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On the axiomatisation of Boolean categories with and without medial

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Lutz Strassburger

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

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