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Encyclopedia > Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of small categories. In mathematics, category theory deals in an abstract way with mathematical structures and relationships between them. ... For other meanings of mathematics or uses of math and maths, see Mathematics (disambiguation) and Math (disambiguation). ... In mathematics, a morphism is an abstraction of a structure-preserving process between two mathematical structures. ... In mathematics, specifically in category theory, the 2-category of small categories is the 2-category whose objects are small categories, whose arrows are functors and whose 2-arrows are natural transformations. ...


Functors were first considered in algebraic topology, where algebraic objects (like the fundamental group) are associated to topological spaces, and algebraic homomorphisms are associated to continuous maps. Nowadays, functors are used throughout modern mathematics to relate various categories. The word "functor" was borrowed by mathematicians from the philosopher Carnap [Mac Lane, p. 30]. Carnap used the term "functor" to stand in relation to functions analogously as predicates stand in relation to properties. [See Carnap, The Logical Syntax of Language, p.13-14, 1937, Routledge & Kegan Paul.] For Carnap then, unlike modern category theory's use of the term, a functor is a linguistic item. For category theorists, a functor is a particular kind of function. Algebraic topology is a branch of mathematics in which tools from abstract algebra are used to study topological spaces. ... In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... Topological spaces are structures that allow one to formalize 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 spaces). ... In mathematics, a continuous function is a function for which, intuitively, small changes in the input result in small changes in the output. ... Rudolf Carnap (May 18, 1891 - September 14, 1970) was a German philosopher. ...

Contents

Definition

Let C and D be categories. A functor F from C to D is a mapping that In mathematics, categories allow one to formalize notions involving abstract structure and processes that preserve structure. ...

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

such that the following two properties hold:

  • F(idX) = idF(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:Yrightarrow Z.

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


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


Covariance and contravariance

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

  • associates to each object X in C an object F(X) in D,
  • associates to each morphism f:Xrightarrow Y in C a morphism F(f):F(Y) rightarrow F(X) in D such that
    • F(idX) = idF(X) for every object X in C,
    • F(g circ f) = F(f) circ F(g) for all morphisms f:Xrightarrow Y and g:Yrightarrow 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 Cop. Some authors prefer to write all expressions covariantly. That is, instead of saying F: Crightarrow 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. In category theory, an abstract branch of mathematics, the dual of a category C is the category formed by reversing all the morphisms of C. That is, we take Cop to be the category with objects that are those of C, but with the morphisms from X to Y in...


Contravariant functors are also occasionally called cofunctors.


Examples

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


Diagonal functor: The diagonal functor is defined as the functor from D to the functor category DC which sends each object in D to the constant functor at that object. In category theory, for any object a in any category C where the product a×a exists, there exists the diagonal morphism δa: a → a×a, satisfying πkδa = ida for k=1,2, where πk is the canonical projection morphism to the k-th component. ...


Limit functor: For a fixed index category J, if every functor JC has a limit (for instance if C is complete), then the limit functor CJC assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor and invoking the Freyd adjoint functor theorem. This requires a suitable version of the axiom of choice. Similar remarks apply to the colimit functor (which is covariant). In mathematics, it is a common practice to index or label a collection of objects by some set I called an index set. ... 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 of mathematics, like products and inverse limits. ... In mathematics, the axiom of choice, or AC, is an axiom of set theory. ...


Power sets: The power set functor P : SetSet maps each set to its power set and each function  f : X to Y to the map which sends U subseteq X to its image f(U) subseteq Y. One can also consider the contravariant power set functor which sends f to the map which sends U to its inverse image in Y. In mathematics, given a set S, the power set (or powerset) of S, written or 2S, is the set of all subsets of S. In axiomatic set theory (as developed e. ... In mathematics, the image of an element x in a set X under the function f : X → Y, denoted by f(x), is the unique y in Y that is associated with x. ...


Dual vector space: The map which assigns to every vector space its dual space and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed field to itself. 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 on V. Dual vector spaces defined on finite-dimensional vector spaces can be used for defining tensors which are studied in tensor algebra. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In abstract algebra, a field is an algebraic structure in which the operations of addition, subtraction, multiplication and division (except division by zero) may be performed, and the same rules hold which are familiar from the arithmetic of ordinary numbers. ...


Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs (X, x0), where X is a topological space and x0 is a point in X. A morphism from (X, x0) to (Y, y0) is given by a continuous map f : XY with f(x0) = y0. In mathematics, a pointed space is a topological space X with a distinguised basepoint x0 in X. Maps of pointed spaces are continuous maps preserving basepoints, i. ... In topology and related areas of mathematics a continuous function is a morphism between topological spaces. ...


To every topological space X with distinguished point x0, one can define the fundamental group based at x0, denoted π1(X, x0). This is the group of homotopy classes of loops based at x0. If f : XY morphism of pointed spaces, then every loop in X with base point x0 can be composed with f to yield a loop in Y with base point y0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from π(X, x0) to π(Y, y0). We thus obtain a functor from the category of pointed topological spaces to the category of groups. In mathematics, the fundamental group is one of the basic concepts of algebraic topology. ... This picture illustrates how the hours on a clock form a group under modular addition. ... The two bold paths shown above are homotopic relative to their endpoints. ... In mathematics, a pointed space is a topological space X with a distinguished basepoint x0 in X. Maps of pointed spaces (based maps) are continuous maps preserving basepoints, i. ... In mathematics, an equivalence relation is a binary relation between two elements of a set which groups them together as being equivalent in some way. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, the category Grp has the class of all groups for objects and group homomorphisms for morphisms. ...


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, a groupoid is a concept (first developed by Heinrich Brandt in 1926) that simultaneously generalises groups, equivalence relations on sets, and actions of groups on sets. ...


Algebra of continuous functions: a contravariant functor from the category of topological spaces (with continuous maps as morphisms) to the category of real associative algebras is given by assigning to every topological space X the algebra C(X) of all real-valued continuous functions on that space. Every continuous map f : XY 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 Möbius strip, an object with only one surface and one edge; such shapes are an object of study in topology. ... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ... A homomorphism between two algebras over a field K, A and B, is a map such that for all k in K and x,y in A, F(kx)=kF(x) F(x+y)=F(x)+F(y) F(xy)=F(x)F(y) Categories: Math stubs | Algebra ...


Tangent and cotangent bundles: The map which sends every differentiable manifold to its tangent bundle and every smooth map to its derivative is a covariant functor from the category of differentiable manifolds to the category of vector bundles. Likewise, the map which sends every differentiable manifold to its cotangent bundle and every smooth map to its pullback is a contravariant functor. Informally, a differentiable manifold is a type of manifold (which is in turn a kind of topological space) that is locally similar enough to Euclidean space to allow one to do calculus. ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space... In mathematics, a smooth function is one that is infinitely differentiable, i. ... For a non-technical overview of the subject, see Calculus. ... In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In differential geometry, the cotangent bundle of a manifold is the vector bundle of all the cotangent spaces at every point in the manifold. ... Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is...


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. A functor from G to Set is nothing but a group action of G on a particular set, i.e. a G-set. Likewise, a functor from G to the category of vector spaces, VectK, is a linear representation of G. In general, a functor GC can be considered as an "action" of G on an object in the category C. This picture illustrates how the hours on a clock form a group under modular addition. ... In mathematics, a symmetry group describes all symmetries of objects. ... In mathematics, especially category theory, the category K-Vect has all vector spaces over a fixed field K as objects and K-linear transformations as morphisms. ... Representation theory is the branch of mathematics that studies properties of abstract groups via their representations as linear transformations of vector spaces. ...


Lie algebras: Assigning to every real (complex) Lie group its real (complex) Lie algebra defines a functor. In mathematics, a Lie group, named after Norwegian mathematician Sophus Lie (IPA pronunciation: , sounds like Lee), is a group which is also a differentiable manifold, with the property that the group operations are compatible with the smooth structure. ... In mathematics, a Lie algebra is an algebraic structure whose main use is in studying geometric objects such as Lie groups and differentiable manifolds. ...


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 × CC which is covariant in both arguments. In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ...


Forgetful functors: The functor U : GrpSet which maps a group to its underlying set and a group homomorphism to its underlying function of sets is a functor. Functors like these, which "forget" some structure, are termed forgetful functors. Another example is the functor RngAb which maps a ring to its underlying additive abelian group. Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms). This picture illustrates how the hours on a clock form a group under modular addition. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... A forgetful functor is a type of functor in mathematics. ... In ring theory, a branch of abstract algebra, a ring is an algebraic structure in which addition and multiplication are defined and have similar properties to those familiar from the integers. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In 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 : SetGrp sends every set X to the free group generated by X. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object. The Cayley graph of the free group on two generators a and b In mathematics, a group G is called free if there is a subset S of G such that any element of G can be written in one and only one way as a product of finitely many... 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. This is a functor which is contravariant in the first and covariant in the second argument, i.e. it is a functor Abop × AbAb (where Ab denotes the category of abelian groups with group homomorphisms). If f : A1A2 and g : B1B2 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. This picture illustrates how the hours on a clock form a group under modular addition. ... Given two groups (G, *) and (H, ·), a group homomorphism from (G, *) to (H, ·) is a function h : G -> H such that for all u and v in G it holds that h(u * v) = h(u) · h(v) From this property, one can deduce that h maps the identity element... In mathematics, the category Ab has the abelian groups as objects and group homomorphisms as morphisms. ... In mathematics, specifically in category theory, Hom-sets, i. ...


Representable functors: We can generalize the previous example to any category C. To every pair X, Y of objects in C one can assign the set Hom(X,Y) of morphisms from X to Y. This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor Cop × CSet. If f : X1X2 and g : Y1Y2 are morphisms in C, then the group homomorphism Hom(f,g) : Hom(X2,Y1) → Hom(X1,Y2) is given by φ mapsto g o φ o f.


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


Presheaves: If X is a topological space, then the open sets in X form a partially ordered set Open(X) under inclusion. Like every partially ordered set, Open(X) forms a small category by adding a single arrow UV if and only if U subseteq V. Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a presheaf of algebras on X. Topological spaces are structures that allow one to formalize concepts such as convergence, connectedness and continuity. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics, especially order theory, a partially ordered set (or poset) is a set equipped with a partial order relation. ... In mathematics, a sheaf F on a given topological space X gives a set or richer structure F(U) for each open set U of X. The structures F(U) are compatible with the operations of restricting the open set to smaller subsets and gluing smaller open sets to obtain... In mathematics, an associative algebra is a vector space (or more generally, a module) which also allows the multiplication of vectors in a distributive and associative manner. ...


Properties

Two important consequences of the functor axioms are: This article is about a logical statement. ...

On any category C one can define the identity functor 1C which maps each object and morphism to itself. One can also compose functors, i.e. if F is a functor from A to B and G is a functor from B to C then one can form the composite functor GF from A to C. Composition of functors is associative where defined. This shows that functors can be considered as morphisms in categories of categories. In mathematics, especially the many applications of category theory, a commutative diagram is a diagram of objects and morphisms such that, when picking two objects, one can follow any path through the diagram and obtain the same result by composition. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...


A small category with a single object is the same thing as a monoid: the morphisms of a one-object category can be thought of as elements of the monoid, and composition in the category is thought of as the monoid operation. Functors between one-object categories correspond to monoid homomorphisms. So in a sense, functors between arbitrary categories are a kind of generalization of monoid homomorphisms to categories with more than one object. In abstract algebra, a branch of mathematics, a monoid is an algebraic structure with a single, associative binary operation and an identity element. ... In abstract algebra, a homomorphism is a structure-preserving map between two algebraic structures (such as groups, rings, or vector spaces). ...


Bifunctors

A bifunctor is a generalization of the functor concept to 'two variables'. The Hom functor is a natural example, and is contravariant in one variable, covariant in the other. In mathematics, specifically in category theory, Hom-sets, i. ...


Relation to other categorical concepts

Let C and D be categories. The collection of all functors CD form the objects of a category: the functor category. Morphisms in this category are natural transformations between functors. In category theory, the functors between two given categories can themselves be turned into a category; the morphisms in this functor category are natural transformations between functors. ... In category theory, an abstract branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i. ...


Functors are often defined by universal properties; examples are the tensor product, the direct sum and direct product of groups or vector spaces, construction of free groups and modules, direct and inverse limits. The concepts of limit and colimit generalize several of the above. In various branches of mathematics, certain constructions are frequently defined or characterised by an abstract property which requires the existence of a unique morphism under certain conditions. ... In mathematics, the tensor product, denoted by , may be applied in different contexts to vectors, matrices, tensors, vector spaces, algebras, topological vector spaces, and modules. ... In abstract algebra, the direct sum is a construction which combines several vector spaces (or groups, or abelian groups, or modules) into a new, bigger one. ... In mathematics, one can often define a direct product of objects already known, giving a new one. ... In mathematics, the direct limit (also called the inductive limit) is a general method of taking limits of directed families of objects. We will first give the definition for algebraic structures like groups and modules, and then the general definition which can be used in any category. ... In mathematics, the inverse limit (also called the projective limit) is a construction which allows one to glue together several related objects, the precise matter of the gluing process being specified by morphisms between the objects. ... 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 of mathematics, like products and inverse limits. ...


Universal constructions often give rise to pairs of adjoint functors. In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ...


See also

Look up functor in Wiktionary, the free dictionary.

Wikipedia does not have an article with this exact name. ... Wiktionary (a portmanteau of wiki and dictionary) is a multilingual, Web-based project to create a free content dictionary, available in over 150 languages. ... A preadditive category is a category that is enriched over the monoidal category of abelian groups. ... In mathematics, an abelian group, also called a commutative group, is a group (G, * ) with the additional property that * commutes: for all a and b in G, a * b = b * a. ... In mathematics, adjoint functors are pairs of functors which stand in a particular relationship with one another. ... In mathematics, certain functors may be derived to obtain other functors closely related to the original ones. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In category theory, an enriched functor is a variant on a special type of mapping between categories. ... In category theory, a functor is essentially surjective when each object of is isomorphic to an object of the form for some object of . ... In homological algebra, an exact functor is one which preserves exact sequences. ... In mathematics, especially in homological algebra and other applications of Abelian category theory, as well as in group theory, an exact sequence is a (finite or infinite) sequence of objects and morphisms between them such that the image of one morphism equals the kernel of the next. ... In category theory, a faithful functor is a functor which is injective when restricted to each set of morphisms with a given source and target. ... An injective function. ... In category theory, a full functor is a functor which is surjective when restricted to each set of morphisms with a given source and target. ... A surjective function. ... Kan extensions are universal constructs in category theory, a branch of mathematics. ... In mathematical analysis, a differentiability class is a classification of functions according to the properties of their derivatives. ...

References

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

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Limit (category theory) - Wikipedia, the free encyclopedia (1907 words)
The importance of adjoint functors lies in the fact that every functor which has a left adjoint (and therefore is a right adjoint) is continuous.
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Every functor which has a right adjoint (and hence is a left adjoint) is cocontinuous.
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