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.
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 NERFINISHED ⓘ |
| appliesTo | univalent universes ⓘ |
| associatedWith | Institute for Advanced Study NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ 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 NERFINISHED ⓘ |
| 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 NERFINISHED ⓘ |
| 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 ⓘ |
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.