On the Axiomatisation of Boolean Categories with and without Medial
Lutz Straßburger · Theory and applications of categories · 2007
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.