The Integers as a Higher Inductive Type

Altenkirch, Thorsten; Scoccola, Luis

doi:10.1145/3373718.3394760

Cited by 9 publications

(4 citation statements)

References 8 publications

(8 reference statements)

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“…The definition of various (higher) algebraic structures in HoTT/UF using HITs is an active line of research; for example, the problem of defining (free higher) groups has been considered by [37,14,3]. For algebraic structures with commutation axioms, there are various techniques which are known folklore.…”

Section: Commutation Axiomsmentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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We develop a constructive theory of finite multisets in Homotopy Type Theory, defining them as free commutative monoids. After recalling basic structural properties of the free commutative-monoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1-HITs. These presentations correspond to two different equational theories invariably including commutation axioms. In this setting, we prove important structural combinatorial properties of finite multisets. These properties are established in full generality without assuming decidable equality on the carrier set. As an application, we present a constructive formalisation of the relational model of classical linear logic and its differential structure. This leads to constructively establishing that free commutative monoids are conical refinement monoids. Thereon we obtain a characterisation of the equality type of finite multisets and a new presentation of the free commutative-monoid construction as a set-quotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutative-monoid construction seen as a combinatorial Fock space.

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Section: Commutation Axiomsmentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2023

Electronic Notes in Theoretical Informatics and Computer Science

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“…In addition, algebraic theories can be modeled as HITs, which allows one to define finite sets as a higher inductive type [32]. Other applications of HITs include homotopical patch theory, which provides a way to model version control systems [12], and modeling data types such as the integers [15,10]. Besides, quotient inductive-inductive types can be used to define the partiality monad [6].…”

Section: Related Workmentioning

confidence: 99%

Constructing Higher Inductive Types as Groupoid Quotients

Weide

2020

Proceedings of the 35th Annual ACM/IEEE Symposium on Logic in Computer Science

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In this paper, we show that all finitary 1-truncated higher inductive types (HITs) can be constructed from the groupoid quotient. We start by defining internally a notion of signatures for HITs, and for each signature, we construct a bicategory of algebras in 1-types and in groupoids. We continue by proving initial algebra semantics for our signatures. After that, we show that the groupoid quotient induces a biadjunction between the bicategories of algebras in 1-types and in groupoids. We finish by constructing a biinitial object in the bicategory of algebras in groupoids. From all this, we conclude that all finitary 1-truncated HITs can be constructed from the groupoid quotient. All the results are formalized over the UniMath library of univalent mathematics in Coq.

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“…Using this equality type for equations, it is a challenging problem to define (higher) algebraic structures. For example, the problem of defining (free higher) groups has been considered by Kraus and Altenkirch [32], Buchholtz, van Doorn, and Rijke [10], Altenkirch and Scoccola [2]. In this paper, we consider the problem of defining and, crucially, studying free commutative monoids in HoTT, using various HITs.…”

Section: Introductionmentioning

confidence: 99%

Free Commutative Monoids in Homotopy Type Theory

Choudhury¹,

Fiore²

2021

Preprint

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We develop a constructive theory of finite multisets, defining them as free commutative monoids in Homotopy Type Theory. We formalise two algebraic presentations of this construction using 1-HITs, establishing the categorical universal property for each and thereby their equivalence. These presentations correspond to equational theories including a commutation axiom. In this setting, we prove important structural combinatorial properties of singleton multisets arising from concatenations and projections of multisets. This is done in generality, without assuming decidable equality on the carrier set. Further, as applications, we present a constructive formalisation of the relational model of differential linear logic and use it to characterise the equality type of multisets. This leads us to a novel conditional equational presentation of the finite-multiset construction and thereby to a sound and complete deduction system for multiset equality.

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The Integers as a Higher Inductive Type

Cited by 9 publications

References 8 publications

Free Commutative Monoids in Homotopy Type Theory

Free Commutative Monoids in Homotopy Type Theory

Constructing Higher Inductive Types as Groupoid Quotients

Free Commutative Monoids in Homotopy Type Theory

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