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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-sets (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) give rise to important functors to the category of sets. 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 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 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 : 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 : \left \ -\right: C^\text \times C \to C to emphasize its product-like nature, or as : \mathop\Rightarrow : C^\text \times C \to C to emphasize its functorial nature, or sometimes merely in lower-case: : \operatorname(-, -) : C^\text \times C \to C . 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 : \operatorname(I, \operatorname(-, -)) \simeq \operatorname(-, -), 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 that ...
, namely, that : \operatorname(X, Y \Rightarrow Z) \simeq \operatorname(X\otimes Y, Z) where \otimes is a bifunctor, the internal product functor defining a monoidal category. The isomorphism is natural 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 Y \Rightarrow Z is called the internal Hom. When \otimes 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 ...
\times, the object Y \Rightarrow Z is called the exponential object, and is often written as Z^Y. 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, 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; likewise, Hom(''A'', –) is a copresheaf. A functor ''F'' : ''C'' → Set that is naturally isomorphic 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 \operatorname_C \colon C \nrightarrow C. The internal hom functor preserves limits; that is, \operatorname(X, -) \colon C \to C sends limits to limits, while \operatorname(-, X) \colon C^\text \to C sends limits in C^\text, 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 su ...
s in C, 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 HomA(''A'', –) is a covariant left-exact functor from A to the category Ab of abelian groups. It is exact if and only if ''A'' is projective.Jacobson (2009), p. 149, Prop. 3.9. Let ''R'' be a ring and ''M'' a left ''R''- module. The functor HomR(''M'', –): Mod-''R'' → Ab is adjoint to the tensor product functor – \otimes''R'' ''M'': Ab → Mod-''R''.


See also

* Ext functor * Functor category * Representable functor


Notes


References

* * *


External links

* * {{DEFAULTSORT:Hom Functor Functors