inductive-inductive type



Deduction and Induction

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/path type
equivalence classquotientquotient type
inductioncolimitinductive type, W-type, M-type
higher inductionhigher colimithigher inductive type
-0-truncated higher colimitquotient inductive type
coinductionlimitcoinductive 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, (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





In type theory, induction-induction is a principle for mutually defining types of the form

A:Type,andB:AType, A \colon Type\,,\;\;\; and \;\;\; B \colon A \to Type \,,

where both AA and BB are defined inductively, such that the constructors for AA can refer to BB and vice versa. In addition, the constructor for BB can refer to the constructor for AA.

Such induction-induction occurs for instance when formalising dependent type theory in type theory.


Inductive-inductive types are related to inductive-recursive types. Importantly, inductive-inductive types have an initial algebra semantics with respect to dialgebras. In Forsberg’s thesis inductive-inductive types are reduced to indexed inductive types in the setting of extensional type theory. This reduction however only provides “simple” elimination rules (not to be confused with simply typed). Indexed inductive types in turn can be reduced to W-types in extensional type theory. See inductive families.

The consistency of the framework used for the elimination (e.g. in the theorem prover Agda) is not so clear, as it allows the definition of a universe containing a code for itself. There is an axiomatisation of the new principle in such a way that the resulting type theory is consistent, as proved by constructing a set-theoretic model; see Forsberg-Setzer 10.

Hugunin provides a reduction of an inductive-inductive type to an inductive type. This construction is conjectured to generalize to all inductive-inductive types. The construction is done in cubical type theory and hence is consistent with homotopy type theory.

Higher inductive inductive types

Experiments with higher inductive inductive types (elaborate versions of higher inductive types) are sections 11.3 “Cauchy reals” and section 11.6 “Conway surreals” of the HoTT book. As they are at the set level, these are instances of quotient inductive-inductive types; see QIIT. An experimental syntax for HIITs by Kaposi and Kovacs.


Parts of the above text are taken from

Last revised on May 30, 2021 at 15:05:54. See the history of this page for a list of all contributions to it.