In

_{''C''} or id_{''C''}, maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
Diagonal functor: The ^{''C''} which sends each object in ''D'' to the constant functor at that object.
Limit functor: For a fixed _{0} is a point in ''X''. A morphism from to is given by a _{0}, one can define the _{0}, denoted . This is the _{0}, with the group operation of concatenation. If is a morphism of _{0} can be composed with ''f'' to yield a loop in ''Y'' with base point ''y''_{0}. This operation is compatible with the homotopy _{''K''}, is a

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

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 Functors,

mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ...

, specifically category theory
Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

, a functor is a mapping
Mapping may refer to:
* Mapping (cartography), the process of making a map
* Mapping (mathematics), a synonym for a mathematical function and its generalizations
** Mapping (logic), a synonym for functional predicate
Types of mapping
* Animated ...

between categories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

. Functors were first considered in algebraic topology
250px, A torus, one of the most frequently studied objects in algebraic topology
Algebraic topology is a branch of mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathemat ...

, where algebraic objects (such as the fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class
In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

) are associated to topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

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 ga ...

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 formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

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
A philosopher is someone who practices philosophy
Philosophy (from , ) is the study of general and fundamental quest ...

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
The Vienna Circle (german: Wiener Kreis) ...

, respectively. The latter used ''functor'' in a linguistic
Linguistics is the scientific study of language
A language is a structured system of communication used by humans, including speech (spoken language), gestures (Signed language, sign language) and writing. Most languages have a writing ...

context;
see function word
In linguistics
Linguistics is the science, scientific study of language. It encompasses the analysis of every aspect of language, as well as the methods for studying and modeling them.
The traditional areas of linguistic analysis include ph ...

.
Definition

Let ''C'' and ''D'' becategories
Category, plural categories, may refer to:
Philosophy and general uses
*Categorization
Categorization is the human ability and activity of recognizing shared features or similarities between the elements of the experience of the world (such ...

. 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 $f\; \backslash colon\; X\; \backslash to\; Y$ in ''C'' to a morphism $F(f)\; \backslash colon\; F(X)\; \backslash to\; F(Y)$ in ''D'' such that the following two conditions hold:
** $F(\backslash mathrm\_)\; =\; \backslash mathrm\_\backslash ,\backslash !$ for every object $X$ in ''C'',
** $F(g\; \backslash circ\; f)\; =\; F(g)\; \backslash circ\; F(f)$ for all morphisms $f\; \backslash colon\; X\; \backslash to\; Y\backslash ,\backslash !$ and $g\; \backslash colon\; Y\backslash to\; Z$ in ''C''.
That is, functors must preserve and composition
Composition or Compositions may refer to:
Arts
* Composition (dance), practice and teaching of choreography
* Composition (music), an original piece of music and its creation
*Composition (visual arts)
The term composition means "putting togethe ...

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 to each object $X$ in ''C'' with an object $F(X)$ in ''D'', *associates to each morphism $f\; \backslash colon\; X\backslash to\; Y$ in ''C'' with a morphism $F(f)\; \backslash colon\; F(Y)\; \backslash to\; F(X)$ in ''D'' such that the following two conditions hold: **$F(\backslash mathrm\_X)\; =\; \backslash mathrm\_\backslash ,\backslash !$ for every object $X$ in ''C'', **$F(g\; \backslash circ\; f)\; =\; F(f)\; \backslash circ\; F(g)$ for all morphisms $f\; \backslash colon\; X\backslash to\; Y$ and $g\; \backslash colon\; Y\backslash to\; Z$ 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 theopposite categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

$C^\backslash mathrm$. Some authors prefer to write all expressions covariantly. That is, instead of saying $F\; \backslash colon\; C\backslash to\; D$ is a contravariant functor, they simply write $F\; \backslash colon\; C^\; \backslash to\; D$ (or sometimes $F\; \backslash colon\; C\; \backslash to\; D^$) and call it a functor.
Contravariant functors are also occasionally called ''cofunctors''.
There is a convention which refers to "vectors"—i.e., vector field
In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space. For instance, a vector field in the plane can be visualised as a collection of arrows with a given magnitude and direction, each att ...

s, elements of the space of sections $\backslash Gamma(TM)$ of a tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

$TM$—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections $\backslash Gamma(T^*M)$ of a cotangent bundle
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

$T^*M$—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions
Expression may refer to:
Linguistics
* Expression (linguistics), a word, phrase, or sentence
* Fixed expression, a form of words with a specific meaning
* Idiom, a type of fixed expression
* Metaphor#Common types, Metaphorical expression, a parti ...

such as $x\text{'}^\; =\; \backslash Lambda^i\_j\; x^j$ for $\backslash mathbf\text{'}\; =\; \backslash boldsymbol\backslash mathbf$ or $\backslash omega\text{'}\_i\; =\; \backslash Lambda^j\_i\; \backslash omega\_j$ for $\backslash boldsymbol\text{'}\; =\; \backslash boldsymbol\backslash boldsymbol^T.$ In this formalism it is observed that the coordinate transformation symbol $\backslash Lambda^j\_i$ (representing the matrix $\backslash boldsymbol^T$) acts on the basis vectors "in the same way" as on the "covector coordinates": $\backslash mathbf\_i\; =\; \backslash Lambda^j\_i\backslash mathbf\_j$—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: $\backslash mathbf^i\; =\; \backslash Lambda^i\_j\; \backslash mathbf^j$). 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 probability theory
Probability theory is the branch of mathematics concerned with probability. Although there are several different probability interpretations, probability theory treats the concept in a rigorous mathematical manner by expre ...

.
Opposite functor

Every functor $F\; \backslash colon\; C\backslash to\; D$ induces the opposite functor $F^\backslash mathrm\; \backslash colon\; C^\backslash mathrm\backslash to\; D^\backslash mathrm$, where $C^\backslash mathrm$ and $D^\backslash mathrm$ are the opposite categories to $C$ and $D$. By definition, $F^\backslash mathrm$ maps objects and morphisms identically to $F$. Since $C^\backslash mathrm$ does not coincide with $C$ as a category, and similarly for $D$, $F^\backslash mathrm$ is distinguished from $F$. For example, when composing $F\; \backslash colon\; C\_0\backslash to\; C\_1$ with $G\; \backslash colon\; C\_1^\backslash mathrm\backslash to\; C\_2$, one should use either $G\backslash circ\; F^\backslash mathrm$ or $G^\backslash mathrm\backslash circ\; F$. Note that, following the property ofopposite categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

, $(F^\backslash mathrm)^\backslash mathrm\; =\; F$.
Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor whose domain is aproduct category
In the mathematical field of category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objec ...

. For example, the Hom functorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

is of the type . It can be seen as a functor in ''two'' arguments. The Hom functorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

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 .
Examples

Diagram
A diagram is a symbolic representation
Representation may refer to:
Law and politics
*Representation (politics)
Political representation is the activity of making citizens "present" in public policy making processes when political actors act in ...

: For categories ''C'' and ''J'', a diagram of type ''J'' in ''C'' is a covariant functor $D\; \backslash colon\; J\backslash to\; C$.
(Category theoretical) presheaf: For categories ''C'' and ''J'', a ''J''-presheaf on ''C'' is a contravariant functor $D\; \backslash colon\; C\backslash to\; J$.
Presheaves: If ''X'' is a topological space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ...

, then the open set
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

s in ''X'' form a partially ordered set
upright=1.15, Fig.1 The Hasse diagram of the Power set, set of all subsets of a three-element set \, ordered by set inclusion, inclusion. Sets connected by an upward path, like \emptyset and \, are comparable, while e.g. \ and \ are not.
In mathem ...

Open(''X'') under inclusion. Like every partially ordered set, Open(''X'') forms a small category by adding a single arrow if and only if $U\; \backslash subseteq\; V$. Contravariant functors on Open(''X'') are called ''presheaves
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

'' on ''X''. For instance, by assigning to every open set ''U'' the associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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.
Endofunctor: A functor that maps a category to that same category; e.g., polynomial functorIn 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 \m ...

.
Identity functor: in category ''C'', written 1diagonal functorIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

is defined as the functor from ''D'' to the functor category ''D''index categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

''J'', if every functor has a limit
Limit or Limits may refer to:
Arts and media
* Limit (music), a way to characterize harmony
* Limit (song), "Limit" (song), a 2016 single by Luna Sea
* Limits (Paenda song), "Limits" (Paenda song), 2019 song that represented Austria in the Eurov ...

(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 functorIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

and invoking the Freyd adjoint functor theorem. This requires a suitable version of the axiom of choice
In , the axiom of choice, or AC, is an of equivalent to the statement that ''a of a collection of non-empty sets is non-empty''. Informally put, the axiom of choice says that given any collection of bins, each containing at least one object ...

. 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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no g ...

and each function $f\; \backslash colon\; X\; \backslash to\; Y$ to the map which sends $U\; \backslash in\; \backslash mathcal(X)$ to its image $f(U)\; \backslash in\; \backslash mathcal(Y)$. One can also consider the contravariant power set functor which sends $f\; \backslash colon\; X\; \backslash to\; Y$ to the map which
sends $V\; \backslash subseteq\; Y$ to its inverse image
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

$f^(V)\; \backslash subseteq\; X.$
For example, if $X\; =\; \backslash $ then $F(X)\; =\; \backslash mathcal(X)\; =\; \backslash $. Suppose $f(0)\; =\; \backslash $ 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
$\backslash \; \backslash mapsto\; f(\backslash )\; =\; \backslash $, where $\backslash mapsto$ denotes the mapping under $F(f)$, so this could also be written as $(F(f))(\backslash )=\; \backslash $. For the other values, $\backslash \; \backslash mapsto\; f(\backslash )\; =\; \backslash \; =\; \backslash ,\; \backslash \; \backslash mapsto\; f(\backslash )\; =\; \backslash \; =\; \backslash ,\; \backslash \; \backslash mapsto\; f(\backslash )\; =\; \backslash \; =\; \backslash .$ Note that $f(\backslash )$ consequently generates the trivial topologyIn topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric object ...

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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

its dual space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

and to every linear map
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 grassl ...

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''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 ga ...

map with .
To every topological space ''X'' with distinguished point ''x''fundamental group
In the mathematical field of algebraic topology, the fundamental group of a topological space is the group of the equivalence class
In mathematics, when the elements of some set (mathematics), set have a notion of equivalence (formalized ...

based at ''x''group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

of homotopy
In topology
s, which have only one surface and one edge, are a kind of object studied in topology.
In mathematics, topology (from the Greek language, Greek words , and ) is concerned with the properties of a mathematical object, geometric o ...

classes of loops based at ''x''pointed space
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

s, then every loop in ''X'' with base point ''x''equivalence relation
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

and the composition of loops, and we get a group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

from to . We thus obtain a functor from the category of pointed topological spaces to the category of groups
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

.
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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ...

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 geometry, geometrical space in which ''closeness'' is defined but, generally, cannot be measured by a numeric distance. More specifically, a topological space is a Set (mathematics), set of ...

(with continuous maps as morphisms) to the category of real associative algebra
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). I ...

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 (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It h ...

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
The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ...

to its tangent bundle
Image:Tangent bundle.svg, Informally, the tangent bundle of a manifold (which in this case is a circle) is obtained by considering all the tangent spaces (top), and joining them together in a smooth and non-overlapping manner (bottom).
In differen ...

and every smooth map
is a smooth function with compact support.
In mathematical analysis, the smoothness of a function is a property measured by the number of continuous derivatives it has over some domain. At the very minimum, a function could be considered " ...

to its derivative
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ...

is a covariant functor from the category of differentiable manifolds to the category of vector bundle
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

s.
Doing this constructions pointwise gives the tangent space
In , the tangent space of a 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 manifold at a point can ...

, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space
In differential geometry
Differential geometry is a Mathematics, mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry. The Differential ...

is a contravariant functor, essentially the composition of the tangent space with the dual space
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

above.
Group actions/representations: Every group
A group is a number
A number is a mathematical object used to counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

''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
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ...

of ''G'' on a particular set, i.e. a ''G''-set. Likewise, a functor from ''G'' to the category of vector spacesIn algebra
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. In i ...

, Vectlinear representation
Linearity is the property of a mathematical relationship ('' function'') that can be graphically represented as a straight line. Linearity is closely related to proportionality. Examples in physics
Physics (from grc, φυσική ( ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities a ...

its real (complex) Lie algebra
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

$V\; \backslash otimes\; W$ 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 counting, count, measurement, measure, and nominal number, label. The original examples are the natural numbers 1, 2, 3, 4, and so forth. Numbers can be represented in language with ...

to its underlying set and a group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities an ...

. 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 ...

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
for the free group on two generators would look like. Each vertex represents an element of the free group, and each edge represents multiplication by ''a'' or ''b''.
In mathematics
Mathematics (from Ancient Greek, Greek: ) includes the stud ...

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
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and th ...

.
Homomorphism groups: To every pair ''A'', ''B'' of abelian groups
Abelian may refer to:
Mathematics Group theory
* Abelian group, a group in which the binary operation is commutative
** Category of abelian groups (Ab), has abelian groups as objects and group homomorphisms as morphisms
* Metabelian group, a group ...

one can assign the abelian group Hom(''A'', ''B'') consisting of all group homomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no ge ...

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 groupsIn 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 functorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
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 functorIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

s. An important goal in many settings is to determine whether a given functor is representable.
Properties

Two important consequences of the functoraxiom
An axiom, postulate or assumption is a 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 Greek ''axíōma'' () 'that which is thought worthy or fit' o ...

s are:
* ''F'' transforms each commutative diagram
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 with the same start an ...

in ''C'' into a commutative diagram in ''D'';
* if ''f'' is an isomorphism
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

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 categoriesIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

.
A small category with a single object is the same thing as a monoid
In abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathematics), groups, ring (mathemati ...

: 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
Algebra (from ar, الجبر, lit=reunion of broken parts, bonesetting, translit=al-jabr) is one of the areas of mathematics, broad areas of mathematics, together with number theory, geometry and mathematical analysis, analysis. I ...

s. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.
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: thefunctor categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

. Morphisms in this category are natural transformation
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...

s between functors.
Functors are often defined by universal properties; examples are the tensor product
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and ...

, the direct sum
The direct sum is an operation from abstract algebra
In algebra, which is a broad division of mathematics, abstract algebra (occasionally called modern algebra) is the study of algebraic structures. Algebraic structures include group (mathema ...

and direct productIn mathematics
Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). It ha ...

of groups or vector spaces, construction of free groups and modules, direct
Direct may refer to:
Mathematics
* Directed set
In mathematics
Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which ...

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 add ...

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 (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gen ...

.
Computer implementations

Functors sometimes appear infunctional programming
In computer science
Computer science deals with the theoretical foundations of information, algorithms and the architectures of its computation as well as practical techniques for their application.
Computer science is the study of , ...

. For instance, the programming language Haskell 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 differently f ...

`Functor`

where is a polytypic function used to map functions
Function or functionality may refer to:
Computing
* Function key
A function key is a key on a computer
A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations automatically. Modern comp ...

(''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 categoryIn category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dire ...

* Kan extensionKan 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 JoyalCatLab

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
Category theory formalizes and its concepts in terms of a called a ', whose nodes are called ''objects'', and whose labelled directed edges are called ''arrows'' (or s). A has two basic properties: the ability to the arrows , and the exi ...

package for Mathematica. Manipulation and visualization of objects, morphisms, categories, functors, natural transformation
In category theory
Category theory formalizes mathematical structure and its concepts in terms of a Graph labeling, labeled directed graph called a ''Category (mathematics), category'', whose nodes are called ''objects'', and whose labelled dir ...

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 Functors,