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, cate ...
,
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 a ...
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 a ...
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 ...
) give rise to important
functor
In mathematics, specifically category theory, a functor is a Map (mathematics), mapping between Category (mathematics), categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) ar ...
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 of m ...
. 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
In mathematics, a category (sometimes called an abstract category to distinguish it from a concrete category) is a collection of "objects" that are linked by "arrows". A category has two basic properties: the ability to compose the arrows asso ...
(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 for ...
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 of m ...
as follows:
:
The functor Hom(–, ''B'') is also called the ''
functor of points In algebraic geometry, a functor represented by a scheme ''X'' is a set-valued contravariant functor on the category of schemes such that the value of the functor at each scheme ''S'' is (up to natural bijections) the set of all morphisms S \to X. T ...
'' 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
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 ma ...
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
In category theory, a branch of mathematics, the opposite category or dual category ''C''op 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 t ...
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
In mathematics, the Yoneda lemma is arguably the most important result in category theory. It is an abstract result on functors of the type ''morphisms into a fixed object''. It is a vast generalisation of Cayley's theorem from group theory (viewi ...
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 In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
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
In mathematics, the category Rel has the class of sets as objects and binary relations as morphisms.
A morphism (or arrow) ''R'' : ''A'' → ''B'' in this category is a relation between the sets ''A'' and ''B'', so .
The composition of two re ...
.
Categories that possess an internal Hom functor are referred to as
closed categories. One has that
:
,
where ''I'' is the
unit object
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 and ri ...
of the closed category. For the case of a
closed monoidal category
In mathematics, especially in category theory, a closed monoidal category (or a ''monoidal closed category'') is a category that is both a monoidal category and a closed category in such a way that the structures are compatible.
A classic exampl ...
, 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 that ...
, namely, that
:
where
is a
bifunctor
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 ma ...
, 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 and r ...
. 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 are p ...
in both ''X'' and ''Z''. In other words, in a closed monoidal category, the internal Hom functor is an
adjoint functor
In mathematics, specifically category theory, adjunction is a relationship that two functors may exhibit, intuitively corresponding to a weak form of equivalence between two related categories. Two functors that stand in this relationship are kno ...
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\ti ...
, the object
is called the
exponential object
In mathematics, specifically in category theory, an exponential object or map object is the categorical generalization of a function space in set theory. Categories with all finite products and exponential objects are called cartesian closed ca ...
, and is often written as
.
Internal Homs, when chained together, form a language, called the
internal language
__NOTOC__
Categorical logic is the branch of mathematics in which tools and concepts from category theory are applied to the study of mathematical logic. It is also notable for its connections to theoretical computer science.
In broad terms, categ ...
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 cal ...
, 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 ma ...
, and the
linear type system
Substructural type systems are a family of type systems analogous to substructural logics where one or more of the structural rules are absent or only allowed under controlled circumstances. Such systems are useful for constraining access to sy ...
, 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
In mathematics, a sheaf is a tool for systematically tracking data (such as sets, abelian groups, rings) attached to the open sets of a topological space and defined locally with regard to them. For example, for each open set, the data could ...
; 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 In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...
(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 In category theory, a branch of mathematics, profunctors are a generalization of relations and also of bimodules.
Definition
A profunctor (also named distributor by the French school and module by the Sydney school) \,\phi from a category C t ...
, and, specifically, it is the identity profunctor
.
The internal hom functor preserves
limits
Limit or Limits may refer to:
Arts and media
* ''Limit'' (manga), a manga by Keiko Suenobu
* ''Limit'' (film), a South Korean film
* Limit (music), a way to characterize harmony
* "Limit" (song), a 2016 single by Luna Sea
* "Limits", a 2019 ...
; that is,
sends limits to limits, while
sends limits in
, that is
colimit
In category theory, a branch of mathematics, the abstract notion of a limit captures the essential properties of universal constructions such as products, pullbacks and inverse limits. The dual notion of a colimit generalizes constructions such ...
s 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 ab ...
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 commut ...
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 bicondi ...
''A'' is
projective.
[Jacobson (2009), p. 149, Prop. 3.9.]
Let ''R'' be a
ring
Ring may refer to:
* Ring (jewellery), a round band, usually made of metal, worn as ornamental jewelry
* To make a sound with a bell, and the sound made by a bell
:(hence) to initiate a telephone connection
Arts, entertainment and media Film and ...
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
In mathematics, the term ''adjoint'' applies in several situations. Several of these share a similar formalism: if ''A'' is adjoint to ''B'', then there is typically some formula of the type
:(''Ax'', ''y'') = (''x'', ''By'').
Specifically, adjoin ...
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 \otimes W ...
functor –
''R'' ''M'': Ab → Mod-''R''.
See also
*
Ext functor
In mathematics, the Ext functors are the derived functors of the Hom functor. Along with the Tor functor, Ext is one of the core concepts of homological algebra, in which ideas from algebraic topology are used to define invariants of algebraic stru ...
*
Functor category In category theory, a branch of mathematics, a functor category D^C is a category where the objects are the functors F: C \to D and the morphisms are natural transformations \eta: F \to G between the functors (here, G: C \to D is another object in t ...
*
Representable functor In mathematics, particularly category theory, a representable functor is a certain functor from an arbitrary category into the category of sets. Such functors give representations of an abstract category in terms of known structures (i.e. sets and ...
Notes
References
*
*
*
External links
*
*
{{DEFAULTSORT:Hom Functor
Functors