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 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.