Functor - Examples

Examples

Diagram: For categories C and J, a diagram of type J in C is a covariant functor .

(Category theoretical) presheaf: For categories C and J, a J-presheaf on C is a contravariant functor .

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 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.

Endofunctor: A functor that maps a category to itself.

Identity functor in category C, written 1_C or id_C, maps an object to itself and a morphism to itself. 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 is covariant).

Power sets: 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

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 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. 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 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 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 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 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 o φ o 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 group homomorphism Hom(f,g) : Hom(X₂,Y₁) → Hom(X₁,Y₂) is given by φ ↦ 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.