  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.
