# Hom functor > Mediated Wiki article. Canonical URL: https://mediated.wiki/source/Hom_functor > Markdown URL: https://mediated.wiki/source/Hom_functor.md > Source: https://en.wikipedia.org/wiki/Hom_functor > Source revision: 1347375777 > License: Creative Commons Attribution-ShareAlike 4.0 International (https://creativecommons.org/licenses/by-sa/4.0/) {{Short description|Functor mapping hom objects to an underlying category}} In [mathematics](/source/mathematics), specifically in [category theory](/source/category_theory), [hom-set](/source/hom-set)s (i.e. sets of [morphism](/source/morphism)s between [objects](/source/object_(category_theory))) give rise to important [functor](/source/functor)s to the [category of sets](/source/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](/source/locally_small_category) (i.e. a [category](/source/category_(mathematics)) for which hom-classes are actually [sets](/source/Set_(mathematics)) and not [proper class](/source/proper_class)es). For all objects ''A'' and ''B'' in ''C'' we define two functors to the [category of sets](/source/category_of_sets) as follows: :{| class=wikitable |- ! Hom(''A'', –) : ''C'' → '''Set''' ! Hom(–, ''B'') : ''C'' → '''Set'''Also commonly denoted ''C''op → '''Set''', where ''C''op denotes the [opposite category](/source/opposite_category), and this encodes the arrow-reversing behaviour of Hom(–, ''B''). |- | This is a [covariant functor](/source/covariant_functor) given by: * Hom(''A'', –) [maps](/source/map_(mathematics)) each object ''X'' in ''C'' to the set of morphisms, Hom(''A'', ''X'') * Hom(''A'', –) maps each morphism ''f'' : ''X'' → ''Y'' to the [function](/source/function_(mathematics)) *: Hom(''A'', ''f'') : Hom(''A'', ''X'') → Hom(''A'', ''Y'') given by *: g \mapsto f \circ g for each ''g'' in Hom(''A'', ''X''). | This is a [contravariant functor](/source/contravariant_functor) given by: * Hom(–, ''B'') maps each object ''X'' in ''C'' to the set of morphisms, Hom(''X'', ''B'') * Hom(–, ''B'') maps each morphism ''h'' : ''X'' → ''Y'' to the function *: Hom(''h'', ''B'') : Hom(''Y'', ''B'') → Hom(''X'', ''B'') given by *: g \mapsto g \circ h for each ''g'' in Hom(''Y'', ''B''). |} The functor Hom(–, ''B'') is also called the ''[functor of points](/source/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](/source/natural_transformation). For any pair of morphisms ''f'' : ''B'' → ''B''′ and ''h'' : ''A''′ → ''A'' the following diagram [commutes](/source/commutative_diagram): center|320px Both paths send ''g'' : ''A'' → ''B'' to ''f''{{Hair space}}∘{{Hair space}}''g''{{Hair space}}∘{{Hair space}}''h'' : ''A''′ → ''B''′. The commutativity of the above diagram implies that Hom(–, –) is a [bifunctor](/source/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](/source/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== {{Main|Yoneda lemma}} Referring to the above commutative diagram, one observes that every morphism : ''h'' : ''A''′ → ''A'' gives rise to a [natural transformation](/source/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](/source/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](/source/full_and_faithful_functors) embedding of the category ''C'' into the [functor category](/source/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{op} \times C \to C to emphasize its product-like nature, or as : \mathop\Rightarrow : C^\text{op} \times C \to C to emphasize its functorial nature, or sometimes merely in lower-case: : \operatorname{hom}(-, -) : C^\text{op} \times C \to C . For examples, see [Category of relations](/source/Category_of_relations). Categories that possess an internal Hom functor are referred to as [closed categories](/source/closed_category). One has that : \operatorname{Hom}(I, \operatorname{hom}(-, -)) \simeq \operatorname{Hom}(-, -), where ''I'' is the [unit object](/source/unit_object) of the closed category. For the case of a [closed monoidal category](/source/closed_monoidal_category), this extends to the notion of [currying](/source/currying), namely, that : \operatorname{Hom}(X, Y \Rightarrow Z) \simeq \operatorname{Hom}(X\otimes Y, Z) where \otimes is a [bifunctor](/source/bifunctor), the '''internal product functor''' defining a [monoidal category](/source/monoidal_category). The isomorphism is [natural](/source/natural_isomorphism) in both ''X'' and ''Z''. In other words, in a closed monoidal category, the internal Hom functor is an [adjoint functor](/source/adjoint_functor) to the internal product functor. The object Y \Rightarrow Z is called the '''internal Hom'''. When \otimes is the [Cartesian product](/source/Cartesian_closed_category) \times, the object Y \Rightarrow Z is called the [exponential object](/source/exponential_object), and is often written as Z^Y. Internal Homs, when chained together, form a language, called the [internal language](/source/internal_language) of the category. The most famous of these are [simply typed lambda calculus](/source/simply_typed_lambda_calculus), which is the internal language of [Cartesian closed categories](/source/Cartesian_closed_categories), and the [linear type system](/source/linear_type_system), which is the internal language of [closed symmetric monoidal categories](/source/closed_monoidal_category). ==Properties== Note that a functor of the form : Hom(–, ''A'') : ''C''op → '''Set''' is a [presheaf](/source/presheaf_(category_theory)); likewise, Hom(''A'', –) is a [copresheaf](/source/copresheaf). A functor ''F'' : ''C'' → '''Set''' that is [naturally isomorphic](/source/natural_isomorphism) to Hom(''A'', –) for some ''A'' in ''C'' is called a [representable functor](/source/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](/source/profunctor), and, specifically, it is the identity profunctor \operatorname{id}_C \colon C \nrightarrow C. The internal hom functor preserves [limits](/source/limit_(category_theory)); that is, \operatorname{hom}(X, -) \colon C \to C sends limits to limits, while \operatorname{hom}(-, X) \colon C^\text{op} \to C sends limits in C^\text{op}, that is [colimit](/source/colimit)s in C, into limits. In a certain sense, this can be taken as the definition of a limit or colimit. The [endofunctor](/source/endofunctor) Hom(''E'', –) : '''Set''' → '''Set''' can be given the structure of a [monad](/source/Monad_(category_theory)); this monad is called the [environment (or reader) monad](/source/Monad_(category_theory)). ==Other properties== If '''A''' is an [abelian category](/source/abelian_category) and ''A'' is an object of '''A''', then Hom'''A'''(''A'', –) is a covariant [left-exact](/source/exact_functor) functor from '''A''' to the category '''Ab''' of [abelian group](/source/abelian_group)s. It is exact [if and only if](/source/if_and_only_if) ''A'' is [projective](/source/projective_module).Jacobson (2009), p. 149, Prop. 3.9. Let ''R'' be a [ring](/source/ring_(mathematics)) and ''M'' a left ''R''-[module](/source/module_(mathematics)). The functor Hom'''R'''(''M'', –): '''Mod'''-''R'' → '''Ab'''{{clarify|reason=Does Mod-R refer to the category of left R-modules here? This needs to be clarified because a commonly used notation is for "Mod-R" to denote the category of right R-modules and "R-Mod" to denote the category of left R-modules.|date=February 2022}} is [adjoint](/source/right_adjoint) to the [tensor product](/source/tensor_product_of_modules) functor – \otimes''R'' ''M'': '''Ab''' → '''Mod'''-''R''. ==See also== * [Ext functor](/source/Ext_functor) * [Functor category](/source/Functor_category) * [Representable functor](/source/Representable_functor) == Notes == ==References== {{refbegin}} * {{Cite book |first=Saunders |last=Mac Lane |author-link=Saunders Mac Lane|title=[Categories for the Working Mathematician](/source/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=https://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|access-date=2009-11-25|edition=Revised|year=2006|orig-year=1984|publisher=[Dover Publications](/source/Dover_Publications)|isbn=978-0-486-45026-1|archive-date=2020-03-21|archive-url=https://web.archive.org/web/20200321030307/http://historical.library.cornell.edu/cgi-bin/cul.math/docviewer?did=Gold010&id=3|url-status=dead}} * {{Cite book| last=Jacobson| first=Nathan| author-link=Nathan Jacobson| date=2009| title=Basic algebra| edition=2nd| volume = 2 | publisher=Dover| isbn = 978-0-486-47187-7}} **{{cite book |url=https://www.heldermann.de/SSPM/SSPM01/sspm01.htm|isbn=978-3-88538-001-6 |series=SSPM (Sigma Series in Pure Mathematics) 01|title=Category Theory |chapter= V. Functors and Natural Transformations|last1=Herrlich |first1=Horst |author-link1=Horst Herrlich |last2=Strecker |first2=George E. |date=1979 |publisher=Heldermann }} {{refend}} == External links == * {{nlab|id=hom-functor|title=Hom functor}} * {{nlab|id=internal-hom|title=Internal Hom}} {{DEFAULTSORT:Hom Functor}} Category:Functors Category:Binary operations --- Adapted from the Wikipedia article [Hom functor](https://en.wikipedia.org/wiki/Hom_functor) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Hom_functor?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.