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

We then define a contravariant functor F from C to D as a mapping that

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

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.

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.

The categorical structure of the functor F assures us that F defines a G-action on the set X.

The ( unique ) representable functor F: → is the Cayley representation of G. In fact, this functor is isomorphic to and so sends to the set which is by definition the " set " G and the morphism g of ( i. e. the element g of G ) to the permutation F < sub > g </ sub > of the set G. We deduce from the Yoneda embedding that the group G is isomorphic to the group

Given a functor F: J → C ( thought of as an object in C < sup > J </ sup >), the limit of F, if it exists, is nothing but a terminal morphism from Δ to F. Dually, the colimit of F is an initial morphism from F to Δ.

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.

* 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.

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

The word functor was borrowed by mathematicians from the philosopher Rudolf Carnap, who used the term in a linguistic context.

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.

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.

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 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.

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.

For example, the Hom functor is of the type C < sup > op </ sup > × C → Set.

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.

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.

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.

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 contravariant functor, is like a covariant functor, except that it " turns morphisms around " (" reverses all the arrows ").

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

The common feature of these examples is that they are expressions of the idea that the derivative is part of a functor.

A functor is an operation on spaces and functions between them.

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.

This variant of the chain rule is not an example of a functor because the two functions being composed are of different types.

The main problem is to prove a signalizer functor theorem for nonsolvable signalizer functors.

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

That is, instead of saying is a contravariant functor, they simply write ( or sometimes ) and call it a functor.

A bifunctor ( also known as a binary functor ) is a functor in two arguments.

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

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

The association G is a functor ( mapping between categories satisfying certain axioms ).

The mapping from rings of integers R to their corresponding class groups is functorial, and the class group can be subsumed under the heading of algebraic K-theory, with K < sub > 0 </ sub >( R ) being the functor assigning to R its ideal class group ; more precisely, K < sub > 0 </ sub >( R )

However, in contrast to the situation common for isomorphisms in an algebraic setting, the composition of the functor and its " inverse " is not necessarily the identity mapping.

In other words, we ask to identify the image of the associated bundle mapping ( which is actually a functor ).

The Hodge conjecture, may be neatly reformulated using motives: it holds iff the Hodge realization mapping any pure motive with rational coefficients ( over a subfield k of ℂ ) to its Hodge structure is a full functor H: M ( k )< sub > ℚ </ sub > → HS < sub > ℚ </ sub > ( rational Hodge structures ).

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.

For example, if is the forgetful functor mapping an abelian group to its underlying set, and is some fixed set ( regarded as a functor from 1 ), then the comma category has objects that are maps from to a set underlying a group.

In mathematics, in the area of category theory, a forgetful functor ( also known as a stripping functor ) ' forgets ' or drops some or all of the input's structure or properties ' before ' mapping to the output.

This is the point where the homotopy category comes into play again: mapping an object A of to ( any ) injective resolution of A extends to a functor