nLab univalent type theory

Contents

Context

Type theory

natural deduction metalanguage, practical foundations

judgement
hypothetical judgement, sequent
- antecedents $\vdash$ consequent, succedents

type theory (dependent, intensional, observational type theory, homotopy type theory)

calculus of constructions

syntax object language

theory, axiom
proposition/type (propositions as types)
definition/proof/program (proofs as programs)
theorem

computational trinitarianism =
propositions as types +programs as proofs +relation type theory/category theory

logic	set theory (internal logic of)	category theory	type theory
proposition	set	object	type
predicate	family of sets	display morphism	dependent type
proof	element	generalized element	term/program
cut rule		composition of classifying morphisms / pullback of display maps	substitution
introduction rule for implication		counit for hom-tensor adjunction	lambda
elimination rule for implication		unit for hom-tensor adjunction	application
cut elimination for implication		one of the zigzag identities for hom-tensor adjunction	beta reduction
identity elimination for implication		the other zigzag identity for hom-tensor adjunction	eta conversion
true	singleton	terminal object/(-2)-truncated object	h-level 0-type/unit type
false	empty set	initial object	empty type
proposition, truth value	subsingleton	subterminal object/(-1)-truncated object	h-proposition, mere proposition
logical conjunction	cartesian product	product	product type
disjunction	disjoint union (support of)	coproduct ((-1)-truncation of)	sum type (bracket type of)
implication	function set (into subsingleton)	internal hom (into subterminal object)	function type (into h-proposition)
negation	function set into empty set	internal hom into initial object	function type into empty type
universal quantification	indexed cartesian product (of family of subsingletons)	dependent product (of family of subterminal objects)	dependent product type (of family of h-propositions)
existential quantification	indexed disjoint union (support of)	dependent sum ((-1)-truncation of)	dependent sum type (bracket type of)
logical equivalence	bijection set	object of isomorphisms	equivalence type
	support set	support object/(-1)-truncation	propositional truncation/bracket type
		n-image of morphism into terminal object/n-truncation	n-truncation modality
equality	diagonal function/diagonal subset/diagonal relation	path space object	identity type/path type
completely presented set	set	discrete object/0-truncated object	h-level 2-type/set/h-set
set	set with equivalence relation	internal 0-groupoid	Bishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
	equivalence class/quotient set	quotient	quotient type
induction		colimit	inductive type, W-type, M-type
higher induction		higher colimit	higher inductive type
-		0-truncated higher colimit	quotient inductive type
coinduction		limit	coinductive type
	preset		type without identity types
	set of truth values	subobject classifier	type of propositions
domain of discourse	universe	object classifier	type universe
modality		closure operator, (idempotent) monad	modal type theory, monad (in computer science)
linear logic		(symmetric, closed) monoidal category	linear type theory/quantum computation
proof net		string diagram	quantum circuit
(absence of) contraction rule		(absence of) diagonal	no-cloning theorem
		synthetic mathematics	domain specific embedded programming language

homotopy levels

semantics

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Constructivism, Realizability, Computability

Definition

Traditionally, there are two different ways to present dependent type theory:

One can present dependent type theory with a hierarchy of type universes such that every type is a term of a universe of the hierarchy. Examples of such dependent type theories include the one found in the HoTT book and the proof assistants Agda, Coq, Lean, and its variants like Cubical Agda.
Alternatively, one can present dependent type theory with a separate type judgment and no type universes at all. Examples of such dependent type theories include the one found in Egbert Rijke‘s Introduction to Homotopy Type Theory.

From these two notions of dependent type theory, there are two notions of a univalent type theory:

In the first case, a univalent type theory is a dependent type theory with identity types, dependent product types, and dependent sum types such that all universes in the universe hierarchy satisfy the univalence axiom.
In the second case, a univalent type theory is a dependent type theory with type variables and with identity types, dependent product types, dependent sum types, and identity types between types such that the univalence axiom holds in the type theory.

These univalent type theories usually have additional types such as the empty type, the unit type, the booleans, and the natural numbers type, making it into a Martin-Löf dependent type theory.

References

Auke Booij, Analysis in Univalent Type Theory, (slides) (thesis)
Martín Hötzel Escardó and Cory Knapp, Partial Elements and Recursion via Dominances in Univalent Type Theory, (LIPIcs:2017:7682)

Last revised on December 17, 2024 at 00:22:55. See the history of this page for a list of all contributions to it.

nLab univalent type theory

Context

Type theory

Constructivism, Realizability, Computability

Constructive mathematics

Realizability

Computability

Foundations

The basis of it all

Set theory

Foundational axioms

Removing axioms

Contents

Definition

See also

References