For an object X in a category, the representable functor y(X) maps any object to the set of morphisms into X, denoted as Hom(-, X). It captures all the relationships (morphisms) from other objects to X.
For an object X in a category, the representable functor y(X) maps any object to the set of morphisms into X, denoted as Hom(-, X). It captures all the relationships (morphisms) from other objects to X.
The enriched Yoneda lemma generalizes the standard Yoneda lemma to enriched categories, where morphisms are elements of a monoidal category rather than just sets. It states that the object of enriched natural transformations from Hom_X to F is isomorphic, as an object of the enriching category, to F(X).
Conditions: The setting is enriched category theory.; Hom_X denotes the representable functor at X.; F is a functor in the same enriched context.
The enriched Yoneda lemma generalizes the standard Yoneda lemma to enriched categories, where morphisms are elements of a monoidal category rather than just sets. It states that the object of enriched natural transformations from Hom_X to F is isomorphic, as an object of the enriching category, to F(X).
Conditions: The setting is enriched category theory.; Hom_X denotes the representable functor at X.; F is a functor in the same enriched context.
The Yoneda lemma states that for any functor F from C^{op} to Set, the value F(X) is isomorphic to the set of natural transformations from the representable functor Hom(-, X) to F. Informally, it means an object is entirely determined by its relationships to other objects.
Conditions: C is a category.; X is an object in C.; F : C^{op} → Set is a functor.
The Yoneda lemma states that for any functor F from C^{op} to Set, the value F(X) is isomorphic to the set of natural transformations from the representable functor Hom(-, X) to F. Informally, it means an object is entirely determined by its relationships to other objects.
Conditions: C is a category.; X is an object in C.; F : C^{op} → Set is a functor.