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.
Keywords: Boolean category, *-autonomous category, proof theory, classical logic, proof nets
2000 MSC: 03B05, 03G05, 03F03, 18D15, 18D35
Theory and Applications of Categories,
Vol. 18, 2007,
No. 18, pp 536-601.
http://www.tac.mta.ca/tac/volumes/18/18/18-18.dvi
http://www.tac.mta.ca/tac/volumes/18/18/18-18.ps
http://www.tac.mta.ca/tac/volumes/18/18/18-18.pdf
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/18/18-18.dvi
ftp://ftp.tac.mta.ca/pub/tac/html/volumes/18/18/18-18.ps
Revised 2007-12-13. Original version at
http://www.tac.mta.ca/tac/volumes/18/18/18-18a.dvi