In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, specifically in
category theory
Category theory is a general theory of mathematical structures and their relations that was introduced by Samuel Eilenberg and Saunders Mac Lane in the middle of the 20th century in their foundational work on algebraic topology. Nowadays, ca ...
,
hom-set
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s (i.e. sets of
morphism
In mathematics, particularly in category theory, a morphism is a structure-preserving map from one mathematical structure to another one of the same type. The notion of morphism recurs in much of contemporary mathematics. In set theory, morphisms ...
s between
objects
Object may refer to:
General meanings
* Object (philosophy), a thing, being, or concept
** Object (abstract), an object which does not exist at any particular time or place
** Physical object, an identifiable collection of matter
* Goal, an ai ...
) give rise to important
functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and m ...
s to the
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 the total functions from ''A'' to ''B'', and the composition o ...
. 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, plural categories, may refer to:
Philosophy and general uses
*Categorization, categories in cognitive science, information science and generally
* Category of being
* ''Categories'' (Aristotle)
* Category (Kant)
* Categories (Peirce) ...
for which hom-classes are actually
sets and not
proper class
Proper may refer to:
Mathematics
* Proper map, in topology, a property of continuous function between topological spaces, if inverse images of compact subsets are compact
* Proper morphism, in algebraic geometry, an analogue of a proper map f ...
es).
For all objects ''A'' and ''B'' in ''C'' we define two functors to the
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 the total functions from ''A'' to ''B'', and the composition o ...
as follows:
:
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 related in a
natural manner. For any pair of morphisms ''f'' : ''B'' → ''B''′ and ''h'' : ''A''′ → ''A'' the following diagram
commutes:
Both paths send ''g'' : ''A'' → ''B'' to ''f''∘''g''∘''h'' : ''A''′ → ''B''′.
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 bifunctor
: Hom(–, –) : ''C''
op × ''C'' → Set
where ''C''
op is the
opposite category to ''C''. The notation Hom
''C''(–, –) is sometimes used for Hom(–, –) in order to emphasize the category forming the domain.
Yoneda's lemma
Referring to the above commutative diagram, one observes that every morphism
: ''h'' : ''A''′ → ''A''
gives rise to a
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 (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
: 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 Set
''C''''op'' (covariant or contravariant depending on which Hom functor is used).
Internal Hom functor
Some categories may possess a functor that behaves like a Hom functor, but takes values in the category ''C'' itself, rather than Set. Such a functor is referred to as the internal Hom functor, and is often written as
:
to emphasize its product-like nature, or as
:
to emphasize its functorial nature, or sometimes merely in lower-case:
:
For examples, see
Category of relations.
Categories that possess an internal Hom functor are referred to as
closed categories
Closed may refer to:
Mathematics
* Closure (mathematics), a set, along with operations, for which applying those operations on members always results in a member of the set
* Closed set, a set which contains all its limit points
* Closed interval, ...
. One has that
:
,
where ''I'' is the
unit object of the closed category. For the case of a
closed monoidal category, this extends to the notion of
currying
In mathematics and computer science, currying is the technique of translating the evaluation of a function that takes multiple arguments into evaluating a sequence of functions, each with a single argument. For example, currying a function f tha ...
, namely, that
:
where
is a
bifunctor, the internal product functor defining a
monoidal category
In mathematics, a monoidal category (or tensor category) is a category \mathbf C equipped with a bifunctor
:\otimes : \mathbf \times \mathbf \to \mathbf
that is associative up to a natural isomorphism, and an object ''I'' that is both a left ...
. The isomorphism is
natural
Nature, in the broadest sense, is the physical world or universe. "Nature" can refer to the phenomena of the physical world, and also to life in general. The study of nature is a large, if not the only, part of science. Although humans ar ...
in both ''X'' and ''Z''. In other words, in a closed monoidal category, the internal Hom functor is an
adjoint functor to the internal product functor. The object
is called the internal Hom. When
is the
Cartesian product
In mathematics, specifically set theory, the Cartesian product of two sets ''A'' and ''B'', denoted ''A''×''B'', is the set of all ordered pairs where ''a'' is in ''A'' and ''b'' is in ''B''. In terms of set-builder notation, that is
: A\t ...
, the object
is called the
exponential object, and is often written as
.
Internal Homs, when chained together, form a language, called the
internal language of the category. The most famous of these are
simply typed lambda calculus
The simply typed lambda calculus (\lambda^\to), a form
of type theory, is a typed interpretation of the lambda calculus with only one type constructor (\to) that builds function types. It is the canonical and simplest example of a typed lambda c ...
, which is the internal language of
Cartesian closed categories
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 mat ...
, and the
linear type system, which is the internal language of
closed symmetric monoidal categories.
Properties
Note that a functor of the form
: Hom(–, ''A'') : ''C''
op → Set
is a
presheaf; likewise, Hom(''A'', –) is a copresheaf.
A functor ''F'' : ''C'' → Set that is
naturally isomorphic
In category theory, a branch of mathematics, a natural transformation provides a way of transforming one functor into another while respecting the internal structure (i.e., the composition of morphisms) of the categories involved. Hence, a natur ...
to Hom(''A'', –) for some ''A'' in ''C'' is called a
representable functor (or representable copresheaf); likewise, a contravariant functor equivalent to Hom(–, ''A'') might be called corepresentable.
Note that Hom(–, –) : ''C''
op × ''C'' → Set is a
profunctor, and, specifically, it is the identity profunctor
.
The internal hom functor preserves
limits; that is,
sends limits to limits, while
sends limits in
, that is
colimits in
, into limits. In a certain sense, this can be taken as the definition of a limit or colimit.
Other properties
If A is an
abelian category
In mathematics, an abelian category is a category in which morphisms and objects can be added and in which kernels and cokernels exist and have desirable properties. The motivating prototypical example of an abelian category is the category of ...
and ''A'' is an object of A, then Hom
A(''A'', –) is a covariant
left-exact functor from A to the category Ab of
abelian group
In mathematics, 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 the order in which they are written. That is, the group operation is comm ...
s. It is exact
if and only if
In logic and related fields such as mathematics and philosophy, "if and only if" (shortened as "iff") is a biconditional logical connective between statements, where either both statements are true or both are false.
The connective is bic ...
''A'' is
projective.
[Jacobson (2009), p. 149, Prop. 3.9.]
Let ''R'' be a
ring and ''M'' a left ''R''-
module
Module, modular and modularity may refer to the concept of modularity. They may also refer to:
Computing and engineering
* Modular design, the engineering discipline of designing complex devices using separately designed sub-components
* Modul ...
. The functor Hom
R(''M'', –): Mod-''R'' → Ab is
adjoint to the
tensor product
In mathematics, the tensor product V \otimes W of two vector spaces and (over the same field) is a vector space to which is associated a bilinear map V\times W \to V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otime ...
functor –
''R'' ''M'': Ab → Mod-''R''.
See also
*
Ext functor
*
Functor category
*
Representable functor
Notes
References
*
*
*
External links
*
*
{{DEFAULTSORT:Hom Functor
Functors