The functor associates to every commutative ring its spectrum, the scheme defined by the prime ideals of the ring.

The functor which associates to each sheaf F the group of global sections F ( X ) is left-exact.

Then the map that associates to a sheaf its global sections is a covariant functor to.

The operation which associates to an object S of E the fibre category F < sub > S </ sub > and to a morphism f the inverse image functor f < sup >*</ sup > is almost a contravariant functor from E to the category of categories.

For example, if one imagines the objects of some category C to be analogous to the open sets of a topological space, then a functor from C to the category of sets gives a set-valued presheaf on C, that is, it associates sets to the objects of C in a way which is compatible with the arrows of C. A subfunctor then associates a subset to each set, again in a compatible way.

More specifically, every morphism in C must be assigned to a morphism in D. In other words, a contravariant functor acts as a covariant functor from the opposite category C < sup > op </ sup > to D.

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.

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 < sup > J </ sup >→ C assigns to each functor its limit.

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.

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.

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.

Lie algebras: Assigning to every real ( complex ) Lie group its real ( complex ) Lie algebra defines a functor.

The free functor F: Set → Grp sends every set X to the free group generated by X.

Another way to describe G-sets is the functor category, where is the groupoid ( category ) with one element and isomorphic to the group G. Indeed, every functor F of this category defines a set X = F and for every g in G ( i. e. for every morphism in ) induces a bijection F < sub > g </ sub >: X → X.

* Function object, or functor or functionoid, a concept of object-oriented programming

Identity functor in category C, written 1 < sub > C </ sub > or id < sub > C </ sub >, maps an object to itself and a morphism to itself.

Diagonal functor: The diagonal functor is defined as the functor from D to the functor category D < sup > C </ sup > which sends each object in D to the constant functor at that object.

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 < sub > K </ sub >, 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.

* Universal constructions are functorial in nature: if one can carry out the construction for every object in a category C then one obtains a functor on C. Furthermore, this functor is a right or left adjoint to the functor U used in the definition of the universal property.

Suppose that U: D → C is a functor from a category D to a category C, and let X be an object of C. Consider the following dual ( opposite ) notions:

This leads to the clarifying concept of natural transformation, a way to " map " one functor to another.

The Yoneda lemma is one of the most famous basic results of category theory ; it describes representable functors in functor categories.

There is one requirement for such an operation to be a functor, namely that the derivative of a composite is the composite of the derivatives.

Note that one can also define a contravariant functor as a covariant functor on the dual category.

The Hom functor is a natural example ; it is contravariant in one argument, covariant in the other.

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 G ∘ F from A to C. Composition of functors is associative where defined.

Similar statements apply to the dual situation of terminal morphisms from U. If such morphisms exist for every X in C one obtains a functor V: C → D which is right-adjoint to U ( so U is left-adjoint to V ).

This functor is left adjoint to the diagonal functor Δ: C → C < sup > J </ sup >, and one has a natural isomorphism

which is natural in the variable N. Here the functor Hom ( N, F –) is the composition of the Hom functor Hom ( N, –) with F. This isomorphism is the unique one which respects the limiting cones.

If F: J → C is a diagram in C and G: C → D is a functor then by composition ( recall that a diagram is just a functor ) one obtains a diagram GF: J → D. A natural question is then:

* Every continuous functor on a small-complete category which satisfies the appropriate solution set condition has a left-adjoint ( the Freyd adjoint functor theorem ).

Abstracting again, a category is itself a type of mathematical structure, so we can look for " processes " which preserve this structure in some sense ; such a process is called a functor.

A ( covariant ) functor F from a category C to a category D, written, consists of:

* The functor category D < sup > C </ sup > has as objects the functors from C to D and as morphisms the natural transformations of such functors.

This shows that the commutator subgroup can be viewed as a functor on the category of groups, some implications of which are explored below.

The abelianization functor is the left adjoint of the inclusion functor from the category of abelian groups to the category of groups.

In category theory, a branch of mathematics, a functor is a special type of mapping between categories.

Formally, a bifunctor is a functor whose domain is a product category.

Endofunctor: A functor that maps a category to itself.