coinductive type


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

logiccategory theorytype theory
trueterminal object/(-2)-truncated objecth-level 0-type/unit type

falseinitial objectempty type

proposition(-1)-truncated objecth-proposition, mere proposition

proofgeneralized elementprogram

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

logical conjunctionproductproduct type

disjunctioncoproduct ((-1)-truncation of)sum type (bracket type of)

implicationinternal homfunction type

negationinternal hom into initial objectfunction type into empty type

universal quantificationdependent productdependent product type

existential quantificationdependent sum ((-1)-truncation of)dependent sum type (bracket type of)

equivalencepath space objectidentity type

equivalence classquotientquotient type

inductioncolimitinductive type, W-type, M-type

higher inductionhigher colimithigher inductive type

completely presented setdiscrete object/0-truncated objecth-level 2-type/preset/h-set

setinternal 0-groupoidBishop set/setoid

universeobject classifiertype of types

modalityclosure operator, (idemponent) 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




The notion of coinductive types is dual to that of inductive types.


Categorical semantics

Where the categorical semantics of an inductive type is an initial algebra for an endofunctor, the semantics of a coinductive type is a terminal coalgebra of an endofunctor.

Coinductive type formation in homotopy type theory

There is an obstacle to the complete dualization of the usual rules for inductive types in homotopy type theory, including dualizing the induction principle to a “coinduction principle”. For this some form of “codependent types” would be needed.

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)

However, it is possible instead to dualize the alternative characterization of inductive types as initial algebras for a notion of coinductive types as terminal coalgebras, and that can be formulated (and often constructed) in ordinary HoTT. In (ACS15) the authors proceed to construct coinductive types from indexed inductive types.


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

  • coinductives, discussion on Homotopy Type Theory group.

Last revised on July 30, 2018 at 06:43:36. See the history of this page for a list of all contributions to it.