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| In [[mathematics]], specifically in [[category theory]], [[hom-set]]s, i.e. sets of [[morphism]]s between objects, give rise to important [[functor]]s to the [[category of sets]]. These functors are called '''hom-functors''' and have numerous applications in category theory and other branches of mathematics.
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| ==Formal definition==
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| Let ''C'' be a [[locally small category]] (i.e. a [[category (mathematics)|category]] for which hom-classes are actually [[Set (mathematics)|sets]] and not [[proper class]]es).
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| For all objects ''A'' and ''B'' in ''C'' we define two functors to the [[category of sets]] as follows:
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| {| class=wikitable
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| |-
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| !Hom(''A'',–) : ''C'' → '''Set'''
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| !Hom(–,''B'') : ''C'' → '''Set'''
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| |-
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| |This is a [[covariant functor]] given by:
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| *Hom(''A'',–) maps each object ''X'' in ''C'' to the set of [[morphism]]s, Hom(''A'', ''X'')
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| *Hom(''A'',–) maps each morphism ''f'' : ''X'' → ''Y'' to the [[function (mathematics)|function]]
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| *: Hom(''A'', ''f'') : Hom(''A'', ''X'') → Hom(''A'', ''Y'') given by
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| *: <math>\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto f\circ g</math> for each ''g'' in Hom(''A'', ''X'').
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| |This is a [[contravariant functor]] given by:
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| *Hom(–,''B'') maps each object ''X'' in ''C'' to the set of [[morphism]]s, Hom(''X'', ''B'')
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| *Hom(–,''B'') maps each morphism ''h'' : ''X'' → ''Y'' to the [[function (mathematics)|function]]
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| *: Hom(''h'', ''B'') : Hom(''Y'', ''B'') → Hom(''X'', ''B'') given by
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| *: <math>\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto g\circ h</math> for each ''g'' in Hom(''Y'', ''B'').
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| |}
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| The functor Hom(–,''B'') is also called the ''[[functor of points]]'' of the object ''B''.
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| 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.
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| The pair of functors Hom(''A'',–) and Hom(–,''B'') are related in a [[natural transformation|natural manner]]. For any pair of morphisms ''f'' : ''B'' → ''B''′ and ''h'' : ''A''′ → ''A'' the following diagram [[commutative diagram|commutes]]:
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| [[File:HomFunctor-01.png|center]]
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| Both paths send ''g'' : ''A'' → ''B'' to ''f'' ∘ ''g'' ∘ ''h''.
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| 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
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| : Hom(–,–) : ''C''<sup>op</sup> × ''C'' → '''Set'''
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| where ''C''<sup>op</sup> is the [[opposite category]] to ''C''. The notation Hom<sub>C</sub>(–,–) is sometimes used for Hom(–,–) in order to emphasize the category forming the domain.
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| ==Yoneda's lemma==
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| {{Main|Yoneda lemma}}
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| Referring to the above commutative diagram, one observes that every morphism
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| :''h'' : ''A''′ → ''A''
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| gives rise to a [[natural transformation]]
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| :Hom(''h'',–) : Hom(''A'',–) → Hom(''A''′,–)
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| and every morphism
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| :''f'' : ''B'' → ''B''′
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| gives rise to a natural transformation
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| :Hom(–,''f'') : Hom(–,''B'') → Hom(–,''B''′)
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| [[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 functor|full]] and [[faithful functor|faithful]] embedding of the category ''C'' into the [[functor category]] '''Set'''<sup>''C''<sup>''Op''</sup></sup> (covariant or contravariant depending on which Hom functor is used).
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| ==Internal Hom functor==
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| 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
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| :<math>\left[-\ -\right] : C^{op} \times C \to C</math>
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| to emphasize its product-like nature, or as
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| :<math>\Rightarrow : C^{op} \times C \to C</math>
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| to emphasize its functorial nature, or sometimes merely in lower-case:
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| :<math>\text{hom}(-, -) : C^{op} \times C \to C</math>
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| Categories that possess an internal Hom functor are referred to as [[closed category|closed categories]]. The [[forgetful functor]] <math>U:C\to\textbf{Set}</math> on such categories takes the internal Hom functor to the external Hom functor. That is,
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| :<math>U \circ \text{hom}(-, -) \simeq \text{Hom}(-, -)</math>
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| where <math>\simeq</math> denotes a [[natural isomorphism]]; the isomorphism is natural in both sites. Alternately, one has that
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| :<math>\text{Hom}(I, \text{hom}(-, -)) \simeq \text{Hom}(-, -)</math>,
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| 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]], namely, that
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| :<math>\text{Hom}(X, Y \Rightarrow Z) \simeq \text{Hom}(X\otimes Y, Z)</math>
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| where <math>\otimes</math> 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 <math>Y \Rightarrow Z</math> is called the '''internal Hom'''. When <math>\otimes</math> is the [[Cartesian closed category|Cartesian product]] <math>\times</math>, the object <math>Y \Rightarrow Z</math> is called the [[exponential object]], and is often written as <math>Z^Y</math>.
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| 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]], and the [[linear type system]], which is the internal language of [[closed monoidal category|closed symmetric monoidal categories]].
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| ==Properties==
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| Note that a functor of the form
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| :Hom(–, C) : ''C''<sup>op</sup> → '''Set'''
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| is a [[presheaf (category theory)|presheaf]]; likewise, Hom(C, –) is a copresheaf.
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| A functor ''F'' : ''C'' → '''Set''' that is [[naturally isomorphic]] to Hom(C, –) is called a [[representable functor]] or sometimes a representable copresheaf; likewise, a contravariant functor equivalent to Hom(–, C) might be called corepresentable.
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| Note that Hom(–, –) : ''C''<sup>op</sup> × ''C'' → '''Set''' is a [[profunctor]], and, specifically, it is the identity profunctor
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| :<math>\text{id}_C \colon C\nrightarrow C</math>,
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| The internal hom functor preserves [[limit (category theory)|limits]]; that is, <math>\text{hom}(X,-):C \to C</math> sends limits to limits, while <math>\text{hom}(-,X):C^\text{op} \to C</math> sends limits to [[colimit]]s. In a certain sense, this can be taken as the definition of a limit or colimit.
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| ==Other properties==
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| If '''A''' is an abelian category and ''A'' is an object of '''A''', then Hom<sub>'''A'''</sub>(''A'',–) is a covariant [[exact functor|left-exact]] functor from '''A''' to the category '''Ab''' of [[abelian group]]s. It is exact if and only if ''A'' is [[projective module|projective]].<ref>Jacobson (2009), p. 149, Prop. 3.9.</ref>
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| Let ''R'' be a [[ring (mathematics)|ring]] and ''M'' a left ''R''-[[module (mathematics)|module]]. The functor Hom<sub>'''Z'''</sub>(''M'',–): '''Ab''' → '''Mod'''-''R'' is right [[adjoint functor|adjoint]] to the [[tensor product of modules|tensor product]] functor – <math>\otimes</math><sub>R</sub> M: '''Mod'''-''R'' → '''Ab'''.
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| ==See also==
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| * [[Ext functor]]
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| * [[Functor category]]
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| * [[Representable functor]]
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| == Notes ==
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| <references/>
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| ==References==
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| * {{Cite book |first=Saunders |last=Mac Lane |authorlink=Saunders Mac Lane|title=Categories for the Working Mathematician | edition=Second |date=September 1998 |publisher=Springer |isbn=0-387-98403-8}}
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| * {{Cite book|first=Robert|last=Goldblatt|title=Topoi, the Categorial Analysis of Logic|url=http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|accessdate=2009-11-25|edition=Revised|year=2006|origyear=1984|publisher=[[Dover Publications]]|isbn=978-0-486-45026-1}}
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| * {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| date=2009| title=Basic algebra| edition=2nd| volume = 2 | series= | publisher=Dover| isbn = 978-0-486-47187-7}}
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| * {{nlab|id=hom-functor|title=Hom functor}}
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| * {{nlab|id=internal-hom|title=Internal Hom}}
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| {{DEFAULTSORT:Hom Functor}}
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| [[Category:Functors]]
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