Interacting Monoidal Structures with Applications in Computing

TL;DR

This paper introduces and explores n-fold monoid and comonoid structures within n-fold monoidal categories, with applications to computing, concurrency, and rewriting, supported by diverse examples from algebra and category theory.

Contribution

It develops the theory of n-fold monoids and comonoids in n-fold monoidal categories, providing new models and examples relevant to computing and higher-dimensional rewriting.

Findings

Presented a categorical model for communication protocols.

Constructed a lax n-fold relational monoid used in rewriting.

Analyzed modules and algebras of the boolean semiring for algebraic structures.

Abstract

With a view on applications in computing, in particular concurrency theory and higher-dimensional rewriting, we develop notions of $n$-fold monoid and comonoid objects in $n$-fold monoidal categories and bicategories. We present a series of examples for these structures from various domains, including a categorical model for a communication protocol and a lax $n$-fold relational monoid, which has previously been used implicitly for higher-dimensional rewriting and which specialises in a natural way to strict $n$-categories. A special set of examples is built around modules and algebras of the boolean semiring, which allows us to deal with semilattices, additively idempotent semirings and quantales using tools from classical algebra.