Choudhury, V. and Fiore, M. (2023) Free Commutative Monoids in Homotopy Type Theory. In: 38th International Conference on Mathematical Foundations of Programming Semantics (MFPS 2022), Ithaca, NY, USA, 11-13 Jul 2022, p. 10492. (doi: 10.46298/entics.10492)
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Abstract
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.
| Item Type: | Conference Proceedings |
|---|---|
| Additional Information: | Research partially supported by EPSRC grant EP/V002309/1. |
| Keywords: | Finite-multiset construction, free commutative-monoid construction, constructive mathematics, homotopy type theory, higher inductive types, category theory, classical linear logic, differential linear logic, conical refinement monoids, combinatorial Fock space, creation/annihilation operators. |
| Status: | Published |
| Refereed: | Yes |
| Glasgow Author(s) Enlighten ID: | Choudhury, Dr Vikraman |
| Authors: | Choudhury, V., and Fiore, M. |
| Subjects: | Q Science > QA Mathematics Q Science > QA Mathematics > QA75 Electronic computers. Computer science |
| College/School: | College of Science and Engineering > School of Computing Science |
| ISSN: | 1571-0661 |
| Copyright Holders: | Copyright © 2023 V. Choudhury, M. Fiore |
| First Published: | First published in Electronic Notes in Theoretical Informatics and Computer Science 1: 10492 |
| Publisher Policy: | Reproduced under a Creative Commons License |
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