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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 Morphism s) of the categories involved. Hence, a natural transformation can be considered to be a "morphism of functors". Indeed this intuition can be formalized to define so called Functor Categories . Natural transformations are, after categories and functors, one of the most basic notions of categorical algebra and consequently appear in the majority of its applications. DEFINITION If ''F'' and ''G'' are If both ''F'' and ''G'' are Contravariant , the horizontal arrows in this diagram are reversed. If η is a natural transformation from ''F'' to ''G'', we also write η : ''F'' → ''G''. This is also expressed by saying the family of morphisms η''X'' : ''F''(''X'') → ''G''(''X'') is natural in ''X''. If, for every object ''X'' in ''C'', the morphism η''X'' is an Isomorphism in ''D'', then η is said to be a natural isomorphism (or sometimes '''natural equivalence''' or '''isomorphism of functors'''). Two functors ''F'' and ''G'' are called ''naturally isomorphic'' or simply ''isomorphic'' if there exists a natural isomorphism from ''F'' to ''G''. An infranatural transformation η from ''F'' to ''G'' is simply a family of morphisms η''X'': ''F''(''X'') → ''G''(''X''). Thus a natural transformation is an infranatural transformation for which η''Y'' o ''F''(''f'') = ''G''(''f'') o η''X'' for every morphism ''f'' : ''X'' → ''Y''. The '''naturalizer''' of η, nat(η), is the largest Subcategory of ''C'' containing all the objects of ''C'' on which η restricts to a natural transformation. EXAMPLES A worked example Statements like :"Every group is naturally isomorphic to its opposite group" abound in modern mathematics. We will now give the precise meaning of this statement as well as its proof. Consider the category Grp of all group becomes a (covariant!) functor from Grp to Grp if we define ''f''op = ''f'' for any group homomorphism ''f'': ''G'' → ''H''. Note that ''f''op is indeed a group homomorphism from ''G''op to ''H''op:
The content of the above statement is: :"The identity functor IdGrp : Grp → Grp is naturally isomorphic to the opposite functor -op : Grp → Grp." To prove this, we need to provide isomorphisms η''G'' : ''G'' → ''G''op for every group ''G'', such that the above diagram commutes. Set η''G''(''a'') = ''a''-1. The formulas (''ab'')-1 = ''b''-1 ''a''-1 and (''a''-1)-1 = ''a'' show that η''G'' is a group homomorphism which is its own inverse. To prove the naturality, we start with a group homomorphism ''f'' : ''G'' → ''H'' and show η''H'' o ''f'' = ''f''op o η''G'', i.e. (''f''(''a''))-1 = ''f''op(''a''-1) for all ''a'' in ''G''. This is true since ''f''op = ''f'' and every group homomorphism has the property (''f''(''a''))-1 = ''f''(''a''-1). Further examples
Every finite dimensional vector space is also isomorphic to its dual space. But this isomorphism relies on an arbitrary choice of basis, and is not natural, though there is an infranatural transformation. More generally, any vector spaces with the same dimensionality are isomorphic, but not naturally so. (Note however that if the space has a Nondegenerate Bilinear Form , then there ''is'' a natural isomorphism between the space and its dual. Here the space is viewed as an object in the category of vector spaces and Transpose s of maps.) Consider the category Ab of Abelian Group s and group homomorphisms. For all abelian groups ''X'', ''Y'' and ''Z'' we have a group isomorphism :Hom(''X''''Y'', ''Z'') → Hom(''X'', Hom(''Y'', ''Z'')). These isomorphisms are "natural" in the sense that they define a natural transformation between the two involved functors Ab × Abop × Abop → Ab. Natural transformations arise frequently in conjunction with Adjoint Functors . Indeed, adjoint functors are defined by a certain natural isomorphism. Additionally, every pair of adjoint functors comes equipped with two natural transformations (generally not isomorphisms) called the ''unit'' and ''counit''. OPERATIONS WITH NATURAL TRANSFORMATIONS If η : ''F'' → ''G'' and ε : ''G'' → ''H'' are natural transformations between functors ''F'',''G'',''H'':''C'' → ''D'', then we can compose them to get a natural transformation εη : ''F'' → ''H''. This is done componentwise: (εη)''X'' = ε''X''η''X''. This "vertical composition" of natural transformation is Associative and has an identity, and allows one to consider the collection of all functors ''C'' → ''D'' itself as a category (see below under Functor Categories ). Natural transformations also have a "horizontal composition". If η:''F'' → ''G'' is a natural transformation between functors ''F'',''G'':''C'' → ''D'' and ε: ''J'' → ''K'' is a natural transformation between functors ''J'',''K'':''D'' → ''E'', then the composition of functors allows a composition of natural transformations η ˆ ε: ''JF'' → ''KG''. This operation is also associative with identity, and the identity coincides with that for vertical composition. The two operations are related by an identity which exchanges vertical composition with horizontal composition. A natural transformation η : ''F'' → ''G'' is a natural isomorphism if and only if there exists a natural transformation ε : ''G'' → ''F'' such that ηε = 1''G'' and εη = 1''F'' (where 1''F'' : ''F'' → ''F'' is the natural transformation assigning to every object ''X'' the identity morphism on ''F''(''X'')). If η : ''F'' → ''G'' is a natural transformation between functors ''F'',''G'':''C'' → ''D'', and ''H'' : ''D'' → ''E'' is another functor, then we can form the natural transformation ''H''η : ''HF'' → ''HG'' by defining (''H''η)''X'' = ''H''(η''X''). If on the other hand ''K'' : ''B'' → ''C'' is a functor, the natural transformation η''K'' : ''FK'' → ''GK'' is defined by (η''K'')''X'' = η''K''(''X''). FUNCTOR CATEGORIES See Also: Functor category If ''C'' is any category and ''I'' is a small category, we can form the Functor Category ''CI'' having as objects all functors from ''I'' to ''C'' and as morphisms the natural transformations between those functors. This is especially useful if ''I'' arises from a Directed Graph . For instance, if ''I'' is the category of the directed graph • → •, then ''CI'' has as objects the morphisms of ''C'', and a morphism between φ : ''U'' → ''V'' and ψ : ''X'' → ''Y'' in ''CI'' is a pair of morphisms ''f'' : ''U'' → ''X'' and ''g'' : ''V'' → ''Y'' in ''C'' such that the "square commutes", i.e. ψ ''f'' = ''g'' φ. More generally, one can build the 2-category Cat whose
The horizontal and vertical compositions are the compositions between natural transformations described previously. A functor category is then simply a hom-set in this category (smallness issues aside). YONEDA LEMMA See Also: Yoneda lemma If ''X'' is an object of the category ''C'', then the assignment ''Y'' Hom''C''(''X'', ''Y'') defines a covariant functor ''F''''X'' : ''C'' → Set. This functor is called ''. HISTORICAL NOTES Saunders Mac Lane , one of the founders of category theory, is said to have remarked, "I didn't invent categories to study functors; I invented them to study natural transformations." Just as the study of Groups is not complete without a study of Homomorphisms , so the study of categories is not complete without the study of Functor s. The reason for Mac Lane's comment is that the study of functors is itself not complete without the study of natural transformations. The context of Mac Lane's remark was the axiomatic theory of the groups defined directly, and those of the singular theory, would be isomorphic. What cannot easily be expressed without the language of natural transformations is how homology groups are compatible with morphisms between objects, and how two equivalent homology theories not only have the same homology groups, but also the same morphisms between those groups. |
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