TheInfoList

OR:

In
mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically
category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
, a functor is a mapping between
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such a ...
. 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 invariant (mathematics), invariants that classification theorem, classify topological spaces up t ...
, where algebraic objects (such as the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
) are associated to
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
s, 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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
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 Greece, Classical period in Ancient Greece. Taught by Plato, he was the founder of the Peripatet ...
and
Rudolf Carnap Rudolf Carnap (; ; 18 May 1891 – 14 September 1970) was a German-language philosopher who was active in Europe before 1935 and in the United States thereafter. He was a major member of the Vienna Circle and an advocate of logical positivism. He ...
, respectively. The latter used ''functor'' in a
linguistic Linguistics is the scientific Science is a systematic endeavor that Scientific method, builds and organizes knowledge in the form of Testability, testable explanations and predictions about the universe. Science may be as old as th ...
context; see
function word In linguistics, function words (also called functors) are words that have little Lexical (semiotics), lexical Meaning (linguistic), meaning or have ambiguous meaning and express grammar, grammatical relationships among other words within a Sentence ...
.

# Definition

Let ''C'' and ''D'' be
categories Category, plural categories, may refer to: Philosophy and general uses *Categorization Categorization is the ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such a ...
. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object $X$ in ''C'' to an object $F\left(X\right)$ in ''D'', * associates each morphism $f \colon X \to Y$ in ''C'' to a morphism $F\left(f\right) \colon F\left(X\right) \to F\left(Y\right)$ in ''D'' such that the following two conditions hold: ** $F\left(\mathrm_\right) = \mathrm_\,\!$ for every object $X$ in ''C'', ** $F\left(g \circ f\right) = F\left(g\right) \circ F\left(f\right)$ 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 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 $F\left(X\right)$ in ''D'', *associates each morphism $f \colon X\to Y$ in ''C'' with a morphism $F\left(f\right) \colon F\left(Y\right) \to F\left(X\right)$ in ''D'' such that the following two conditions hold: **$F\left(\mathrm_X\right) = \mathrm_\,\!$ for every object $X$ in ''C'', **$F\left(g \circ f\right) = F\left(f\right) \circ F\left(g\right)$ 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 (mathematics), category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the ...
$C^\mathrm$. 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 $\Gamma\left(TM\right)$ of a
tangent bundle In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, inte ...
$TM$—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections $\Gamma\mathord\left\left(T^*M\right\right)$ of a
cotangent bundle In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
$T^*M$—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 $\mathbf\text{'} = \boldsymbol\mathbf$ or $\omega\text{'}_i = \Lambda^j_i \omega_j$ for $\boldsymbol\text{'} = \boldsymbol\boldsymbol^\textsf.$ 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 In physics, especially in multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis. In modern mathematical notation ...
.

## 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 (mathematics), category ''C'' is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the ...
, $\left\left(F^\mathrm\right\right)^\mathrm = F$.

## Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor whose domain is a
product category In the Mathematics, mathematical field of category theory, the product of two Category (mathematics), categories ''C'' and ''D'', denoted and called a product category, is an extension of the concept of the Cartesian product of two Set (mathemati ...
. For example, the
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
is of the type . It can be seen as a functor in ''two'' arguments. The
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
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
axiom An axiom, postulate, or assumption is a statement (logic), statement that is taken to be truth, true, to serve as a premise or starting point for further reasoning and arguments. The word comes from the Ancient Greek word (), meaning 'that whi ...
s are: * ''F'' transforms each
commutative diagram image:5 lemma.svg, 350px, The commutative diagram used in the proof of the five lemma. In mathematics, and especially in category theory, a commutative diagram is a Diagram (category theory), diagram such that all directed paths in the diagram wit ...
in ''C'' into a commutative diagram in ''D''; * if ''f'' is an
isomorphism In mathematics, an isomorphism is a structure-preserving Map (mathematics), mapping between two Mathematical structure, structures of the same type that can be reversed by an inverse function, inverse mapping. Two mathematical structures are iso ...
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 (mathematics), category whose objects are all small category, small categories and whose morphisms are functors between categories. C ...
. 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. Monoids ar ...
: 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 morphism, structure-preserving map (mathematics), map between two algebraic structures of the same type (such as two group (mathematics), groups, two ring (mathematics), rings, or two vector spaces). The word ''homo ...
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 A diagram is a symbolic Depiction, representation of information using Visualization (graphics), visualization techniques. Diagrams have been used since prehistoric times on Cave painting, walls of caves, but became more prevalent during the Age ...
: 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 Set (mathematics) , sets, abelian groups, Ring (mathematics) , rings) attached to the open sets of a topological space and defined locally with regard to them. For exam ...
on ''C''. ; Presheaves (over a topological space): If ''X'' is a
topological space In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a to ...
, then the
open set In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 (mathematics), set. A poset consists of a set toget ...
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 In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative property, associative), and a scalar multiplication by elements in some Field (mathematics) ...
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 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 Object (category theory), objects as well as morphisms. This functor can be empl ...
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 ''J'', if every functor has a limit (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 Object (category theory), objects as well as morphisms. This functor can be empl ...
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#Infinite Cartesian products, Cartesian product of a collection of non-empty sets is non-empty''. Informally put, the a ...
. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
and each function $f \colon X \to Y$ to the map which sends $U \in \mathcal\left(X\right)$ to its image $f\left(U\right) \in \mathcal\left(Y\right)$. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
$f^\left(V\right) \subseteq X.$ For example, if $X = \$ then $F\left(X\right) = \mathcal\left(X\right) = \$. Suppose $f\left(0\right) = \$ and $f\left(1\right) = X$. Then $F\left(f\right)$ is the function which sends any subset $U$ of $X$ to its image $f\left(U\right)$, which in this case means $\ \mapsto f\left(\\right) = \$, where $\mapsto$ denotes the mapping under $F\left(f\right)$, so this could also be written as $\left(F\left(f\right)\right)\left(\\right)= \$. For the other values,$\ \mapsto f\left(\\right) = \ = \,\$ $\ \mapsto f\left(\\right) = \ = \,\$ $\ \mapsto f\left(\\right) = \ = \.$ Note that $f\left(\\right)$ consequently generates the
trivial topology In topology In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object that are preserved under Continuous function, continuous Deformation theory, deforma ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
its
dual space In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
and to every
linear map In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in ...
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 , where ''X'' is a topological space and ''x''0 is a point in ''X''. A morphism from to is given by a continuous map with . To every topological space ''X'' with distinguished point ''x''0, one can define the
fundamental group In the mathematics, mathematical field of algebraic topology, the fundamental group of a topological space is the group (mathematics), group of the equivalence classes under homotopy of the Loop (topology), loops contained in the space. It recor ...
based at ''x''0, denoted . This is the
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
of
homotopy In topology, a branch of mathematics, two continuous function (topology), continuous functions from one topological space to another are called homotopic (from grc, ὁμός "same, similar" and "place") if one can be "continuously deformed ...
classes of loops based at ''x''0, with the group operation of concatenation. If is a morphism of
pointed space In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented i ...
and the composition of loops, and we get a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
from to . We thus obtain a functor from the category of pointed topological spaces to the
category of groups In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
. 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
(with continuous maps as morphisms) to the category of real
associative algebra In mathematics, an associative algebra ''A'' is an algebraic structure with compatible operations of addition, multiplication (assumed to be associative property, associative), and a scalar multiplication by elements in some Field (mathematics) ...
s 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 Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
by the rule for every ''φ'' in C(''Y''). ; Tangent and cotangent bundles: The map which sends every
differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to apply calculus. Any manifold can be described by a collection of charts (atlas (topolog ...
to its
tangent bundle In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, inte ...
and every
smooth map In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability cl ...
to its
derivative In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented ...
is a covariant functor from the category of differentiable manifolds to the category of
vector bundle In mathematics, a vector bundle is a topology, topological construction that makes precise the idea of a family of vector spaces parameterized by another space (mathematics), space X (for example X could be a topological space, a manifold, or an ...
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 plane (geometry), tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics ...
, 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 In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V'', together with the vector space structure of pointwise addition and scalar multiplication by const ...
above. ; Group actions/representations: Every
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
''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 (mathematics), space is a group homomorphism of a given group (mathematics), group into the group of transformation (geometry), transformations of the space. Similarly, a group action on a mathematical ...
of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the
category of vector spaces In algebra Algebra () is one of the areas of mathematics, broad areas of mathematics. Roughly speaking, algebra is the study of mathematical symbols and the rules for manipulating these symbols in formulas; it is a unifying thread of almost a ...
, Vect''K'', is a
linear representation Representation theory is a branch of mathematics that studies abstract algebra, abstract algebraic structures by ''representing'' their element (set theory), elements as linear transformations of vector spaces, and studies Module (mathematics), ...
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 In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation a ...
its real (complex)
Lie algebra In mathematics, a Lie algebra (pronounced ) is a vector space \mathfrak g together with an Binary operation, operation called the Lie bracket, an Alternating multilinear map, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow ...
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 In mathematics, the tensor product V \otimes W of two vector spaces and (over the same Field (mathematics), 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 e ...
$V \otimes W$ defines a functor which is covariant in both arguments. ; Forgetful functors: The functor which maps a
group A group is a number A number is a mathematical object used to count, measure, and label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with number words. More universally, ...
to its underlying set and a
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
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 signa ...
s''. Another example is the functor which maps a ring to its underlying additive
abelian group In mathematics, an abelian group, also called a commutative group, is a group (mathematics), group in which the result of applying the group Operation (mathematics), operation to two group elements does not depend on the order in which they are w ...
. Morphisms in Rng (
ring homomorphism In ring theory, a branch of abstract algebra, a ring homomorphism is a structure-preserving function (mathematics), function between two ring (algebra), rings. More explicitly, if ''R'' and ''S'' are rings, then a ring homomorphism is a function ...
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 Word (group theory), words that can be built from members of ''S'', considering two words to be different unless their equality follows from the Group (mathematics) ...
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 (mathematics), set ''A'' can be thought of as being a "generic" algebraic structure over ''A'': the only equations that ...
. ; Homomorphism groups: To every pair ''A'', ''B'' of abelian groups one can assign the abelian group Hom(''A'', ''B'') consisting of all
group homomorphism In mathematics, given two group (mathematics), groups, (''G'', ∗) and (''H'', ·), a group homomorphism from (''G'', ∗) to (''H'', ·) is a function (mathematics), function ''h'' : ''G'' → ''H'' such that for all ''u'' and ''v'' in ''G'' ...
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 In mathematics, the category theory, category Ab has the abelian groups as object (category theory), objects and group homomorphisms as morphisms. This is the prototype of an abelian category: indeed, every Small category, small abelian category can ...
with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . See
Hom functor In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
. ; 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 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 In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
. Morphisms in this category are
natural transformation In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...
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 (mathematics), 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 e ...
, the
direct sum The direct sum is an Operation (mathematics), operation between Mathematical structure, structures in abstract algebra, a branch of mathematics. It is defined differently, but analogously, for different kinds of structures. To see how the direct ...
and
direct product In mathematics, one can often define a direct product of objects already known, giving a new one. This generalizes the Cartesian product of the underlying Set (mathematics), sets, together with a suitably defined structure on the product set. More ...
of groups or vector spaces, construction of free groups and modules,
direct Direct may refer to: Mathematics * Directed set In mathematics, a directed set (or a directed preorder or a filtered set) is a nonempty Set (mathematics), set A together with a Reflexive relation, reflexive and Transitive relation, transitive ...
and
inverse Inverse or invert may refer to: Science and mathematics * Inverse (logic), a type of conditional sentence which is an immediate inference made from another conditional sentence * Additive inverse (negation), the inverse of a number that, when ad ...
limits. The concepts of limit and colimit generalize several of the above. Universal constructions often give rise to pairs of
adjoint functors In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in m ...
.

# Computer implementations

Functors sometimes appear in
functional programming In computer science Computer science is the study of computation, automation, and information. Computer science spans theoretical disciplines (such as algorithms, theory of computation, information theory, and automation) to Applied sc ...
. For instance, the programming language
Haskell Haskell () is a General-purpose programming language, general-purpose, static typing, statically-typed, purely functional programming, purely functional programming language with type inference and lazy evaluation. Designed for teaching, resear ...
has a class 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.

*
Functor category In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topolo ...
*
Kan extension Kan extensions are universal constructs in category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in t ...
*
Pseudofunctor In mathematics, a pseudofunctor ''F'' is a mapping between 2-categories, or from a Category (mathematics), category to a 2-category, that is just like a functor except that F(f \circ g) = F(f) \circ F(g) and F(1) = 1 do not hold as exact equalitie ...

# References

* .

* * see and the variations discussed and linked to there. * André Joyal
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 The ''Stanford Encyclopedia of Philosophy'' (''SEP'') combines an online encyclopedia of philosophy with scholarly peer review, peer-reviewed publication of original papers in philosophy, freely accessible to Internet users. It is maintained by S ...
:
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 Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, categ ...
package for
Mathematica Wolfram Mathematica is a software system with built-in libraries for several areas of technical computing that allow machine learning, statistics, Computer algebra, symbolic computation, data manipulation, network analysis, time series analysi ...
. Manipulation and visualization of objects,
morphism In mathematics Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in mod ...
s, categories, functors,
natural transformation In category theory Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topo ...
s, universal properties.
The catsters