nLab codependent type theory

Contents

Context

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
propositionsetobjecttype
predicatefamily of setsdisplay morphismdependent type
proofelementgeneralized elementterm/program
cut rulecomposition of classifying morphisms / pullback of display mapssubstitution
cut elimination for implicationcounit for hom-tensor adjunctionbeta reduction
introduction rule for implicationunit 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 setinternal homfunction type
negationfunction set into empty setinternal hom into initial objectfunction type into empty type
universal quantificationindexed cartesian productdependent productdependent product type
existential quantificationindexed disjoint union (support of)dependent sum ((-1)-truncation of)dependent sum type (bracket type of)
logical equivalencebijectionisomorphism/adjoint equivalenceequivalence of types
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

semantics

Deduction and Induction

Foundations

foundations

The basis of it all

 Set theory

set theory

Foundational axioms

foundational axiom

Removing axioms

Contents

Idea

When attempting to define coinductive types, there is an obstacle to the complete dualization of the usual rules for inductive types in dependent type theory, including dualizing the induction principle to a “coinduction principle”. For this some form of “codependent types” would be needed. In the context of homotopy type theory, Ahrens, Capriotti, and Spadotti write:

The universal property defining (internal) coinductive types in HoTT is dual to the one defining (internal) inductive types. One might hence assume that their existence is equivalent to a set of type-theoretic rules dual (in a suitable sense) to those given for external W-types… However, the rules for external W-types cannot be dualized in a naïve way, due to some asymmetry of HoTT related to dependent types as maps into a “type of types” (a universe) (ACS15)

Codependent type theory is a hypothetical type theory where one should be able to define codependent types, and coinductive types with a coinduction principle.

References

  • Benedikt Ahrens, Paolo Capriotti, Régis Spadotti, Non-wellfounded trees in Homotopy Type Theory, (arXiv:1504.02949)

Created on June 3, 2022 at 15:39:22. See the history of this page for a list of all contributions to it.