Symmetric Monoidal Structure with Local Character is a Property

TL;DR

This paper introduces a unifying categorical notion of symmetric monoidal structure with local character, demonstrating it is a property (uniquely determined) across various categories, including infinite-dimensional and non-free cases.

Contribution

It generalizes previous results by defining a new notion of symmetric monoidal structure with local character and proving it is a property in broader categorical contexts.

Findings

Symmetric monoidal structure with local character is a property in various categories.

The notion applies to infinite-dimensional relations over quantales.

It extends to finitely generated modules over principal ideal domains.

Abstract

In previous work we proved that, for categories of free finite-dimensional modules over a commutative semiring, linear compact-closed symmetric monoidal structure is a property, rather than a structure. That is, if there is such a structure, then it is uniquely defined (up to monoidal equivalence). Here we provide a novel unifying category-theoretic notion of symmetric monoidal structure with local character, which we prove to be a property for a much broader spectrum of categorical examples, including the infinite-dimensional case of relations over a quantale and the non-free case of finitely generated modules over a principal ideal domain.