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.
All labels observed (1)
| Label | Occurrences |
|---|---|
| Yoneda lemma canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T6834031 — resolving that mention is where its identity was fixed. The disambiguator weighed these candidate entities and picked the highlighted one (or “None”, minting a new entity). This is how homonymy is resolved: the same surface form can point to different entities.
NED1
Entity disambiguation (via context triple)
gpt-5-mini-2025-08-07
Target entity: Yoneda lemma Context triple: [Categories for the Working Mathematician, hasConcept, Yoneda lemma]
-
A.
category theory
Category theory is a branch of mathematics that studies abstract structures and relationships between them using the language of objects and morphisms, providing a unifying framework across many areas of math and theoretical computer science.
-
B.
Curry–Howard correspondence
The Curry–Howard correspondence is a foundational principle in logic and computer science that establishes a deep analogy between proofs and programs, and between logical propositions and types in programming languages.
-
C.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
D.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
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E.
Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
- F. None of above. chosen
- G. Unsure - the case is ambiguous/there is not enough information to decide.
NED2
Entity disambiguation (via description)
gpt-5-mini-2025-08-07
Target entity: Yoneda lemma Target entity description: 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.
-
A.
category theory
Category theory is a branch of mathematics that studies abstract structures and relationships between them using the language of objects and morphisms, providing a unifying framework across many areas of math and theoretical computer science.
-
B.
Curry–Howard correspondence
The Curry–Howard correspondence is a foundational principle in logic and computer science that establishes a deep analogy between proofs and programs, and between logical propositions and types in programming languages.
-
C.
Ky Fan’s lemma
Ky Fan’s lemma is a combinatorial topological result that generalizes Tucker’s lemma and provides conditions guaranteeing the existence of certain balanced or fully labeled simplices in labeled triangulations of spheres or simplices.
-
D.
Grothendieck universe
A Grothendieck universe is a set-theoretic construct large enough to contain all the usual objects and operations of mathematics, used to rigorously handle "large" categories while avoiding paradoxes.
-
E.
Poincaré lemma
The Poincaré lemma is a fundamental result in differential geometry and topology stating that every closed differential form on a star-shaped (or more generally, contractible) domain is locally exact.
- F. None of above. chosen
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 ⓘ |
How these facts were elicited
The pipeline generated the facts above by prompting gpt-5.1 with this entity's name + description and the instruction below.
Instruction
You are a knowledge base construction expert. Given a subject entity and a description of it, return factual statements that you know for the subject as a JSON list of dictionaries(triples), where keys must be "subject", "predicate" and "object". The number of facts may be very high, between 25 to 50 or more, for very popular subjects. For less popular subjects, the number of facts can be very low, like 5 or 10. # Requirements - If you don't know the subject at all, return an empty list. - If the subject is not a named entity, return an empty list. - Include at least one triple where predicate is "instanceOf". - Do not get too wordy. - Separate several objects into multiple triples with one object.
Input
Subject: Yoneda lemma Description of subject: 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.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.