Quantum Logic, Dagger Kernel Categories and Inverse Baer*-Categories

This paper investigates dagger kernel categories which are considered first by Crown (J Nat Sci Math 15:11–25, 1975) and used by Heunen and Jacobs (Order 27:177–212, 2010) in their study of quantum logic from the perspective of categorical logic. The inverse Baer*-categories with splitting projections as special dagger kernel categories have a central place in our investigations. The inverse Baer*-categories with splitting and closed projections are Boolean and therefore the subobject lattices of such categories are representing classical logics. Examples are presented at every stage of our investigations