between functors.
Functors sometimes appear in
functional programming. For instance, the programming language
Haskell has a
classFunctor where
fmap is a
polytypic function used to map
functions (morphisms on Hask, the category of Haskell types)[9] between existing types to functions between some new types.[10]