On the Axiomatisation of Boolean Categories with and without Medial

Details On the Axiomatisation of Boolean Categories with and without Medial On the Axiomatisation of Boolean Categories with and without Medial

2005-12-22

arxiv.org/abs/cs/0512086v3

Computer Science

Logic in Computer Science

66 pages LaTeX, using TAC style, v3: minor changes as requested by the referee

Scientific paper

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.

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