nLab impredicative dependent type theory

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

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

Edit this sidebar

Disambiguation

There are in general two ways to define a dependent type theory:

One approach assumes a single separate type judgment and defines types via the single type judgment.
A second approach assumes an infinite number of universe levels $i$ and then defines an infinite number of type universes and, in the case for Coquand universes, type judgments indexed by $i$ , whereby types are then terms of the universes for Russell universes or identified with terms of the universes for Coquand universes via judgmental isomorphisms.

These two ways to define a dependent type theory leads to many different ways to make a dependent type theory impredicative. We assume throughout that there are some notion of dependent sum types, dependent product types, and identity types/path types.

For dependent type theories in the first sense, an impredicative dependent type theory is a dependent type theory with a type of all propositions defined via natural deduction inference rules. These dependent type theories are higher-order logics and are the internal logics of (the $(\infty, 1)$ -categorical analogues of) elementary toposes. In general, (the $(\infty, 1)$ -categorical analogues of) elementary toposes do not have any object classifiers.
For dependent type theories in the second sense, there are multiple notions of impredicative dependent type theory:
- For impredicative dependent type theory in the sense of a dependent type theory for which the base universe $U_0$ satisfies impredicative polymorphism; these impredicative type theories are incommensurable with the notion of impredicative dependent type theory using a single type judgment, since one can prove that the type of all $U_0$ -small propositions is not $U_0$ -small.
- For impredicative dependent type theory in the sense of a dependent type theory which has the propositional resizing axiom; these impredicative dependent type theories are stronger than the notion of impredicative dependent type theory using a single type judgment since these are the internal logics of $(\infty, 1)$ -categories with an infinite number of inaccessible object classifiers.

Related concepts

strongly predicative dependent type theory

category: disambiguation

Last revised on December 20, 2024 at 17:20:54. See the history of this page for a list of all contributions to it.