Санаттар арасындағы байланыс және функциялар

Mapping between categories
the mathematical concept

In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words category and functor were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used functor in a linguistic context;
see function word.

Every functor induces the opposite functor , where and are the opposite categories to and By definition, maps objects and morphisms in the identical way as does Since does not coincide with as a category, and similarly for , is distinguished from For example, when composing with , one should use either or Note that, following the property of opposite category, .

A bifunctor (also known as a binary functor) is a functor whose domain is a product category. For example, the Hom functor is of the type Cop × C → Set. It can be seen as a functor in two arguments. The Hom functor 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 1=n = 2.

Diagram For categories C and J, a diagram of type J in C is a covariant functor (Category theoretical) presheafFor categories C and J, a J presheaf on C is a contravariant functor In the special case when J is Set, the category of sets and functions, D is called a presheaf on C.
Presheaves (over a topological space) 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 U → V if and only if Contravariant functors on Open(X) are called presheaves on X. For instance, by assigning to every open set U the associative algebra of real valued continuous functions on U, one obtains a presheaf of algebras on X. Constant functor The functor C → D 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. term=Endofunctor A functor that maps a category to that same category; e. g., polynomial functor. term=Identity functor In category C, written 1C or idC, maps an object to itself and a morphism to itself. The identity functor is an endofunctor. 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. 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 CJ → C 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 assigns to every functor its colimit, and is covariant). Power sets functor The power set functor P : Set → Set maps each set to its power set and each function to the map which sends to its image One can also consider the contravariant power set functor which sends to the map which sends to its inverse image For example, if then Suppose and Then is the function which sends any subset of to its image , which in this case means , where denotes the mapping under , so this could also be written as For the other values, Note that consequently generates the trivial topology on Also note that although the function in this example mapped to the power set of , that need not be the case in general. 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. 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 : X → Y with 1=f(x0) = y0. 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, with the group operation of concatenation. If f : X → Y is a 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 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. 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 : X → Y induces an algebra homomorphism C(f) : C(Y) → C(X) by the rule 1=C(f)(φ) = φ ∘ f for every φ in C(Y). 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. Doing this constructions pointwise gives the tangent space, a covariant functor from the category of pointed differentiable manifolds to the category of real vector spaces. Likewise, cotangent space is a contravariant functor, essentially the composition of the tangent space with the dual space above. Group actions/representations Every group 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 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 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. Tensor products If C denotes the category of vector spaces over a fixed field, with linear maps as morphisms, then the tensor product defines a functor C × C → C which is covariant in both arguments. 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. Functors like these, which "forget" some structure, are termed forgetful functors. Another example is the functor Rng → Ab which maps a ring to its underlying additive abelian group. Morphisms in Rng (ring homomorphisms) become morphisms in Ab (abelian group homomorphisms). 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. Functions get mapped to group homomorphisms between free groups. Free constructions exist for many categories based on structured sets. See free object. 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 × Ab → Ab (where Ab denotes the category of abelian groups with group homomorphisms). 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 ↦ g ∘ φ ∘ f. See Hom functor. 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 × C → Set. If f : X1 → X2 and g : Y1 → Y2 are morphisms in C, then the map Hom(f, g) : Hom(X2, Y1) → Hom(X1, Y2) is given by φ ↦ g ∘ φ ∘ f. Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.

Let C and D be categories. The collection of all functors from C to D forms the objects of a category: the functor category. Morphisms in this category are natural transformations between functors. 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. Universal constructions often give rise to pairs of adjoint functors.

Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class Functor where fmap is a polytypic function used to map functions (morphisms on Hask, the category of Haskell types) between existing types to functions between some new types.