Yoneda lemma
E621111
The Yoneda lemma is a fundamental result in category theory that characterizes objects by their sets of morphisms into them, providing a powerful bridge between abstract categories and concrete set-valued functors.
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
result in category theory
ⓘ
theorem ⓘ |
| appearsIn |
Saunders Mac Lane's "Categories for the Working Mathematician"
NERFINISHED
ⓘ
standard textbooks on category theory ⓘ |
| appliesTo | locally small categories ⓘ |
| bearsOn | foundations of modern category-theoretic mathematics ⓘ |
| category | mathematics ⓘ |
| characterizes | objects of a category by their hom-sets into them ⓘ |
| field | category theory NERFINISHED ⓘ |
| formalizes | the idea that an object is determined by its relationships to all other objects ⓘ |
| generalizedBy |
co-Yoneda lemma
NERFINISHED
ⓘ
enriched Yoneda lemma NERFINISHED ⓘ |
| hasConsequence |
natural transformations between representable functors correspond to morphisms between representing objects
ⓘ
objects are determined by their Hom-functors up to isomorphism ⓘ |
| hasFormulation |
contravariant version
ⓘ
covariant version ⓘ |
| hasKeyConcept |
Hom-set
ⓘ
Yoneda embedding NERFINISHED ⓘ functor category ⓘ natural transformation ⓘ presheaf ⓘ representable functor ⓘ |
| implies | Yoneda embedding is fully faithful NERFINISHED ⓘ |
| involves |
contravariant Hom-functors
ⓘ
covariant Hom-functors ⓘ |
| namedAfter | Nobuo Yoneda NERFINISHED ⓘ |
| provides | a fully faithful embedding of a category into a functor category ⓘ |
| relates |
objects in a category
ⓘ
set-valued functors on that category ⓘ |
| states |
natural transformations from Hom(-,C) to a functor F correspond bijectively to elements of F(C)
ⓘ
natural transformations from Hom(C,-) to a functor F correspond bijectively to elements of F(C) ⓘ |
| toolFor |
algebraic geometry via functor of points
ⓘ
categorical algebra ⓘ defining limits and colimits via representable functors ⓘ homological algebra ⓘ studying adjoint functors ⓘ topos theory ⓘ |
| typicalCodomain | Set-valued functors on C ⓘ |
| typicalDomain | small category C ⓘ |
| usedFor |
characterizing objects up to isomorphism by their Hom-functors
ⓘ
defining Yoneda embedding ⓘ defining and studying universal properties ⓘ enriched category theory generalizations ⓘ foundations of categorical semantics in logic and computer science ⓘ representable functors ⓘ studying presheaf categories ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.