  From every pair of adjoint functors it is possible to produce a (possibly
trivial) equivalence of categories by restricting to the subcategories where
the unit and counit are isomorphisms. If we do this for the adjunction between
effect algebras and abstract convex sets, we get the surprising result that the
equivalent subcategories consist of reflexive order-unit spaces and reflexive
base-norm spaces, respectively. These are the convex sets that can occur as
state spaces in generalized probabilistic theories satisfying both the
no-restriction hypothesis and its dual. The linearity of the morphisms is
automatic. If we add a compact topology to either the states or the effects, we
can obtain a duality for all Banach order-unit spaces or all Banach base-norm
spaces, but not both at the same time.

Logic in Computer Science