On the Axiomatisation of Boolean Categories with and without Medial

TL;DR

This paper explores the foundational axioms of Boolean categories, focusing on the medial map and its relation to proof systems, and presents a category of proof nets as a well-behaved example.

Contribution

It introduces a series of axiomatisations for Boolean categories based on *-autonomous categories and analyzes the medial map's role.

Findings

No canonical axiomatisation exists for Boolean categories.

A series of increasingly strong axiomatisations are proposed.

A category of proof nets exemplifies a well-behaved Boolean category.

Abstract

The term ``Boolean category'' should be used for describing an object that is to categories what a Boolean algebra is to posets. More specifically, a Boolean category should provide the abstract algebraic structure underlying the proofs in Boolean Logic, in the same sense as a Cartesian closed category captures the proofs in intuitionistic logic and a *-autonomous category captures the proofs in linear logic. However, recent work has shown that there is no canonical axiomatisation of a Boolean category. In this work, we will see a series (with increasing strength) of possible such axiomatisations, all based on the notion of *-autonomous category. We will particularly focus on the medial map, which has its origin in an inference rule in KS, a cut-free deductive system for Boolean logic in the calculus of structures. Finally, we will present a category of proof nets as a particularly well-behaved example of a Boolean category.