On the axiomatisation of Boolean categories with and without medial

Lutz Strassburger

Theory and applications of categories, Tome 18 (2007), pp. 536-601

Résumé

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.

Détail

Classification : 03B05, 03G05, 03F03, 18D15, 18D35
Keywords: Boolean category, *-autonomous category, proof theory, classical logic, proof nets