Free Commutative Monoids in Homotopy Type Theory
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 commutativemonoid construction, we formalise and establish the categorical universal property of two, necessarily equivalent, algebraic presentations of free commutative monoids using 1HITs. 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 commutativemonoid construction as a setquotient of the list construction. These developments crucially rely on the commutation relation of creation/annihilation operators associated with the free commutativemonoid construction seen as a combinatorial Fock space.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 DOI:
 10.48550/arXiv.2110.05412
 arXiv:
 arXiv:2110.05412
 Bibcode:
 2021arXiv211005412C
 Keywords:

 Computer Science  Logic in Computer Science;
 Mathematics  Combinatorics;
 Mathematics  Category Theory;
 Mathematics  Logic;
 03G30;
 F.4.1
 EPrint:
 Appeared in MFPS'22