This paper uncovers the fundamental relationship between total and partial
computation in the form of an equivalence of certain categories. This
equivalence involves on the one hand effectuses, which are categories for total
computation, introduced by Jacobs for the study of quantum/effect logic. On the
other hand, it involves what we call FinPACs with effects; they are finitely
partially additive categories equipped with effect algebra structures, serving
as categories for partial computation. It turns out that the Kleisli category
of the lift monad (-)+1 on an effectus is always a FinPAC with effects, and
this construction gives rise to the equivalence. Additionally, state-and-effect
triangles over FinPACs with effects are presented.Comment: In Proceedings QPL 2015, arXiv:1511.0118
