univalence axiom
E860089
The univalence axiom is a principle in homotopy type theory asserting that equivalent mathematical structures can be identified, providing a foundation for a new, homotopical approach to the foundations of mathematics.
All labels observed (1)
| Label | Occurrences |
|---|---|
| univalence axiom canonical | 4 |
How this entity was disambiguated
This entity first appeared as the object of triple T10388495 — 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: univalence axiom Context triple: [Vladimir Voevodsky, notableIdea, univalence axiom]
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A.
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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B.
Yoneda lemma
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.
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C.
Brouwer–Heyting–Kolmogorov interpretation
The Brouwer–Heyting–Kolmogorov interpretation is a foundational explanation of intuitionistic logic that interprets logical connectives and proofs in terms of explicit constructions and algorithms rather than classical truth values.
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D.
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.
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E.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
- 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: univalence axiom Target entity description: The univalence axiom is a principle in homotopy type theory asserting that equivalent mathematical structures can be identified, providing a foundation for a new, homotopical approach to the foundations of mathematics.
-
A.
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.
-
B.
Yoneda lemma
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.
-
C.
Brouwer–Heyting–Kolmogorov interpretation
The Brouwer–Heyting–Kolmogorov interpretation is a foundational explanation of intuitionistic logic that interprets logical connectives and proofs in terms of explicit constructions and algorithms rather than classical truth values.
-
D.
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.
-
E.
axiom of choice
The axiom of choice is a fundamental principle in set theory asserting that one can select an element from each set in any collection of nonempty sets, with far-reaching consequences across mathematics.
- F. None of above. chosen
Statements (46)
| Predicate | Object |
|---|---|
| instanceOf |
axiom
ⓘ
principle in homotopy type theory ⓘ |
| aimsToReplace | equality of sets by equivalence of types at the foundational level ⓘ |
| appearsIn |
Homotopy Type Theory: Univalent Foundations of Mathematics
ⓘ
linked to:
homotopy type theory
|
| appliesTo | univalent universes ⓘ |
| associatedWith | Institute for Advanced Study ⓘ |
| consequence |
equalities can be regarded as paths
ⓘ
identity types behave like paths in spaces ⓘ terms can be regarded as points in spaces ⓘ types can be regarded as spaces ⓘ |
| contrastWith |
extensional type theory without univalence
ⓘ
set-theoretic foundations ⓘ |
| coreIdea |
equivalent types can be identified
ⓘ
isomorphic structures are equal at the level of types ⓘ |
| enables |
invariance of constructions under equivalence
ⓘ
transport of structures along equivalences ⓘ |
| field |
foundations of mathematics
ⓘ
homotopy type theory ⓘ type theory ⓘ |
| formalStatement | for types A and B, the canonical map (A = B) → (A ≃ B) is an equivalence ⓘ |
| implies | function extensionality ⓘ |
| influenced |
design of homotopy type theory
ⓘ
development of cubical models of type theory ⓘ |
| interpretedIn |
Kan complexes
ⓘ
simplicial sets ⓘ |
| introducedBy | Vladimir Voevodsky ⓘ |
| logicalStrength |
compatible with constructive mathematics
ⓘ
compatible with intuitionistic logic ⓘ |
| modelType | homotopical models of type theory ⓘ |
| motivation |
align equality of types with equivalence of spaces
ⓘ
treat isomorphic mathematical structures as identical ⓘ |
| philosophicalView | structuralism in mathematics ⓘ |
| publishedBy |
Univalent Foundations Program
ⓘ
linked to:
univalent foundations program
|
| relatedTo |
higher inductive types
ⓘ
homotopy equivalence ⓘ ∞-groupoids ⓘ |
| relatesConcept |
equivalence of types
ⓘ
identity of types ⓘ |
| requires | universe types ⓘ |
| status |
independent of standard Martin-Löf type theory
ⓘ
not derivable from ordinary intensional type theory ⓘ |
| technicalForm | univalence for a universe U states that Id_U(A,B) ≃ Equiv(A,B) ⓘ |
| usedIn |
cubical type theory
ⓘ
proof assistants based on homotopy type theory ⓘ univalent foundations ⓘ |
| yearProposed | 2006 ⓘ |
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: univalence axiom Description of subject: The univalence axiom is a principle in homotopy type theory asserting that equivalent mathematical structures can be identified, providing a foundation for a new, homotopical approach to the foundations of mathematics.
Referenced by (4)
Full triples — surface form annotated when it differs from this entity's canonical label.