|
|
| (One intermediate revision by one other user not shown) |
| Line 1: |
Line 1: |
| 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.
| | Hi there. Let me start by introducing the writer, her name is Myrtle Cleary. Hiring has been my profession for some time but I've currently utilized for an additional one. Years ago he moved to North Dakota and his family members enjoys it. To gather cash is what her family members and her enjoy.<br><br>My web site :: std testing at home - [http://www.videoworld.com/user/SMaloney similar web page], |
| | |
| ==Formal definition==
| |
| 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). | |
| | |
| For all objects ''A'' and ''B'' in ''C'' we define two functors to the [[category of sets]] as follows:
| |
| {| class=wikitable
| |
| |-
| |
| !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]]s, Hom(''A'', ''X'')
| |
| *Hom(''A'',–) maps each morphism ''f'' : ''X'' → ''Y'' to the [[function (mathematics)|function]]
| |
| *: Hom(''A'', ''f'') : Hom(''A'', ''X'') → Hom(''A'', ''Y'') given by
| |
| *: <math>\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto f\circ g</math> 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]]s, Hom(''X'', ''B'')
| |
| *Hom(–,''B'') maps each morphism ''h'' : ''X'' → ''Y'' to the [[function (mathematics)|function]]
| |
| *: Hom(''h'', ''B'') : Hom(''Y'', ''B'') → Hom(''X'', ''B'') given by
| |
| *: <math>\definecolor{gray}{RGB}{249,249,249}\pagecolor{gray} g \mapsto g\circ h</math> 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 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]]:
| |
| [[File:HomFunctor-01.png|center]]
| |
| 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''<sup>op</sup> × ''C'' → '''Set'''
| |
| 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.
| |
| | |
| ==Yoneda's lemma==
| |
| {{Main|Yoneda 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 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).
| |
| | |
| ==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
| |
| :<math>\left[-\ -\right] : C^{op} \times C \to C</math>
| |
| to emphasize its product-like nature, or as
| |
| :<math>\Rightarrow : C^{op} \times C \to C</math>
| |
| to emphasize its functorial nature, or sometimes merely in lower-case:
| |
| :<math>\text{hom}(-, -) : C^{op} \times C \to C</math>
| |
| 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,
| |
| :<math>U \circ \text{hom}(-, -) \simeq \text{Hom}(-, -)</math>
| |
| where <math>\simeq</math> denotes a [[natural isomorphism]]; the isomorphism is natural in both sites. Alternately, one has that
| |
| :<math>\text{Hom}(I, \text{hom}(-, -)) \simeq \text{Hom}(-, -)</math>,
| |
| 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
| |
| | |
| :<math>\text{Hom}(X, Y \Rightarrow Z) \simeq \text{Hom}(X\otimes Y, Z)</math>
| |
| 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>.
| |
| | |
| 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]].
| |
| | |
| ==Properties==
| |
| Note that a functor of the form
| |
| :Hom(–, C) : ''C''<sup>op</sup> → '''Set'''
| |
| is a [[presheaf (category theory)|presheaf]]; likewise, Hom(C, –) is a copresheaf.
| |
| | |
| 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.
| |
| | |
| Note that Hom(–, –) : ''C''<sup>op</sup> × ''C'' → '''Set''' is a [[profunctor]], and, specifically, it is the identity profunctor
| |
| :<math>\text{id}_C \colon C\nrightarrow C</math>,
| |
| | |
| 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.
| |
| | |
| ==Other properties==
| |
| 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>
| |
| | |
| 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'''.
| |
| | |
| ==See also==
| |
| * [[Ext functor]]
| |
| * [[Functor category]]
| |
| * [[Representable functor]]
| |
| | |
| == Notes ==
| |
| <references/>
| |
| | |
| ==References==
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{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}}
| |
| * {{nlab|id=hom-functor|title=Hom functor}}
| |
| * {{nlab|id=internal-hom|title=Internal Hom}}
| |
| | |
| {{DEFAULTSORT:Hom Functor}}
| |
| [[Category:Functors]]
| |