Quotient Inductive-Inductive Types (QIITs)

• QIITs are generalized inductive types that simultaneously define multiple dependent sorts and impose equality conditions via quotienting.
• They provide an initial-algebra semantics linking recursion, induction, and universal properties to formalize constructive type-theoretic objects.
• Canonical applications include the algebraic presentation of type theories, partiality monads, and predicative DCPO algebras, showcasing their versatility.

Quotient inductive-inductive types (QIITs) are generalized inductive types that combine simultaneous definitions of mutually dependent sorts with equality constructors that impose identifications directly in the generated structure. In the literature, they are presented both as a generalization of inductive-inductive definitions by quotienting and as the set-truncated fragment of higher inductive-inductive types. Their role is twofold: they provide a specification formalism for algebraic and type-theoretic objects whose data, relations, and equations must be generated together, and they support an initial-algebra view that connects recursion, induction, and universal properties. Representative uses include algebraic presentations of type theories, constructive definitions of reals, ordinals, surreal numbers, partiality monads, and, more recently, predicative constructions of initial DCPO algebras (Altenkirch et al., 2016, Kovács et al., 2020, Collem et al., 12 Sep 2025).

1. Definitional core

A QIIT combines two ingredients. First, it extends inductive-inductive definitions, where multiple sorts are defined simultaneously and later sorts may be indexed by earlier ones. Second, it extends quotient or HIT-style definitions by allowing equality constructors that identify terms generated by the point constructors. In one formulation, QIITs are “generalized inductive types which allow sorts to be indexed over previously declared sorts, and allow usage of equality constructors”; in another, a QIIT “allows one to simultaneously define an inductive type and an inductive relation on that type, together with equations on the type” (Kovács et al., 2020, Collem et al., 12 Sep 2025).

This places QIITs strictly beyond ordinary inductive types and ordinary inductive-inductive types. Ordinary inductive types have a single sort and point constructors only. Inductive-inductive types allow several mutually defined sorts, but not quotienting by equations. QIITs add both indexed sorts and equality constructors. A standard example is the category presentation

$$\begin{aligned} &Ob : Set \ &Mor : Ob \to Ob \to Set \ &\circ : Mor\,J\,K \to Mor\,I\,J \to Mor\,I\,K \ &id : Mor\,I\,I \ &ass : (f\circ g)\circ h = f\circ(g\circ h) \ &idl : id\circ f = f \ &idr : f\circ id = f, \end{aligned}$$

where Mor is indexed over Ob, and the categorical laws are imposed by equality constructors rather than by a later quotient (Kovács et al., 2020).

A recurring QIIT pattern is the simultaneous generation of a carrier $A$, a relation such as $\sqsubseteq$ on $A$, point constructors for elements of $A$, constructors for proofs $x \sqsubseteq y$, and path constructors enforcing equations or truncation. Expository examples include finite sets, sorted lists, and strictly sorted lists. For finite sets, the presentation uses point constructors such as $[]$ and $x :: xs$, path constructors enforcing commutativity and duplication, and a truncation constructor $\isSet(FinSet)$ (Collem et al., 12 Sep 2025).

In the set-truncated setting, QIITs are also described as a restricted form of HIIT. The partiality-monad construction explicitly states that it uses a set-truncated higher inductive-inductive type and “call[s] such a set-truncated HIIT a QIIT” (Altenkirch et al., 2016). This characterization is important because many prominent examples are not homotopical objects such as spheres, but set-level quotient constructions whose equalities are nevertheless most naturally built into the inductive definition.

2. Initial-algebra semantics and elimination

A central semantic proposal for QIITs interprets them as initial objects in categories of algebras built from a specification of sorts and constructors. For a sort signature $H_0,\ldots,H_{n-1}$ with $A$0, one forms a base category $A$1 by iteratively extending $A$2 with families over $A$3. The resulting base categories are complete, and constructor specifications are then added on top of them (Altenkirch et al., 2016).

A constructor is specified by an argument functor

$A$4

and a target functor

$A$5

that is relatively continuous. The target may describe either ordinary point data or an equality type. Equality constructors are handled by relative continuity of equality functors; point constructors are handled by relatively continuous base target functors. The corresponding category of algebras has objects $A$6 with

$A$7

and morphisms preserving that structure. These algebra categories are complete (Altenkirch et al., 2016).

Within this framework, the expected elimination principle is captured categorically by section induction. An object $A$8 is section inductive if every morphism $A$9 admits a section $\sqsubseteq$0 with $\sqsubseteq$1. The principal theorem states that, in the category of QIIT algebras,

$\sqsubseteq$2

Initiality yields recursion and uniqueness; section induction yields dependent elimination and computation behavior. The equivalence identifies the universal property of a QIIT with the elimination principle practitioners expect from its constructors (Altenkirch et al., 2016).

A closely related semantics is developed using type-theoretic signatures. In that setting, a signature is interpreted uniformly into algebras, morphisms, displayed algebras, and sections—often summarized as AMDS semantics. For finitary quotient inductive-inductive signatures, the paper proves that in a cwf with $\sqsubseteq$3 and weak $\sqsubseteq$4, an object supports induction iff it is initial (Kovács, 2023). This recasts QIITs as semantic targets of a general specification language rather than as isolated ad hoc definitions.

3. Signature calculi and metatheory

The syntax of QIIT-like specifications has been studied through theories of signatures. One influential formulation uses a small type theory in which a signature is a context. The theory includes a universe U, decoding, inductive function spaces, equality types, path induction, equality between sorts, and external parameters. This makes it possible to specify simultaneous inductive definitions together with quotient-style equations and even equations between type constructors (Kaposi et al., 2019).

The signature language is designed so that point constructors, equality constructors, and higher equalities appear uniformly as declarations in a context. Logical-relation translations then compute the associated algebras, displayed algebras, homomorphisms, and sections. In that framework, recursion is obtained from homomorphisms, and induction is obtained from sections of displayed algebras. The work is explicit that it does not yet provide full categorical semantics or initial-algebra construction; those are left for future work (Kaposi et al., 2019).

A later metatheoretic development extends QIITs to large constructors, large elimination, recursive equations, and infinitary constructors. There, signatures are again presented through a theory of signatures, but the formalism is bootstrapped so that the theory of signatures can describe its own signature, modulo universe sizes. A signature at external universe level $\sqsubseteq$5 is defined by

$\sqsubseteq$6

The semantics models each signature as a finitely complete CwF of algebras; the paper notes that flCwFs are biequivalent to finitely complete categories (Kovács et al., 2020).

The infinitary setting introduces a technical shift. Compared to finitary QIITs, one must prove invariance under algebra isomorphisms. The reason given is that infinitary $\sqsubseteq$7-like types, identity types, and recursive equations are not preserved strictly by substitution; they are preserved only up to isomorphism. To manage this, the semantics uses weak CwF morphisms, type isomorphisms, context iso-cleaving, type iso-cleaving, and CwF isofibrations. The same work proves, by a term model construction, that every QIIT is constructible from the syntax of the theory of signatures (Kovács et al., 2020).

4. Encodable fragments and constructive realizations

Not every quotient-inductive or quotient inductive-inductive construction is handled in the same way. A significant restricted fragment is given by QW-types, described as the quotient analogue of W-types. A QW-type is an initial algebra for a strictly positive equational theory: one starts from a signature

$\sqsubseteq$8

forms the associated polynomial endofunctor

$\sqsubseteq$9

and imposes a specified system of equations between algebraic terms (Fiore et al., 2019).

The resulting object carries the expected recursion, homomorphism, and uniqueness principles. However, QW-types are explicitly presented as a proper fragment rather than a full account of QIITs. They cover “a wide range of QITs,” but not “the fully general conditional/path-dependent QIIT schemes.” In particular, they do not allow constructors to take previous equality proofs as arguments (Fiore et al., 2019).

A common misconception is that quotient-inductive structures can always be obtained by taking an ordinary inductive object and quotienting it afterward. The QW-type construction directly rejects that simplification. The paper states that its construction “does not simply quotient a W-type”; instead, it defines the carrier and the equivalence relation simultaneously through an inductive-inductive definition, using strictly positive occurrences of Hofmann-style quotient types (Fiore et al., 2019). This is conceptually close to the general QIIT methodology, where the data and the relation are co-generated.

The infinitary case requires an additional termination argument. Here the construction employs Abel’s size types, which are described as a constructive abstraction of what classically would be accomplished with transfinite ordinals. Size-indexed approximations $A$0, $A$1, and $A$2 ensure that recursive definitions for eliminators and computation principles are accepted by Agda. The development is formalized in Agda, but its computation rules are propositional rather than definitional, and the construction uses UIP and heterogeneous equality, so it is “not directly compatible with univalence / HoTT” (Fiore et al., 2019).

5. Canonical examples and applications

Several examples have become canonical in the QIIT literature because they require simultaneous generation of data, equations, and dependent structure. The early semantic work identifies the Cauchy reals, the partiality monad, and internal syntax of type theory as central examples. The internal-syntax case is especially archetypal: Con and Ty are defined mutually, with point constructors for empty context, extension, base type, and sigma-types, together with a path constructor identifying nested context extension with extension by a $A$3-type (Altenkirch et al., 2016).

Permutation quotients provide another standard pattern. For trees branching over an arbitrary type $A$4, one may define

$A$5

together with an equality constructor

$A$6

so that the tree constructor is respected up to permutation of subtrees (Altenkirch et al., 2016). This illustrates how quotienting by symmetry can be internalized as path constructors rather than postulated as an external quotient.

The partiality monad is one of the most detailed QIIT applications. Instead of quotienting Capretta’s delay monad by weak bisimilarity, the construction simultaneously defines a type $A$7 of partial elements and a relation $A$8 on $A$9. The type is generated by

$A$0

together with

$A$1

while the relation is generated as a propositional order with reflexivity, transitivity, bottom, and least-upper-bound rules for increasing $A$2-chains. The tuple $A$3 is initial in the category $A$4 of partiality algebras over $A$5 (Altenkirch et al., 2016).

The significance of this construction is explicitly constructive. Chapman et al. had shown that proving a monad structure on the quotient of the delay monad by weak bisimilarity requires countable choice. The QIIT construction avoids that step by deriving the monad structure from initiality. Under countable choice, the paper proves equivalence between the QIIT-based partiality monad, a quotient of monotone sequences, and the delay monad quotiented by weak bisimilarity (Altenkirch et al., 2016). This suggests a general methodological advantage of QIITs: when quotients and algebraic structure interact badly in ordinary type theory, building the quotient structure into the inductive definition can recover the intended universal property directly.

6. QIITs in predicative domain theory

A recent domain-theoretic application uses QIITs as the mechanism for constructing initial DCPO algebras in a predicative, type-theoretic setting. The framework starts from a signature $A$6 consisting of a family of operation symbols, for each symbol $A$7 a monomial

$A$8

and a family of formal inequalities. A prealgebra for $A$9 is a DCPO $x \sqsubseteq y$0 equipped with Scott-continuous operations

$x \sqsubseteq y$1

and an algebra is a prealgebra satisfying the formal inequalities (Collem et al., 12 Sep 2025).

The main result is that the initial algebra for any suitable DCPO signature exists and is built as a QIIT. The initial object $x \sqsubseteq y$2 is defined so that the QIIT simultaneously generates:

• the underlying type;
• the order relation $x \sqsubseteq y$3;
• the signature operations;
• the equations and inequalities required by the signature.

The DCPO structure is part of the inductive definition itself. The QIIT includes constructors for reflexivity and transitivity of $x \sqsubseteq y$4, propositionality, antisymmetry, set-truncation, and directed least upper bounds

$x \sqsubseteq y$5

for every directed family $x \sqsubseteq y$6. It also includes, for each operation symbol $x \sqsubseteq y$7, an operation constructor

$x \sqsubseteq y$8

continuity constructors forcing Scott continuity by preservation of directed suprema, and inequality constructors asserting

$x \sqsubseteq y$9

for each formal inequality and valuation $[]$0 (Collem et al., 12 Sep 2025).

The universal properties are integrated with the construction. For every algebra $[]$1, there is a unique algebra morphism

$[]$2

satisfying definitional computation rules for operations and directed suprema, and there is also an induction principle for proposition-valued families preserved by directed suprema and operations. From recursion and induction, the paper proves that $[]$3 is initial in $[]$4 (Collem et al., 12 Sep 2025).

This domain-theoretic use of QIITs is notable for two reasons. First, the construction is predicative: it avoids impredicative completion and does not proceed by first building a DCPO and then imposing equations externally. Second, it unifies several standard constructions. The paper shows that coalesced sums, smash products, and free DCPOs all fit the same signature-and-QIIT pattern. The free-construction scheme specializes to partiality and to power domains; in the Plotkin powerdomain case, the QIIT must encode not only associativity, commutativity, and idempotence of union, but also continuity of union with respect to directed suprema. The entire development is formalized in Cubical Agda (Collem et al., 12 Sep 2025).

A plausible implication is that QIITs have matured from a semantic proposal for quotient-like dependent syntax into a reusable constructional device for predicative semantics of computation. The literature does not present this as a single finished theory: some works emphasize semantics and initiality, some emphasize signature calculi and induction principles, some isolate constructive fragments such as QW-types, and some develop specific applications such as partiality or DCPO algebras. Taken together, these works characterize QIITs as a unifying interface between inductive generation, internal quotienting, and algebraic universal properties.