Higher Inductive Types as Homotopy-Initial Algebras

Sojakova, Kristina

doi:10.1145/2676726.2676983

Cited by 32 publications

(40 citation statements)

References 15 publications

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“…So this extension represents one of our main contributions. QW-types generalise prior developments; the internal encodings for particular subclasses of 1-HITs given by Sojakova [25] and Swan [27] are direct instances of QW-types, as the next two examples show. [20,Section 9] give an example of a HIT not constructible in type theory from only pushouts and N. Their HIT F can be thought of as a set of notations for countable ordinals.…”

Section: Quotient-inductive Typesmentioning

confidence: 71%

“…That unordered countably-branching trees are a QW-type is significant since no previous work on various subclasses of QITs (or indeed QIITs [18,9]) supports infinitary QITs [5,25,27,11,18,9]. See Example 5 for another, more substantial infinitary QW-type.…”

Section: Quotient-inductive Typesmentioning

confidence: 99%

“…W-suspensions[25] are an instance of QW-types. The data for a W-suspension is: A , C : Set, a type family B : A → Set and functions l, r : C → A .…”

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confidence: 99%

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Constructing Infinitary Quotient-Inductive Types

Fiore

Pitts

Steenkamp

2020

Lecture Notes in Computer Science

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This paper introduces an expressive class of quotient-inductive types, called QW-types. We show that in dependent type theory with uniqueness of identity proofs, even the infinitary case of QW-types can be encoded using the combination of inductive-inductive definitions, Hofmann-style quotient types, and Abel's size types. The latter, which provide a convenient constructive abstraction of what classically would be accomplished with transfinite ordinals, are used to prove termination of the recursive definitions of the elimination and computation properties of our encoding of QW-types. The development is formalized using the Agda theorem-prover.

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Section: Quotient-inductive Typesmentioning

confidence: 71%

Section: Quotient-inductive Typesmentioning

confidence: 99%

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Constructing Infinitary Quotient-Inductive Types

Fiore

Pitts

Steenkamp

2020

Lecture Notes in Computer Science

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“…In [van der Weide 2016] a semantics is given for the same class of HITs but with no recursive equality constructors. Sojakova [Sojakova 2015] defines a subset of HITs called W-suspensions by an internal coding scheme similar to W-types. She proves that the induction principle is equivalent to homotopy initiality.…”

Section: Overview Of the Rest Of The Papermentioning

confidence: 99%

Constructing quotient inductive-inductive types

Kaposi

Kovács

Altenkirch

2019

Proc. ACM Program. Lang.

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Quotient inductive-inductive types (QIITs) generalise inductive types in two ways: a QIIT can have more than one sort and the later sorts can be indexed over the previous ones. In addition, equality constructors are also allowed. We work in a setting with uniqueness of identity proofs, hence we use the term QIIT instead of higher inductive-inductive type. An example of a QIIT is the well-typed (intrinsic) syntax of type theory quotiented by conversion. In this paper first we specify finitary QIITs using a domain-specific type theory which we call the theory of signatures. The syntax of the theory of signatures is given by a QIIT as well. Then, using this syntax we show that all specified QIITs exist and they have a dependent elimination principle. We also show that algebras of a signature form a category with families (CwF) and use the internal language of this CwF to show that dependent elimination is equivalent to initiality.

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“…Higher inductive types were first conceived by participants at the 2011 Oberwolfach workshop on homotopical interpretations of ITT. While an informal description of a general class was given in the HoTT Book [Univalent Foundations Program 2013, ğ6.13], the first rigorous presentation of a large syntactic class was Sojakova's W-quotients [Sojakova 2015], a generalization of W-types which added a path constructor. Sojakova showed that these types are homotopy-initial algebras, building on work on universal characterizations of ordinary inductive types in HoTT by Awodey et al [2012].…”

Section: Free Heterogeneous Compositionmentioning

confidence: 99%

Higher inductive types in cubical computational type theory

Cavallo

Harper

2019

Proc. ACM Program. Lang.

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Homotopy type theory proposes higher inductive types (HITs) as a means of defining and reasoning about inductively-generated objects with higher-dimensional structure. As with the univalence axiom, however, homotopy type theory does not specify the computational behavior of HITs. Computational interpretations have now been provided for univalence and specific HITs by way of cubical type theories, which use a judgmental infrastructure of dimension variables. We extend the cartesian cubical computational type theory introduced by Angiuli et al. with a schema for indexed cubical inductive types (CITs), an adaptation of higher inductive types to the cubical setting. In doing so, we isolate the canonical values of a cubical inductive type and prove a canonicity theorem with respect to these values.

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Higher Inductive Types as Homotopy-Initial Algebras

Cited by 32 publications

References 15 publications

Constructing Infinitary Quotient-Inductive Types

Constructing Infinitary Quotient-Inductive Types

Constructing quotient inductive-inductive types

Higher inductive types in cubical computational type theory

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