Convexity, Duality and Effects

Abstract : This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that we call 'convex functors'; they include what are usually called probability distribution functors. These relationships take the form of three adjunctions. Two of these three are 'dual' adjunctions for convex sets, one time with the Boolean truth values {0,1} as dualising object, and one time with the probablity values [0,1]. The third adjunction is between effect algebras and convex functors.