nLab weak type theory



Type theory

natural deduction metalanguage, practical foundations

  1. type formation rule
  2. term introduction rule
  3. term elimination rule
  4. computation rule

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

syntax object language

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

logicset theory (internal logic of)category theorytype theory
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
introduction rule for implicationcounit for hom-tensor adjunctionlambda
elimination rule for implicationunit for hom-tensor adjunctionapplication
cut elimination for implicationone of the zigzag identities for hom-tensor adjunctionbeta reduction
identity elimination for implicationthe other zigzag identity for hom-tensor adjunctioneta conversion
truesingletonterminal object/(-2)-truncated objecth-level 0-type/unit type
falseempty setinitial objectempty type
proposition, truth valuesubsingletonsubterminal object/(-1)-truncated objecth-proposition, mere proposition
logical conjunctioncartesian productproductproduct type
disjunctiondisjoint union (support of)coproduct ((-1)-truncation of)sum type (bracket type of)
implicationfunction set (into subsingleton)internal hom (into subterminal object)function type (into h-proposition)
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian product (of family of subsingletons)dependent product (of family of subterminal objects)dependent product type (of family of h-propositions)
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijection setobject of isomorphismsequivalence type
support setsupport object/(-1)-truncationpropositional truncation/bracket type
n-image of morphism into terminal object/n-truncationn-truncation modality
equalitydiagonal function/diagonal subset/diagonal relationpath space objectidentity type/path type
completely presented setsetdiscrete object/0-truncated objecth-level 2-type/set/h-set
setset with equivalence relationinternal 0-groupoidBishop set/setoid with its pseudo-equivalence relation an actual equivalence relation
equivalence class/quotient setquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive type
presettype without identity types
set of truth valuessubobject classifiertype of propositions
domain of discourseuniverseobject classifiertype universe
modalityclosure operator, (idempotent) monadmodal type theory, monad (in computer science)
linear logic(symmetric, closed) monoidal categorylinear type theory/quantum computation
proof netstring diagramquantum circuit
(absence of) contraction rule(absence of) diagonalno-cloning theorem
synthetic mathematicsdomain specific embedded programming language

homotopy levels


Deduction and Induction

Constructivism, Realizability, Computability



The basis of it all

 Set theory

set theory

Foundational axioms

foundational axioms

Removing axioms



Weak type theory or propositional type theory is a dependent type theory without judgmental conversion; all the computation rules/ β \beta -conversion rules and uniqueness rules/ η \eta -conversion rules for types use identity types instead of judgmental equality. As a result, the results in weak type theory are more general than in models which use judgmental equality for computational and uniqueness rules, since judgmental equality implies typal equality, while the converse isn’t necessarily the case.

Formally, weak type theories come into two general versions:

The latter includes weak versions of Martin-Löf type theory, cubical type theory, and observational type theory, as well as extensions thereof such as type theory with shapes and simplicial type theory. Hypothetically, the latter would also be seen in proof assistants, where the base programming language used to implement the weak type theory, such as Coq or Agda, already has a judgmental equality.

A hybrid of weak and non-weak type theories can occur in two-level type theory and variants thereof like Homotopy Type System, where one level has weak types and the other one has strict types.

Open problems

There are plenty of questions which are currently unresolved in weak type theory. Van der Berg and Besten listed the following problems in the context of objective type theory, but equally this applies to any weak type theory:

Other problems include

See also open problems in homotopy type theory

See also


On quadratic type checking in weak type theory (objective type theory):

Talks at Strength of Weak Type Theory, hosted by DutchCATS:

Project to convert extensional type theory to weak type theory:

Last revised on May 16, 2024 at 23:11:06. See the history of this page for a list of all contributions to it.