A functor associates to every object of one category an object of another category, and to every morphism in the first category a morphism in the second.

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.

A commutative diagram in a category C can be interpreted as a functor from an index category J to C ; one calls the functor a diagram.

This makes F into a functor from commutative R-algebras S to groups.

Given any finite flat commutative group scheme G over S, its Cartier dual is the group of characters, defined as the functor that takes any S-scheme T to the abelian group of group scheme homomorphisms from the base change G < sub > T </ sub > to G < sub > m, T </ sub > and any map of S-schemes to the canonical map of character groups.

This functor is representable by a finite flat S-group scheme, and Cartier duality forms an additive involutive antiequivalence from the category of finite flat commutative S-group schemes to itself.

Then T is a right exact functor from Mod-R to the category of abelian groups Ab ( in the case when R is commutative, it is a right exact functor from Mod-R to Mod-R ) and its left derived functors L < sub > n </ sub > T are defined.

In the language of category theory, the functor that sends a commutative monoid M to its Grothendieck group N is left adjoint to the forgetful functor from the category of abelian groups to the category of commutative monoids.

Formally, the right Kan extension of along consists of a functor and a natural transformation which is couniversal with respect to the specification, in the sense that for any functor and natural transformation, a unique natural transformation is defined and fits into a commutative diagram

This gives rise to the alternate description: the left Kan extension of along consists of a functor and a natural transformation which are universal with respect to this specification, in the sense that for any other functor and natural transformation, a unique natural transformation exists and fits into a commutative diagram:

Another example is the functor Rng → Ab which maps a ring to its underlying additive abelian group.

For a simple example, if the rings R and S are represented by the one-object preadditive categories R and S, then a ring homomorphism from R to S is represented by an additive functor from R to S, and conversely.

One can check that U is a functor from the category of rings to the category of groups: every ring homomorphism f: R → S induces a group homomorphism U ( f ): U ( R ) → U ( S ), since f maps units to units.

This functor has a left adjoint which is the integral group ring construction.

* Any ring R can be considered as a one-object preadditive category ; the category of left modules over R is the same as the additive functor category Add ( R, Ab ) ( where Ab denotes the category of abelian groups ), and the category of right R-modules is Add ( R < sup > op </ sup >, Ab ).

By dualizing the homology chain complex ( i. e. applying the functor Hom (-, R ), R being any ring ) we obtain a cochain complex with coboundary map.

If R is a ring and T is a right R-module, we can define a functor H < sub > T </ sub > from the abelian category of all left R-modules to Ab by using the tensor product over R: H < sub > T </ sub >( X ) = T ⊗ X.

The group-valued functor of F can also be described using the formal group ring H of F. For simplicity we will assume that F is 1-dimensional ; the general case is similar.

The functor category of all additive functors from this category to the category of abelian groups is isomorphic to the category of left modules over the ring.

* For any abelian group A, one can form the corresponding diagonalizable group D ( A ), defined as a functor by setting D ( A )( T ) to be the set of abelian group homomorphisms from A to invertible global sections of O < sub > T </ sub > for each S-scheme T. If S is affine, D ( A ) can be formed as the spectrum of a group ring.

As a functor, it sends an S-scheme T to the group of invertible n by n matrices whose entries are global sections of T. Over an affine base, one can construct it as a quotient of a polynomial ring in n < sup > 2 </ sup > + 1 variables by an ideal encoding the invertibility of the determinant.

The functor K < sub > 0 </ sub > takes a ring A to the Grothendieck group of the set of isomorphism classes of its finitely generated projective modules, regarded as a monoid under direct sum.

Any ring homomorphism A → B gives a map K < sub > 0 </ sub >( A ) → K < sub > 0 </ sub >( B ) by mapping ( the class of ) a projective A-module M to M ⊗< sub > A </ sub > B, making K < sub > 0 </ sub > a covariant functor.

Deleting "*" and 1 yields a functor to the category of abelian groups, which assigns to each ring R the underlying additive abelian group of R. To each morphism of rings is assigned the same function considered merely as a morphism of addition between the underlying groups.

In each of the above cases, the functor sends each space to its tangent bundle and it sends each function to its derivative.

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.

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.

be the forgetful functor which assigns to each algebra its underlying vector space.

Mapping each object A in C to its associated hom-functor h < sup > A </ sup > = Hom ( A ,–) and each morphism f: B → A to the corresponding natural transformation Hom ( f ,–) determines a contravariant functor h < sup >–</ sup > from C to Set < sup > C </ sup >, the functor category of all ( covariant ) functors from C to Set.

There is a covariant functor from the category of abelian groups to the category of torsion groups that sends every group to its torsion subgroup and every homomorphism to its restriction to the torsion subgroup.