Free Commutative Monoids in Homotopy Type Theory

TL;DR

This paper develops a constructive theory of finite multisets as free commutative monoids within Homotopy Type Theory, establishing their categorical properties and applications to linear logic and combinatorial Fock space.

Contribution

It formalizes two equivalent algebraic presentations of free commutative monoids using 1-HITs and proves their structural properties without assuming decidable equality.

Findings

Constructive formalization of the relational model of classical linear logic.

Proof that free commutative monoids are conical refinement monoids.

New presentation of free commutative-monoid as a set-quotient of list construction.

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.