Duality for Convexity

Abstract

This paper studies convex sets categorically, namely as algebras of a distribution monad. It is shown that convex sets occur in two dual adjunctions, namely one with preframes via the Boolean truth values {0,1} as dualising object, and one with effect algebras via the (real) unit interval [0,1] as dualising object. These effect algebras are of interest in the foundations of quantum mechanics.