In mathematics

Mathematics

Mathematics is the study of quantity, space, structure, and change. Mathematicians seek out patterns and formulate new conjectures. Mathematicians resolve the truth or falsity of conjectures by mathematical proofs, which are arguments sufficient to convince other mathematicians of their validity...

, specifically in category theory

Category theory

Category theory is an area of study in mathematics that examines in an abstract way the properties of particular mathematical concepts, by formalising them as collections of objects and arrows , where these collections satisfy certain basic conditions...

, hom-sets, i.e. sets of morphism

Morphism

In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics...

s between objects, give rise to important functor

Functor

In category theory, a branch of mathematics, a functor is a special type of mapping between categories. Functors can be thought of as homomorphisms between categories, or morphisms when in the category of small categories....

s to the category of sets

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

. These functors are called hom-functors and have numerous applications in category theory and other branches of mathematics.

Formal definition

Let C be a locally small category (i.e. a category

Category (mathematics)

In mathematics, a category is an algebraic structure that comprises "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows associatively and the existence of an identity arrow for each object. A simple example is the category of sets, whose...

 for which hom-classes are actually sets and not proper classes).

For all objects A and B in C we define two functors to the category of sets

Category of sets

In the mathematical field of category theory, the category of sets, denoted as Set, is the category whose objects are sets. The arrows or morphisms between sets A and B are all functions from A to B...

 as follows:

Hom(A,–) : C → Set

Hom(–,B) : C → Set

This is a covariant functor given by: Hom(A,–) maps each object X in C to the set of morphism Morphism In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics... s, Hom(A, X) Hom(A,–) maps each morphism f : X → Y to the function Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... Hom(A, f) : Hom(A, X) → Hom(A, Y) given by for each g in Hom(A, X).

This is a contravariant functor given by: Hom(–,B) maps each object X in C to the set of morphism Morphism In mathematics, a morphism is an abstraction derived from structure-preserving mappings between two mathematical structures. The notion of morphism recurs in much of contemporary mathematics... s, Hom(X, B) Hom(–,B) maps each morphism h : X → Y to the function Function (mathematics) In mathematics, a function associates one quantity, the argument of the function, also known as the input, with another quantity, the value of the function, also known as the output. A function assigns exactly one output to each input. The argument and the value may be real numbers, but they can... Hom(h, B) : Hom(Y, B) → Hom(X, B) given by for each g in Hom(Y, B).

The functor Hom(–,B) is also called the functor of points of the object B.

Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant functor. This is an artifact of the way in which one must compose the morphisms.

The pair of functors Hom(A,–) and Hom(–,B) are obviously related in a natural manner. For any pair of morphisms f : B → B′ and h : A′ → A the following diagram commutes

Commutative diagram

In mathematics, and especially in category theory, a commutative diagram is a diagram of objects and morphisms such that all directed paths in the diagram with the same start and endpoints lead to the same result by composition...

:

Both paths send g : A → B to f ∘ g ∘ h.

The commutativity of the above diagram implies that Hom(–,–) is a bifunctor from C × C to Set which is contravariant in the first argument and covariant in the second. Equivalently, we may say that Hom(–,–) is a covariant bifunctor

Hom(–,–) : C^op × C → Set

where C^op is the opposite category

Opposite category

In category theory, a branch of mathematics, the opposite category or dual category Cop of a given category C is formed by reversing the morphisms, i.e. interchanging the source and target of each morphism. Doing the reversal twice yields the original category, so the opposite of an opposite...

 to C.

Yoneda's lemma

Referring to the above commutative diagram, one observes that every morphism

h : A′ → A

gives rise to a natural transformation

Natural transformation

In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition...

Hom(h,–) : Hom(A,–) → Hom(A′,–)

and every morphism

f : B → B′

gives rise to a natural transformation

Hom(–,f) : Hom(–,B) → Hom(–,B′)

Yoneda's lemma implies that every natural transformation between Hom functors is of this form. In other words, the Hom functors give rise to a full and faithful embedding of the category C into the functor category

Functor category

In category theory, a branch of mathematics, the functors between two given categories form a category, where the objects are the functors and the morphisms are natural transformations between the functors...

 Set^C (covariant or contravariant depending on which Hom functor is used).

Other properties

If A is an abelian category and A is an object of A, then Hom_A(A,–) is a covariant left-exact

Exact functor

In homological algebra, an exact functor is a functor, from some category to another, which preserves exact sequences. Exact functors are very convenient in algebraic calculations, roughly speaking because they can be applied to presentations of objects easily...

 functor from A to the category Ab of abelian group

Abelian group

In abstract algebra, an abelian group, also called a commutative group, is a group in which the result of applying the group operation to two group elements does not depend on their order . Abelian groups generalize the arithmetic of addition of integers...

s. It is exact if and only if A is projective

Projective module

In mathematics, particularly in abstract algebra and homological algebra, the concept of projective module over a ring R is a more flexible generalisation of the idea of a free module...

.

Let R be a ring

Ring (mathematics)

In mathematics, a ring is an algebraic structure consisting of a set together with two binary operations usually called addition and multiplication, where the set is an abelian group under addition and a semigroup under multiplication such that multiplication distributes over addition...

 and M a left R-module

Module (mathematics)

In abstract algebra, the concept of a module over a ring is a generalization of the notion of vector space, wherein the corresponding scalars are allowed to lie in an arbitrary ring...

. The functor Hom_Z(M,–): Ab → Mod-R is right adjoint to the tensor product

Tensor product of modules

In mathematics, the tensor product of modules is a construction that allows arguments about bilinear maps to be carried out in terms of linear maps...

 functor – _R M: Mod-R → Ab.

See also

• Representable functor
  Representable functor
  In mathematics, particularly category theory, a representable functor is a functor of a special form from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures In mathematics, particularly category theory, a...
  
  
• Ext functor
  Ext functor
  In mathematics, the Ext functors of homological algebra are derived functors of Hom functors. They were first used in algebraic topology, but are common in many areas of mathematics.- Definition and computation :...
  
  
• Currying
  Currying
  In mathematics and computer science, currying is the technique of transforming a function that takes multiple arguments in such a way that it can be called as a chain of functions each with a single argument...
  
  
• Cartesian closed category
  Cartesian closed category
  In category theory, a category is cartesian closed if, roughly speaking, any morphism defined on a product of two objects can be naturally identified with a morphism defined on one of the factors. These categories are particularly important in mathematical logic and the theory of programming, in...