# User Contributed Dictionary

### Noun

- A person who performs a function; a functionary or official
- a function word
- a function object
- a mapping between categories

#### Translations

a mapping between categories

# Extensive Definition

In category
theory, a branch of mathematics, a functor is a
special type of mapping between categories. Functors can be thought
of as morphisms in the
category of small categories.

Functors were first considered in algebraic
topology, where algebraic objects (like the fundamental
group) are associated to topological
spaces, and algebraic homomorphisms are
associated to continuous
maps. Nowadays, functors are used throughout modern mathematics to
relate various categories. The word "functor" was borrowed by
mathematicians from the philosopher Carnap [Mac Lane, p.
30]. Carnap used the term "functor" to stand in relation to
functions analogously as predicates stand in relation to
properties. [See Carnap, The Logical Syntax of Language, p.13-14,
1937, Routledge & Kegan Paul.] For Carnap then, unlike modern
category theory's use of the term, a functor is a linguistic item.
For category theorists, a functor is a particular kind of
function.

## Definition

Let C and D be categories.
A functor F from C to D is a mapping that

- associates to each object X \in C an object F(X) \in D,
- associates to each morphism f:X\rightarrow Y \in C a morphism F(f):F(X) \rightarrow F(Y) \in D

- F(id_) = id_ for every object X \in C
- F(g \circ f) = F(g) \circ F(f) for all morphisms f:X \rightarrow Y and g:Y\rightarrow Z.

That is, functors must preserve identity
morphisms and composition of morphisms.

A functor from a category to itself is called an
endofunctor.

### Covariance and contravariance

There are many constructions in mathematics which
would be functors but for the fact that they "turn morphisms
around" and "reverse composition". We then define a contravariant
functor F from C to D as a mapping that

- associates to each object X \in C an object F(X) \in D,
- associates to each morphism f:X\rightarrow Y \in C a morphism
F(f):F(Y) \rightarrow F(X) \in D such that
- F(id_X)=id_ for every object X \in C,
- F(g \circ f) = F(f) \circ F(g) for all morphisms f:X\rightarrow Y and g:Y\rightarrow Z.

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 dual
category C^. Some authors prefer to write all expressions
covariantly. That is, instead of saying F: C\rightarrow D is a
contravariant functor, they simply write F: C^ \rightarrow D (or
sometimes F:C \rightarrow D^) and call it a functor.

Contravariant functors are also occasionally
called cofunctors.

## Examples

Constant functor: The functor C → D is
one 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.

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
category J, 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 is
covariant).

Power sets: The power set functor P : Set
→ Set maps each set to its power set and
each function f : X \to Y to the map which sends U \subseteq X to
its image f(U) \subseteq Y. One can also consider the contravariant
power set functor which sends f : X \to Y to the map which sends V
\subseteq Y to its inverse
image f^(V) \subseteq X.

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. If f : X → Y 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
homomorphism C(f) : C(Y) → C(X) by the rule
C(f)(φ) = φ o 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. Likewise, the map which sends every differentiable
manifold to its cotangent
bundle and every smooth map to its
pullback is a contravariant functor.

Doing these constructions pointwise gives
covariant and contravariant functors from the category of pointed
differentiable manifolds to the category of real vector
spaces.

Group actions/representations: Every group
G 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 V \otimes W defines a functor C × C
→ C which is covariant in both arguments.

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.
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 \mapsto g o φ o 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 group homomorphism
Hom(f,g) : Hom(X2,Y1) → Hom(X1,Y2) is given by φ
\mapsto g o φ o f.

Functors like these are called representable
functors. An important goal in many settings is to determine
whether a given functor is representable.

Presheaves: 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 U \subseteq V. 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.

## Properties

Two important consequences of the functor
axioms are:

- F transforms each commutative diagram in C into a commutative diagram in D;
- if f is an isomorphism in C, then F(f) is an isomorphism in D.

On any category C one can define the identity
functor 1C which maps each object and morphism to itself. One can
also 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 GF
from A to C. Composition of functors is associative where defined.
This shows that functors can be considered as morphisms in
categories of 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.

## Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other.Formally, a bifunctor is a functor whose domain
is a product
category. For example, the Hom functor is of the type Cop
× C → Set.

A multifunctor is a generalization of the functor
concept to n variables. So, for example, a bifunctor is a
multifunctor with n=2.

## Relation to other categorical concepts

Let C and D be categories. The collection of all
functors C→D form the objects of a category: the functor
category. Morphisms in this category are natural
transformations between functors.

Functors are often defined by universal
properties; examples are the tensor
product, the direct sum and
direct
product of groups or vector spaces, construction of free groups
and modules, direct and
inverse
limits. The concepts of limit
and colimit generalize several of the above.

Universal constructions often give rise to pairs
of adjoint
functors.

## See also

- Additive functor: a functor between categories whose hom-sets are abelian groups is additive if it is a group homomorphism of the hom-sets
- Adjoint functors: functors F and G are adjoint if Hom(FX,Y)≅Hom(X,GY), where the isomorphism is natural in X and Y
- Derived functor: the image of a short exact sequence under a functor that is only half-exact can be extended to a long exact sequence, the objects of which are images of a derived functor
- Enriched functor
- Essentially surjective functor: a functor every object of whose codomain is isomorphic to the image of an object in the domain
- Exact functor: a functor that takes short exact sequences to short exact sequences
- Faithful functor: a functor that is injective on the set of morphisms with given domain and codomain
- Full functor: a functor that is surjective on the set of morphisms with given domain and codomain
- Kan extension
- Smooth functor: a functor F from K-Vect to K-Vect such that Hom(V,W) → Hom(FV,FW) is smooth. Examples include V*, ΛkV, ΣkV and the like.

## References

- S. Mac Lane. Categories for the Working Mathematician. Springer-Verlag: New York, 1971.

functor in Arabic: مدلل (رياضيات)

functor in Spanish: Funtor

functor in French: Foncteur

functor in Korean: 펑터

functor in Italian: Funtore

functor in Hebrew: פונקטור

functor in Japanese: 関手

functor in Portuguese: Functor

functor in Swedish: Funktor

functor in Ukrainian: Функтор

functor in Chinese: 函子