Functor

For functors as a synonym of "function objects" in computer programming, see Function object. A function object, also called a functor or functional, is a Computer programming construct allowing an object to be invoked or called as if For functors in linguistics, see Function word.

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. In Mathematics, category theory deals in an abstract way with mathematical Structures and relationships between them it abstracts from sets Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Functors can be thought of as morphisms in the category of small categories. In Mathematics, a morphism is an abstraction derived from structure-preserving mappings between two Mathematical structures The study of morphisms and In Mathematics, specifically in Category theory, the category of small categories, denoted by Cat, is the category whose objects are all

Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Algebraic topology is a branch of Mathematics which uses tools from Abstract algebra to study Topological spaces The basic goal is to find algebraic In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector In Mathematics, a continuous function is a function for which intuitively small changes in the input result in small changes in the output Nowadays, functors are used throughout modern mathematics to relate various categories. The word "functor" was borrowed by mathematicians from the philosopher Carnap [Mac Lane, p. Rudolf Carnap ( May 18, 1891 &ndash September 14, 1970) was an influential German -born philosopher who was active in 30]. Carnap used the term "functor" to stand in relation to functions analogously as predicates stand in relation to properties. [See Carnap, The Logical Syntax of Language, p. 13-14, 1937, Routledge & Kegan Paul. ] For Carnap then, unlike modern category theory's use of the term, a functor is a linguistic item. For category theorists, a functor is a particular kind of function.

1 Definition
- 1.1 Covariance and contravariance
2 Examples
3 Properties
4 Bifunctors and multifunctors
5 Relation to other categorical concepts
6 See also
7 References

Definition

Let C and D be categories. In Mathematics, a category is a fundamental and abstract way to describe mathematical entities and their relationships A functor F from C to D is a mapping that

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

such that the following two conditions hold:

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

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

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

Covariance and contravariance

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

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

Note that contravariant functors reverse the direction of composition.

Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a covariant functor on the dual category $C o p$ . In Category theory, an abstract branch of Mathematics, the dual category or opposite category C op of a category C is the Some authors prefer to write all expressions covariantly. That is, instead of saying $F: C\rightarrow D$ is a contravariant functor, they simply write $F: C^{op} \rightarrow D$ (or sometimes $F:C \rightarrow D^{op}$ ) and call it a functor.

Contravariant functors are also occasionally called cofunctors.

Examples

Constant functor: The functor C → D is one which maps every object of C to a fixed object X in D and every morphism in C to the identity morphism on X. Such a functor is called a constant or selection functor.

Diagonal functor: The diagonal functor is defined as the functor from D to the functor category D^C which sends each object in D to the constant functor at that object. In Category theory, for any object a in any category C where the product a × a exists there exists the diagonal morphism

Limit functor: For a fixed index category J, if every functor J→C has a limit (for instance if C is complete), then the limit functor C^J→C assigns to each functor its limit. In Category theory, a branch of mathematics a diagram is the categorical analogue of a Indexed family in Set theory. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version of the axiom of choice. In Mathematics, the axiom of choice, or AC, is an Axiom of Set theory. Similar remarks apply to the colimit functor (which is covariant).

Power sets: The power set functor P : Set → Set maps each set to its power set and each function $f : X \to Y$ to the map which sends $U \subseteq X$ to its image $f(U) \subseteq Y$ . In Mathematics, given a set S, the power set (or powerset) of S, written \mathcal{P}(S P ( S) One can also consider the contravariant power set functor which sends $f : X \to Y$ to the map which sends $V \subseteq Y$ to its inverse image $f^{-1}(V) \subseteq X$ . In Mathematics, the image of a preimage under a given function is the set of all possible function outputs when taking each element of the preimage

Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, any Vector space V has a corresponding dual vector space (or just dual space for short consisting of all Linear functionals In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Abstract algebra, a field is an Algebraic structure in which the operations of Addition, Subtraction, Multiplication and division

Fundamental group: Consider the category of pointed topological spaces, i. In Mathematics, a pointed space is a Topological space X with a distinguished basepoint x 0 in X. e. topological spaces with distinguished points. The objects are pairs (X, x₀), where X is a topological space and x₀ is a point in X. A morphism from (X, x₀) to (Y, y₀) is given by a continuous map f : X → Y with f(x₀) = y₀. In Topology and related areas of Mathematics a continuous function is a Morphism between Topological spaces Intuitively this is a function

To every topological space X with distinguished point x₀, one can define the fundamental group based at x₀, denoted π₁(X, x₀). In Mathematics, the fundamental group is one of the basic concepts of Algebraic topology. This is the group of homotopy classes of loops based at x₀. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Topology, two continuous functions from one Topological space to another are called homotopic ( Greek homos = identical If f : X → Y morphism of pointed spaces, then every loop in X with base point x₀ can be composed with f to yield a loop in Y with base point y₀. In Mathematics, a pointed space is a Topological space X with a distinguished basepoint x 0 in X. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π(X, x₀) to π(Y, y₀). In Mathematics, an equivalence relation is a Binary relation between two elements of a set which groups them together as being "equivalent" In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function We thus obtain a functor from the category of pointed topological spaces to the category of groups. In Mathematics, the category Grp has the class of all groups for objects and Group homomorphisms for Morphisms As such

In the category of topological spaces (without distinguished point), one considers homotopy classes of generic curves, but they cannot be composed unless they share an endpoint. Thus one has the fundamental groupoid instead of the fundamental group, and this construction is functorial. In Mathematics, especially in Category theory and Homotopy theory

Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Topology ( Greek topos, "place" and logos, "study" is the branch of Mathematics that studies the properties of In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive Every continuous map f : X → Y induces an algebra homomorphism C(f) : C(Y) → C(X) by the rule C(f)(φ) = φ o f for every φ in C(Y). A homomorphism between two algebras over a field K, A and B, is a map FA\rightarrow B such that for all k

Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles. A differentiable manifold is a type of Manifold that is locally similar enough to Euclidean space to allow one to do Calculus. In Mathematics, the tangent bundle of a smooth (or differentiable manifold M, denoted by T ( M) or just TM, is the In Mathematical analysis, a differentiability class is a classification of functions according to the properties of their Derivatives Higher order differentiability In Calculus, a branch of mathematics the derivative is a measurement of how a function changes when the values of its inputs change In Mathematics, a vector bundle is a topological construction which makes precise the idea of a family of Vector spaces parameterized by another space Likewise, the map which sends every differentiable manifold to its cotangent bundle and every smooth map to its pullback is a contravariant functor. In Mathematics, especially Differential geometry, the cotangent bundle of a Smooth manifold is the Vector bundle of all the Cotangent spaces Suppose that φ: M → N is a Smooth map between smooth manifolds M and N; then there is an associated Linear map from

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

Group actions/representations: Every group G can be considered as a category with a single object whose morphisms are the elements of G. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element A functor from G to Set is then nothing but a group action of G on a particular set, i. In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. e. a G-set. Likewise, a functor from G to the category of vector spaces, Vect_K, is a linear representation of G. In Mathematics, especially Category theory, the category K-Vect has all Vector spaces over a fixed field K as objects In the mathematical field of Representation theory, group representations describe abstract groups in terms of Linear transformations of In general, a functor G → C can be considered as an "action" of G on an object in the category C. If C is a group, then this action is a group homomorphism.

Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor. In Mathematics, a Lie group (ˈliː sounds like "Lee" is a group which is also a Differentiable manifold, with the property that the group In Mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable Manifolds Lie

Tensor products: If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product $V \otimes W$ defines a functor C × C → C which is covariant in both arguments. In Mathematics, a linear map (also called a linear transformation, or linear operator) is a function between two Vector spaces that In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector

Forgetful functors: The functor U : Grp → Set which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function Functors like these, which "forget" some structure, are termed forgetful functors. In Mathematics, in the area of Category theory, a forgetful functor is a type of Functor. Another example is the functor Rng → Ab which maps a ring to its underlying additive abelian group. In Mathematics, a ring is an Algebraic structure which generalizes the algebraic properties of the Integers though the rational, real An abelian group, also called a commutative group, is a group satisfying the additional requirement that the product of elements does not depend on their order (the Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms). In Ring theory or Abstract algebra, a ring homomorphism is a function between two rings which respects the operations of addition and multiplication

Free functors: Going in the opposite direction of forgetful functors are free functors. The free functor F : Set → Grp sends every set X to the free group generated by X. In Mathematics, a group G is called free if there is a Subset S of G such that any element of G can be 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.

Homomorphism groups: To every pair A, B of abelian groups one can assign the abelian group Hom(A,B) consisting of all group homomorphisms from A to B. In Mathematics, a group is a set of elements together with an operation that combines any two of its elements to form a third element In Mathematics, given two groups ( G, * and ( H, · a group homomorphism from ( G, * to ( H, · is a function This is a functor which is contravariant in the first and covariant in the second argument, i. e. it is a functor Ab^op × Ab → Ab (where Ab denotes the category of abelian groups with group homomorphisms). In Mathematics, the category Ab has the Abelian groups as objects and Group homomorphisms as Morphisms This is the prototype If f : A₁ → A₂ and g : B₁ → B₂ are morphisms in Ab, then the group homomorphism Hom(f,g) : Hom(A₂,B₁) → Hom(A₁,B₂) is given by φ $\mapsto$ g o φ o f. See Hom functor. In Mathematics, specifically in Category theory, Hom-sets ie sets of Morphisms between objects give rise to important Functors to the Category

Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X,Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i. e. it is a functor C^op × C → Set. If f : X₁ → X₂ and g : Y₁ → Y₂ are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X₂,Y₁) → Hom(X₁,Y₂) is given by φ $\mapsto$ g o φ o f.

Functors like these are called representable functors. In Mathematics, especially in Category theory, a representable functor is a Functor of a special form from an arbitrary category into the An important goal in many settings is to determine whether a given functor is representable.

Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion. Topological spaces are mathematical structures that allow the formal definition of concepts such as Convergence, connectedness, and continuity. In Metric topology and related fields of Mathematics, a set U is called open if intuitively speaking starting from any point x in In Mathematics, especially Order theory, a partially ordered set (or poset) formalizes the intuitive concept of an ordering sequencing or arrangement Like every partially ordered set, Open(X) forms a small category by adding a single arrow U → V if and only if $U \subseteq V$ . Contravariant functors on Open(X) are called presheaves on X. In Mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the Open sets of a Topological space. 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. In Mathematics, an associative algebra is a Vector space (or more generally a module) which also allows the multiplication of vectors in a distributive

Properties

Two important consequences of the functor axioms are:

F transforms each commutative diagram in C into a commutative diagram in D;
if f is an isomorphism in C, then F(f) is an isomorphism in D. In traditional Logic, an axiom or postulate is a proposition that is not proved or demonstrated but considered to be either self-evident, or subject In mathematics and especially in Category theory a commutative diagram is a Diagram of objects also known as vertices, and Morphisms also In Abstract algebra, an isomorphism ( Greek: ἴσος isos "equal" and μορφή morphe "shape" is a bijective

On any category C one can define the identity functor 1_C which maps each object and morphism to itself. One can also compose functors, i. e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor GF from A to C. Composition of functors is associative where defined. This shows that functors can be considered as morphisms in categories of categories.

A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. In Abstract algebra, a branch of Mathematics, a monoid is an Algebraic structure with a single Associative Binary operation Functors between one-object categories correspond to monoid homomorphisms. In Abstract algebra, a homomorphism is a structure-preserving map between two Algebraic structures (such as groups rings or Vector So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object.

Bifunctors and multifunctors

A bifunctor (also known as a binary functor) is a functor in two arguments. The Hom functor is a natural example; it is contravariant in one argument, covariant in the other. In Mathematics, specifically in Category theory, Hom-sets ie sets of Morphisms between objects give rise to important Functors to the Category

Formally, a bifunctor 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 C × D and called a product category For example, the Hom functor is of the type C^op × C → Set.

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

Relation to other categorical concepts

Let C and D be categories. The collection of all functors C→D form the objects of a category: the functor category. In Category theory, a branch of Mathematics, the Functors between two given categories can themselves be turned into a category the morphisms in this functor Morphisms in this category are natural transformations between functors. In Category theory, a branch of Mathematics, a natural transformation provides a way of transforming one Functor into another while respecting the internal

Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. In various branches of Mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique Morphism In Mathematics, the tensor product, denoted by \otimes may be applied in different contexts to vectors matrices, Tensors Vector The symbol \oplus \! denotes direct sum it is also the astrological and astronomical symbol for Earth, and a symbol for the Exclusive disjunction In Mathematics, one can often define a direct product of objectsalready known giving a new one In Mathematics, the direct limit (also called the inductive limit) is a general method of taking colimits of "directed families of objects" In Mathematics, the inverse limit (also called the projective limit) is a construction which allows one to "glue together" several related objects the precise The concepts of limit and colimit generalize several of the above. In Category theory, a branch of Mathematics, the abstract notion of a limit captures the essential properties of universal constructions that are used in various parts

Universal constructions often give rise to pairs of adjoint functors.

References

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

Dictionary

-noun

(Can we verify⁽⁺⁾ this sense?) A person who performs a function; a functionary or official
(grammar) a function word
(computing) a function object
(mathematics) a mapping between categories

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