The Yoneda Lemma
( F ◦ G )( x ) = G ( F ( x )), and likewise identity functors on a category by id C ( x ) = x for any object or morphism x . This allows us define the category Cat whose objects are categories and morphisms are functors, but this will not be necessary for our purposes.
1.4.3 Natural transformations and the functor category We have now seen maps between sets and maps between categories. Natural transformations allow us to define maps between functors. Definition 5 Given two functors F,G : C → D, 3 to specify a natural transformation Φ : F → G, one specifies, for each X ∈ Ob( C ) , a morphism Φ X : F ( X ) → G ( X ) , called the X component of Φ , such that, for any morphism f : X → Y in C, the equation F ( f ) ◦ Φ Y = Φ X ◦ G ( f ) holds. This equation is known as the naturality condition . It should be clear that the components, as maps from the output of one functor to the corresponding output of the other, essentially allow one to transform F into G . The naturality condition then states that the transformation respects the structure of the category D ; i.e. morphisms can be composed before or after the transformation. Once again, we can formalize the intuition that natural transformations are maps between functors, with the notion of a functor category. Definition 6 The category of functors from C to D, denoted D C , has as objects all functors from C to D, and has natural transformations as morphisms. The composition of two natural transformations Φ and Ψ is given by (Φ ◦ Ψ) X = Φ X ◦ Ψ X , and identity transformations are given by (id F ) X = id X .
Associativity of composition here is guaranteed by the associativity of morphism composition in D , so this does indeed form a category. In a functor category, we will denote the hom-set by Nat( F,G ) rather than Hom( F,G ), to emphasize that the morphisms are natural transformations.
Definition 7 A natural isomorphism is an isomorphism in a functor category.
Finally, we observe that a natural transformation Φ is a natural isomorphism iff each component Φ X is an isomorphism. This is clear since identity and composition are both defined componentwise, and isomorphism is defined in terms only of composition and identity.
1.4.4 The product category The product category is a way of combining two categories into one without loss of information by considering ordered pairs, much like the Cartesian product of sets. They are in fact both special cases of the categorical notion of a product, but we will not discuss that here. For those unfamiliar with the Cartesian product, the product A × B of sets A and B simply denotes the set of all ordered pairs ( a,b ) where a ∈ A and b ∈ B .
Definition 8 For any categories C and D, the product category C × D is given by the Cartesian product on objects Ob( C × D ) = Ob( C )×Ob( D ) and on homsets Hom(( X,Y ) , ( X ′ ,Y ′ )) = Hom( X,X ′ )×Hom( Y,Y ′ ) . Identities are given by id ( X,Y ) = (id X , id Y ) and composition is given by ( f,g ) ◦ ( f ′ ,g ′ ) = ( f ◦ f ′ ,g ◦ g ′ ) .
3 This is a standard abbreviation of ‘ F : C → D and G : C → D ’.
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