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In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all ( small) categories.
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.
Let C and D be categories. A functor F from C to D is a mapping that
such that the following two properties hold:
That is, functors must preserve identity morphisms and composition of morphism.
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
such 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 dual category Cop. Some authors prefer to write all expressions covariantly. That is, instead of saying F : C → D is a contravariant functor, they simply write F : Cop → D (or sometimes F : C → Dop) and call it a functor.
Contravariant functors are also occasionally called cofunctors.
Constant functor: A very boring 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.
Power sets: The power set functor P : Set → Set maps each set to its power set and each function f : X → Y to the map which sends U ⊆ X to its image f(U) ⊆ 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.
Dual vectorspace: The map which assigns to every vector space its dual spaceIn mathematics, the existence of a 'dual' vector space reflects in an abstract way the relationship between row vectors (1×n) and column vectors (n×1). The construction can also take place for infinite-dimensional spaces and gives rise to important ways o and to every linear map its dual or transpose is a contravariant functor from the category of all vector spaces over a fixed fieldIn 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 associative, commutative, and distributive rules hold, which are famil 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 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 groupIn mathematics, a group is a set, together with a binary operation satisfying certain axioms, detailed below. The branch of mathematics which studies groups is called group theory. The historical origin of group theory goes back to the works of Evariste G of homotopy classes of loops based at x0. 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. This operation is compatible with the homotopy equivalence relation and the composition of loops, and we get a group homomorphismGiven 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 e of G to the identity elem 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.
Algebra of continuous functions: a contravariant functor from the category of topological spacesTopology is the study or science of places. It derives its name from the Greek words τοπος meaning place and λογος meaning study, talk. See also earth science, geography, human geography, g (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 C(f)(φ) = φ o 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. Likewise, the map which sends every differentiable manifold to its cotangent bundle and every smooth map to its pullback is a contravariant functor.
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 (or groupoid) 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 G → C can be considered as an "action" of G on an object in the category C.
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 V ⊗ W 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 homomophisms 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 o φ o f.
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 Mor(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 group homomorphism Mor(f,g) : Mor(X2,Y1) → Mor(X1,Y2) is given by φ |-> g o φ o f.
Functors like these are called representable functors. A major goal in many settings is to determine whether a given functor is representable.
Universal constructions: Functors are often defined by universal properties; examples are the tensor product discussed above, 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.
Pre-Sheaves: If X is a topological space, then the open sets in X can be considered as the objects of a category CX; there is a morphism from U to V if and only if U is a subset of V. In itself, this category is not very exciting, but the functors from CXop into other categories, the so-called pre-sheaves on X, are interesting. For instance, by assigning to every open set U the associative algebra of real-valued continuous functions on U, one obtains a pre-sheaf of algebras on X.
This motivating example is generalized by considering pre-sheaves on arbitrary categories: a pre-sheaf on C is a functor defined on Cop. The Yoneda lemma explains that often a category C can be extended by considering a category of pre-sheaves on C.