nLab 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

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





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 (cf. the HoTT book real numbers). 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.

Another example of a higher inductive-inductive type is a univalent Tarski universe, where the where a type UU is defined inductively together with a type family a:UT(a)typea:U \vdash T(a) \; \mathrm{type} and has constructors

  • an element h(a,b,f):fiber(transport T(a,b),f)h(a, b, f):\mathrm{fiber}(\mathrm{transport}^T(a, b), f) for all a:Ua:U, b:Ub:U, f:T(a)T(b)f:T(a) \simeq T(b)
  • an identity ua(a,b,f,c):c= fiber(transport T(a,b),fh(a,b,f)\mathrm{ua}(a, b, f, c):c =_{\mathrm{fiber}(\mathrm{transport}^T(a, b), f} h(a, b, f) for all a:Ua:U, b:Ub:U, f:T(a)T(b)f:T(a) \simeq T(b) and c:fiber(transport T(a,b),f)c:\mathrm{fiber}(\mathrm{transport}^T(a, b), f)

where transport T(a,b):(a= Ub)(T(a)T(b))\mathrm{transport}^T(a, b):(a =_U b) \to (T(a) \simeq T(b)) is the transport function for all a:Ua:U and b:Ub:U and

fiber(transport T(a,b),f) p:a= Ubtransport T(a,b)(p)= T(a)T(b)f\mathrm{fiber}(\mathrm{transport}^T(a, b), f) \coloneqq \sum_{p:a =_U b} \mathrm{transport}^T(a, b)(p) =_{T(a) \simeq T(b)} f

This means that h(a,b,f)h(a, b, f) is the center of contraction of the fiber of transport across TT at the equivalence f:T(a)T(b)f:T(a) \simeq T(b).


Plain inductive-inductive types

Higher inductive-inductive types:

For the example of constructing the real numbers (cf. HoTT book real number)

Approaches to a general definition:

Discussion of examples in cubical type theory (cubical Agda):

  • Jasper Hugunin, Constructing Inductive-Inductive Types in Cubical Type Theory, in Foundations of Software Science and Computation Structures. FoSSaCS 2019, Lecture Notes in Computer Science 11425 (2019) [doi:10.1007/978-3-030-17127-8_17]

Last revised on February 10, 2023 at 08:09:36. See the history of this page for a list of all contributions to it.