homotopy type theory
E1041770
branch of mathematical logic
foundational framework for mathematics
mathematics book
research area in type theory
Homotopy type theory is a branch of mathematical logic and foundations that interprets types as spaces and equalities as paths, connecting type theory with homotopy theory and higher category theory.
All labels observed (1)
| Label | Occurrences |
|---|---|
| homotopy type theory canonical | 1 |
How this entity was disambiguated
This entity first appeared as the object of triple T13444198 — 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: homotopy type theory Context triple: [Per Martin-Löf, influenced, homotopy type theory]
-
A.
univalent foundations program
The univalent foundations program is a research initiative that redefines the foundations of mathematics using homotopy type theory, emphasizing computationally verifiable proofs and new connections between logic, topology, and category theory.
-
B.
univalence axiom
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.
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C.
calculus of constructions
The calculus of constructions is a powerful type theory and foundational formal system that unifies higher-order logic and typed lambda calculus, serving as the basis for several modern proof assistants.
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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.
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.
- 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: homotopy type theory Target entity description: Homotopy type theory is a branch of mathematical logic and foundations that interprets types as spaces and equalities as paths, connecting type theory with homotopy theory and higher category theory.
-
A.
univalent foundations program
The univalent foundations program is a research initiative that redefines the foundations of mathematics using homotopy type theory, emphasizing computationally verifiable proofs and new connections between logic, topology, and category theory.
-
B.
univalence axiom
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.
-
C.
calculus of constructions
The calculus of constructions is a powerful type theory and foundational formal system that unifies higher-order logic and typed lambda calculus, serving as the basis for several modern proof assistants.
-
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.
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.
- F. None of above. chosen
Statements (50)
| Predicate | Object |
|---|---|
| instanceOf |
branch of mathematical logic
ⓘ
foundational framework for mathematics ⓘ mathematics book ⓘ research area in type theory ⓘ |
| about | homotopy type theory NERFINISHED ⓘ |
| aimsToProvide | new foundations for mathematics ⓘ |
| associatedWith |
Institute for Advanced Study
NERFINISHED
ⓘ
Univalent Foundations program NERFINISHED ⓘ |
| basedOn | Martin-Löf dependent type theory NERFINISHED ⓘ |
| coreConcept |
higher inductive types
ⓘ
homotopy levels ⓘ identity types ⓘ n-types ⓘ path induction ⓘ truncation levels ⓘ univalence axiom NERFINISHED ⓘ |
| developedIn | 21st century ⓘ |
| fieldOfStudy |
higher category theory
ⓘ
homotopy theory ⓘ type theory ⓘ |
| hasAxiom | univalence axiom ⓘ |
| hasModelIn |
Kan complexes
ⓘ
simplicial sets ⓘ ∞-groupoids ⓘ |
| hasProperty |
internalizes homotopical reasoning in type theory
ⓘ
supports higher-dimensional algebraic structures ⓘ treats isomorphic structures as equal via univalence ⓘ |
| implementedIn |
Agda
NERFINISHED
ⓘ
Coq NERFINISHED ⓘ Cubical Agda NERFINISHED ⓘ Lean NERFINISHED ⓘ cubical type theory ⓘ |
| influencedBy |
constructive type theory
ⓘ
higher category theory ⓘ homotopy theory NERFINISHED ⓘ |
| influences |
computer-assisted theorem proving
ⓘ
formalized mathematics ⓘ univalent foundations ⓘ |
| interprets |
equalities as paths
ⓘ
higher equalities as homotopies between paths ⓘ terms as points in spaces ⓘ types as spaces ⓘ |
| notablePublication | Homotopy Type Theory: Univalent Foundations of Mathematics NERFINISHED ⓘ |
| relatesTo |
higher categories
ⓘ
model categories ⓘ simplicial sets ⓘ ∞-groupoids ⓘ |
| supports |
computer-checked proofs
ⓘ
constructive mathematics ⓘ |
| usedIn | proof assistants ⓘ |
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: homotopy type theory Description of subject: Homotopy type theory is a branch of mathematical logic and foundations that interprets types as spaces and equalities as paths, connecting type theory with homotopy theory and higher category theory.
Referenced by (1)
Full triples — surface form annotated when it differs from this entity's canonical label.