Functor

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.
Definition.
Let "C" and "D" be categories. A functor "F" from "C" to "D" is a mapping that
That is, functors must preserve identity morphisms and composition of morphisms.
Covariance and contravariance.
There are many constructions in mathematics that would be functors but for the fact that they "turn morphisms around" and "reverse composition". We then define a contravariant functor "F" from "C" to "D" as a mapping that
Note that contravariant functors reverse the direction of composition.
Ordinary functors are also called covariant functors in order to distinguish them from contravariant ones. Note that one can also define a contravariant functor as a "covariant" functor on the opposite category formula_19. Some authors prefer to write all expressions covariantly. That is, instead of saying formula_20 is a contravariant functor, they simply write formula_21 (or sometimes formula_22) and call it a functor.
Contravariant functors are also occasionally called "cofunctors".
There is a convention which refers to "vectors"—i.e., vector fields, elements of the space of sections formula_23 of a tangent bundle formula_24—as "contravariant" and to "covectors"—i.e., 1-forms, elements of the space of sections formula_25 of a cotangent bundle formula_26—as "covariant". This terminology originates in physics, and its rationale has to do with the position of the indices ("upstairs" and "downstairs") in expressions such as formula_27 for formula_28 or formula_29 for formula_30 In this formalism it is observed that the coordinate transformation symbol formula_31 (representing the matrix formula_32) acts on the basis vectors "in the same way" as on the "covector coordinates": formula_33—whereas it acts "in the opposite way" on the "vector coordinates" (but "in the same way" as on the basis covectors: formula_34). 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.
Opposite functor.
Every functor formula_20 induces the opposite functor formula_36, where formula_19 and formula_38 are the opposite categories to formula_39 and formula_40. By definition, formula_41 maps objects and morphisms identically to formula_42. Since formula_19 does not coincide with formula_39 as a category, and similarly for formula_40, formula_41 is distinguished from formula_42. For example, when composing formula_48 with formula_49, one should use either formula_50 or formula_51. Note that, following the property of opposite category, formula_52.
Bifunctors and multifunctors.
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 . 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 .
Examples.
Diagram: For categories "C" and "J", a diagram of type "J" in "C" is a covariant functor formula_53.
(Category theoretical) presheaf: For categories "C" and "J", a "J"-presheaf on "C" is a contravariant functor formula_54.
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 if and only if formula_55. 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 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 functor.
Identity functor: in category "C", written 1"C" or id"C", maps an object to itself and a morphism to itself. The identity functor is an endofunctor.
Diagonal functor: The diagonal functor is defined as the functor from "D" to the functor category "D""C" which sends each object in "D" to the constant functor at that object.
Limit functor: For a fixed index category "J", if every functor has a limit (for instance if "C" is complete), then the limit functor assigns to each functor its limit. The existence of this functor can be proved by realizing that it is the right-adjoint to the diagonal functor 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 maps each set to its power set and each function formula_56 to the map which sends formula_57 to its image formula_58. One can also consider the contravariant power set functor which sends formula_59 to the map which
sends formula_60 to its inverse image formula_61
For example, if formula_62 then formula_63. Suppose formula_64 and formula_65. Then formula_66 is the function which sends any subset formula_67 of formula_1 to its image formula_69, which in this case means
formula_70, where formula_71 denotes the mapping under formula_66, so this could also be written as formula_73. For the other values, formula_74 Note that formula_75 consequently generates the trivial topology on formula_1. Also note that although the function formula_77 in this example mapped to the power set of formula_1, that need not be the case in general.
Fundamental group: Consider the category of pointed topological spaces, i.e. topological spaces with distinguished points. The objects are pairs , where "X" is a topological space and "x"0 is a point in "X". A morphism from to is given by a continuous map with .
To every topological space "X" with distinguished point "x"0, one can define the fundamental group based at "x"0, denoted . This is the group of homotopy classes of loops based at "x"0, with the group operation of concatenation. If is a morphism of pointed spaces, then every loop in "X" with base point "x"0 can be composed with "f" to yield a loop in "Y" with base point "y"0. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphism from to . 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 induces an algebra homomorphism by the rule 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, Vect"K", is a linear representation 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 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 formula_79 defines a functor which is covariant in both arguments.
Forgetful functors: The functor 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 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 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 (where Ab denotes the category of abelian groups with group homomorphisms). If and are morphisms in Ab, then the group homomorphism : is given by . 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 of morphisms from "X" to "Y". This defines a functor to Set which is contravariant in the first argument and covariant in the second, i.e. it is a functor . If and are morphisms in "C", then the map is given by .
Functors like these are called representable functors. An important goal in many settings is to determine whether a given functor is representable.
Properties.
Two important consequences of the functor axioms are:
One can compose functors, i.e. if "F" is a functor from "A" to "B" and "G" is a functor from "B" to "C" then one can form the composite functor from "A" to "C". Composition of functors is associative where defined. Identity of composition of functors is the identity functor. This shows that functors can be considered as morphisms in categories of categories, for example in the category of small categories.
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.
Relation to other categorical concepts.
Let "C" and "D" be categories. The collection of all functors from "C" to "D" forms the objects of a category: the functor category. 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.
Computer implementations.
Functors sometimes appear in functional programming. For instance, the programming language Haskell has a class codice_1 where codice_2 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.