On the Axiomatisation of Boolean Categories with and without Medial
<|reference_start|>On the Axiomatisation of Boolean Categories with and without Medial: 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.<|reference_end|>
@article{strassburger2005on,
    title={On the Axiomatisation of Boolean Categories with and without Medial},
    author={Lutz Strassburger},
    journal={arXiv preprint arXiv:cs/0512086},
    year={2005},
    archivePrefix={arXiv},
    eprint={cs/0512086},
    primaryClass={cs.LO}
}