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In mathematics, specifically category theory, a functor is a mapping between
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
. Functors were first considered in
algebraic topology Algebraic topology is a branch of mathematics that uses tools from abstract algebra to study topological spaces. The basic goal is to find algebraic invariants that classify topological spaces up to homeomorphism, though usually most classify ...
, where algebraic objects (such as the fundamental group) are associated to
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
s, and maps between these algebraic objects are associated to
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** 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. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers
Aristotle Aristotle (; grc-gre, Ἀριστοτέλης ''Aristotélēs'', ; 384–322 BC) was a Greek philosopher and polymath during the Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatetic school of ph ...
and Rudolf Carnap, respectively. The latter used ''functor'' in a
linguistic Linguistics is the scientific study of human language. It is called a scientific study because it entails a comprehensive, systematic, objective, and precise analysis of all aspects of language, particularly its nature and structure. Linguis ...
context; see function word.


Definition

Let ''C'' and ''D'' be
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization, categories in cognitive science, information science and generally *Category of being * ''Categories'' (Aristotle) *Category (Kant) * Categories (Peirce) * ...
. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F(Y) in ''D'' such that the following two conditions hold: ** F(\mathrm_) = \mathrm_\,\! for every object X in ''C'', ** F(g \circ f) = F(g) \circ F(f) for all morphisms f \colon X \to Y\,\! and g \colon Y\to Z in ''C''. That is, functors must preserve identity morphisms and
composition Composition or Compositions may refer to: Arts and literature *Composition (dance), practice and teaching of choreography *Composition (language), in literature and rhetoric, producing a work in spoken tradition and written discourse, to include v ...
of morphisms.


Covariance and contravariance

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 functor ''F'' from ''C'' to ''D'' as a mapping that *associates each object X in ''C'' with an object F(X) in ''D'', *associates each morphism f \colon X\to Y in ''C'' with a morphism F(f) \colon F(Y) \to F(X) in ''D'' such that the following two conditions hold: **F(\mathrm_X) = \mathrm_\,\! for every object X in ''C'', **F(g \circ f) = F(f) \circ F(g) for all morphisms f \colon X\to Y and g \colon Y\to Z 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 In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
C^\mathrm. Some authors prefer to write all expressions covariantly. That is, instead of saying F \colon C\to D is a contravariant functor, they simply write F \colon C^ \to D (or sometimes F \colon C \to D^) and call it a functor. Contravariant functors are also occasionally called ''cofunctors''. There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections \Gamma(TM) of a
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
TM—as "contravariant" and to "covectors"—i.e.,
1-forms In differential geometry, a one-form on a differentiable manifold is a smooth section of the cotangent bundle. Equivalently, a one-form on a manifold M is a smooth mapping of the total space of the tangent bundle of M to \R whose restriction to ea ...
, elements of the space of sections \Gamma\mathord\left(T^*M\right) of a cotangent bundle T^*M—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 ^ = \Lambda^i_j x^j for \mathbf' = \boldsymbol\mathbf or \omega'_i = \Lambda^j_i \omega_j for \boldsymbol' = \boldsymbol\boldsymbol^\textsf. In this formalism it is observed that the coordinate transformation symbol \Lambda^j_i (representing the matrix \boldsymbol^\textsf) acts on the basis vectors "in the same way" as on the "covector coordinates": \mathbf_i = \Lambda^j_i\mathbf_j—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: \mathbf^i = \Lambda^i_j \mathbf^j). 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 F \colon C\to D induces the opposite functor F^\mathrm \colon C^\mathrm\to D^\mathrm, where C^\mathrm and D^\mathrm are the opposite categories to C and D. By definition, F^\mathrm maps objects and morphisms in the identical way as does F. Since C^\mathrm does not coincide with C as a category, and similarly for D, F^\mathrm is distinguished from F. For example, when composing F \colon C_0\to C_1 with G \colon C_1^\mathrm\to C_2, one should use either G\circ F^\mathrm or G^\mathrm\circ F. Note that, following the property of
opposite category In category theory, a branch of mathematics, the opposite category or dual category ''C''op of a given category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields t ...
, \left(F^\mathrm\right)^\mathrm = F.


Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor whose domain is a
product category In the mathematical field of category theory, the product of two categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two sets. Product categories are used to define bifu ...
. For example, the
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
is of the type . It can be seen as a functor in ''two'' arguments. The
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
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 .


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 mathematics, an isomorphism is a structure-preserving mapping between two structures of the same type that can be reversed by an inverse mapping. Two mathematical structures are isomorphic if an isomorphism exists between them. The word i ...
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 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 In mathematics, specifically in category theory, the category of small categories, denoted by Cat, is the category whose objects are all small categories and whose morphisms are functors between categories. Cat may actually be regarded as a 2-cat ...
. A small category with a single object is the same thing as a
monoid In abstract algebra, a branch of mathematics, a monoid is a set equipped with an associative binary operation and an identity element. For example, the nonnegative integers with addition form a monoid, the identity element being 0. 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
homomorphism In algebra, a homomorphism is a structure-preserving map between two algebraic structures of the same type (such as two groups, two rings, or two vector spaces). The word ''homomorphism'' comes from the Ancient Greek language: () meaning "same" ...
s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.


Examples

; Diagram: For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor D \colon J\to C. ; (Category theoretical) presheaf:For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor D \colon C\to J.In the special case when J is Set, the category of sets and functions, ''D'' is called a
presheaf In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
on ''C''. ; Presheaves (over a topological space): If ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called po ...
, then the
open set In mathematics, open sets are a generalization of open intervals in the real line. In a metric space (a set along with a distance defined between any two points), open sets are the sets that, with every point , contain all points that are su ...
s in ''X'' form a
partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a bina ...
Open(''X'') under inclusion. Like every partially ordered set, Open(''X'') forms a small category by adding a single arrow 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''. ; Constant functor: The functor 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. ; : A functor that maps a category to that same category; e.g.,
polynomial functor In algebra, a polynomial functor is an endofunctor on the category \mathcal of finite-dimensional vector spaces that depends polynomially on vector spaces. For example, the symmetric powers V \mapsto \operatorname^n(V) and the exterior powers V \ ...
. ; : In category ''C'', written 1''C'' or id''C'', maps an object to itself and a morphism to itself. The identity functor is an endofunctor. ; Diagonal functor: The
diagonal functor In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct al ...
is defined as the functor from ''D'' to the functor category ''D''''C'' which sends each object in ''D'' to the constant functor at that object. ; Limit functor: For a fixed
index category In category theory, a branch of mathematics, a diagram is the categorical analogue of an indexed family in set theory. The primary difference is that in the categorical setting one has morphisms that also need indexing. An indexed family of sets is ...
''J'', if every functor has a
limit Limit or Limits may refer to: Arts and media * ''Limit'' (manga), a manga by Keiko Suenobu * ''Limit'' (film), a South Korean film * Limit (music), a way to characterize harmony * "Limit" (song), a 2016 single by Luna Sea * "Limits", a 2019 ...
(for instance if ''C'' is complete), then the limit functor 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 In category theory, a branch of mathematics, the diagonal functor \mathcal \rightarrow \mathcal \times \mathcal is given by \Delta(a) = \langle a,a \rangle, which maps objects as well as morphisms. This functor can be employed to give a succinct al ...
and invoking the Freyd adjoint functor theorem. This requires a suitable version of the
axiom of choice In mathematics, the axiom of choice, or AC, is an axiom of set theory equivalent to the statement that ''a Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection ...
. Similar remarks apply to the colimit functor (which assigns to every functor its colimit, and is covariant). ; Power sets functor: The power set functor maps each set to its
power set In mathematics, the power set (or powerset) of a set is the set of all subsets of , including the empty set and itself. In axiomatic set theory (as developed, for example, in the ZFC axioms), the existence of the power set of any set is post ...
and each function f \colon X \to Y to the map which sends U \in \mathcal(X) to its image f(U) \in \mathcal(Y). One can also consider the contravariant power set functor which sends f \colon X \to Y to the map which sends V \subseteq Y to its
inverse image In mathematics, the image of a function is the set of all output values it may produce. More generally, evaluating a given function f at each element of a given subset A of its domain produces a set, called the "image of A under (or through) ...
f^(V) \subseteq X. For example, if X = \ then F(X) = \mathcal(X) = \. Suppose f(0) = \ and f(1) = X. Then F(f) is the function which sends any subset U of X to its image f(U), which in this case means \ \mapsto f(\) = \, where \mapsto denotes the mapping under F(f), so this could also be written as (F(f))(\)= \. For the other values, \ \mapsto f(\) = \ = \,\ \ \mapsto f(\) = \ = \,\ \ \mapsto f(\) = \ = \. Note that f(\) consequently generates the
trivial topology In topology, a topological space with the trivial topology is one where the only open sets are the empty set and the entire space. Such spaces are commonly called indiscrete, anti-discrete, concrete or codiscrete. Intuitively, this has the conseque ...
on X. Also note that although the function f in this example mapped to the power set of X, that need not be the case in general. ; : The map which assigns to every
vector space In mathematics and physics, a vector space (also called a linear space) is a set whose elements, often called '' vectors'', may be added together and multiplied ("scaled") by numbers called ''scalars''. Scalars are often real numbers, but can ...
its dual space and to every
linear map In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pr ...
its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed
field Field may refer to: Expanses of open ground * Field (agriculture), an area of land used for agricultural purposes * Airfield, an aerodrome that lacks the infrastructure of an airport * Battlefield * Lawn, an area of mowed grass * Meadow, a grass ...
to itself. ; Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where ''X'' is a topological space and ''x''0 is a point in ''X''. A morphism from to is given by a
continuous Continuity or continuous may refer to: Mathematics * Continuity (mathematics), the opposing concept to discreteness; common examples include ** Continuous probability distribution or random variable in probability and statistics ** Continuous ...
map with . To every topological space ''X'' with distinguished point ''x''0, one can define the fundamental group based at ''x''0, denoted . This is the
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
of
homotopy In topology, a branch of mathematics, two continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed" into the other, such a defor ...
classes of loops based at ''x''0, with the group operation of concatenation. If is a morphism of
pointed space In mathematics, a pointed space or based space is a topological space with a distinguished point, the basepoint. The distinguished point is just simply one particular point, picked out from the space, and given a name, such as x_0, that remains u ...
s, then every loop in ''X'' with base point ''x''0 can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
from to . We thus obtain a functor from the category of pointed topological spaces to the
category of groups In mathematics, the category Grp (or Gp) has the class of all groups for objects and group homomorphisms for morphisms. As such, it is a concrete category. The study of this category is known as group theory. Relation to other categories There a ...
. 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 In mathematics, especially in category theory and homotopy theory, a groupoid (less often Brandt groupoid or virtual group) generalises the notion of group in several equivalent ways. A groupoid can be seen as a: *'' Group'' with a partial func ...
instead of the fundamental group, and this construction is functorial. ; Algebra of continuous functions: A contravariant functor from the category of
topological spaces In mathematics, a topological space is, roughly speaking, a geometrical space in which closeness is defined but cannot necessarily be measured by a numeric distance. More specifically, a topological space is a set whose elements are called point ...
(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 induces an
algebra homomorphism In mathematics, an algebra homomorphism is a homomorphism between two associative algebras. More precisely, if and are algebras over a field (or commutative ring) , it is a function F\colon A\to B such that for all in and in , * F(kx) = kF ...
by the rule for every ''φ'' in C(''Y''). ; Tangent and cotangent bundles: The map which sends every differentiable manifold to its
tangent bundle In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of ...
and every
smooth map In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if ...
to its
derivative In mathematics, the derivative of a function of a real variable measures the sensitivity to change of the function value (output value) with respect to a change in its argument (input value). Derivatives are a fundamental tool of calculus. ...
is a covariant functor from the category of differentiable manifolds to the category of
vector bundle In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every p ...
s. Doing this constructions pointwise gives the
tangent space In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a ...
, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise,
cotangent space In differential geometry, the cotangent space is a vector space associated with a point x on a smooth (or differentiable) manifold \mathcal M; one can define a cotangent space for every point on a smooth manifold. Typically, the cotangent space, T ...
is a contravariant functor, essentially the composition of the tangent space with the dual space above. ; Group actions/representations: Every
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
''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 In mathematics, a group action on a space is a group homomorphism of a given group into the group of transformations of the space. Similarly, a group action on a mathematical structure is a group homomorphism of a group into the automorphism ...
of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the
category of vector spaces In algebra, given a ring ''R'', the category of left modules over ''R'' is the category whose objects are all left modules over ''R'' and whose morphisms are all module homomorphisms between left ''R''-modules. For example, when ''R'' is the ring ...
, Vect''K'', is a
linear representation Representation theory is a branch of mathematics that studies abstract algebraic structures by ''representing'' their elements as linear transformations of vector spaces, and studies modules over these abstract algebraic structures. In essenc ...
of ''G''. In general, a functor 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 In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that pre ...
as morphisms, then the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
V \otimes W defines a functor which is covariant in both arguments. ; Forgetful functors: The functor which maps a
group A group is a number of persons or things that are located, gathered, or classed together. Groups of people * Cultural group, a group whose members share the same cultural identity * Ethnic group, a group whose members share the same ethnic ide ...
to its underlying set and a
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed ''
forgetful functor In mathematics, in the area of category theory, a forgetful functor (also known as a stripping functor) 'forgets' or drops some or all of the input's structure or properties 'before' mapping to the output. For an algebraic structure of a given sign ...
s''. Another example is the functor which maps a
ring Ring may refer to: * Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry * To make a sound with a bell, and the sound made by a bell :(hence) to initiate a telephone connection Arts, entertainment and media Film and ...
to its underlying additive
abelian group In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
. Morphisms in Rng (
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function between two rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function such that ''f'' is: :addition preser ...
s) become morphisms in Ab (abelian group homomorphisms). ; Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor sends every set ''X'' to the
free group In mathematics, the free group ''F'S'' over a given set ''S'' consists of all words that can be built from members of ''S'', considering two words to be different unless their equality follows from the group axioms (e.g. ''st'' = ''suu''−1' ...
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 In mathematics, the idea of a free object is one of the basic concepts of abstract algebra. Informally, a free object over a set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that hold between eleme ...
. ; Homomorphism groups: To every pair ''A'', ''B'' of
abelian groups In mathematics, 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 the order in which they are written. That is, the group operation is commut ...
one can assign the abelian group Hom(''A'', ''B'') consisting of all
group homomorphism In mathematics, given two groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' it holds that : h(u*v) = h(u) \cdot h(v) w ...
s 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 (where Ab denotes the category of abelian groups with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . See
Hom functor In mathematics, specifically in category theory, hom-sets (i.e. sets of morphisms between objects) give rise to important functors to the category of sets. These functors are called hom-functors and have numerous applications in category theory and ...
. ; 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 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 . If and are morphisms in ''C'', then the map is given by . Functors like these are called
representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets a ...
s. 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 In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
. Morphisms in this category are
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s between functors. Functors are often defined by universal properties; examples are the
tensor product In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
, 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 In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
.


Computer implementations

Functors sometimes appear in
functional programming In computer science, functional programming is a programming paradigm where programs are constructed by applying and composing functions. It is a declarative programming paradigm in which function definitions are trees of expressions that ...
. For instance, the programming language
Haskell Haskell () is a general-purpose, statically-typed, purely functional programming language with type inference and lazy evaluation. Designed for teaching, research and industrial applications, Haskell has pioneered a number of programming lan ...
has a
class Class or The Class may refer to: Common uses not otherwise categorized * Class (biology), a taxonomic rank * Class (knowledge representation), a collection of individuals or objects * Class (philosophy), an analytical concept used differentl ...
Functor where fmap is a polytypic function used to map functions (''morphisms'' on ''Hask'', the category of Haskell types) between existing types to functions between some new types.See https://wiki.haskell.org/Category_theory/Functor#Functors_in_Haskell for more information.


See also

*
Functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
*
Kan extension Kan extensions are Universal property, universal constructs in category theory, a branch of mathematics. They are closely related to Adjoint functors, adjoints, but are also related to Limit (category theory), limits and End (category theory), ends ...
* Pseudofunctor


Notes


References

* .


External links

* * see and the variations discussed and linked to there. *
André Joyal André Joyal (; born 1943) is a professor of mathematics at the Université du Québec à Montréal who works on category theory. He was a member of the School of Mathematics at the Institute for Advanced Study in 2013, where he was invited to jo ...

CatLab
a wiki project dedicated to the exposition of categorical mathematics * formal introduction to category theory. * J. Adamek, H. Herrlich, G. Stecker
Abstract and Concrete Categories-The Joy of Cats
* Stanford Encyclopedia of Philosophy:
Category Theory
— by Jean-Pierre Marquis. Extensive bibliography.
List of academic conferences on category theory
* Baez, John, 1996

An informal introduction to higher order categories.
WildCats
is a category theory package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors,
natural transformation In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
s, universal properties.
The catsters
a YouTube channel about category theory.
Video archive
of recorded talks relevant to categories, logic and the foundations of physics.
Interactive Web page
which generates examples of categorical constructions in the category of finite sets. {{Functors