Quantum Logic in Dagger Kernel Categories

Authors:
Chris Heunen;Bart Jacobs
Affiliations:
Institute for Computing and Information Sciences (iCIS), Radboud University Nijmegen, The Netherlands;Institute for Computing and Information Sciences (iCIS), Radboud University Nijmegen, The Netherlands
Venue:
Electronic Notes in Theoretical Computer Science (ENTCS)
Year:
2011

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Abstract

This paper investigates quantum logic from the perspective of categorical logic, and starts from minimal assumptions, namely the existence of involutions/daggers and kernels. The resulting structures turn out to (1) encompass many examples of interest, such as categories of relations, partial injections, Hilbert spaces (also modulo phase), and Boolean algebras, and (2) have interesting categorical/logical properties, in terms of kernel fibrations, such as existence of pullbacks, factorisation, and orthomodularity. For instance, the Sasaki hook and and-then connectives are obtained, as adjoints, via the existential-pullback adjunction between fibres.