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.
There is a bijection between the set of morphisms from X to Y, Hom(X,Y), and the set of natural transformations from the representable functor Hom(-,X) to Hom(-,Y). This means that maps between objects correspond exactly to natural transformations between their representable functors.
There is a bijection between the set of morphisms from X to Y, Hom(X,Y), and the set of natural transformations from the representable functor Hom(-,X) to Hom(-,Y). This means that maps between objects correspond exactly to natural transformations between their representable functors.
The path space of a topological space X is the set of maps from the interval to X. Crucially, it is not just a set; it is itself a topological space, carrying its own topology.
Conditions: The ambient context is algebraic topology.; X is a topological space.
The path space of a topological space X is the set of maps from the interval to X. Crucially, it is not just a set; it is itself a topological space, carrying its own topology.
Conditions: The ambient context is algebraic topology.; X is a topological space.
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.