Duality for Convexity

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.

💡 Research Summary

The paper presents a categorical study of convex sets by viewing them as algebras of the distribution monad 𝔻. The distribution monad assigns to any set X the set 𝔻X of finitely supported probability distributions on X, equipped with the usual unit (Dirac delta) and multiplication (flattening of nested distributions). An algebra for 𝔻 is a map α : 𝔻X → X satisfying the monad laws; concretely, α computes the “average” of a probability distribution. The authors show that such algebras are precisely convex sets equipped with an affine combination operation, thereby giving a clean monadic characterisation of convexity.

Having identified convex sets as 𝔻‑algebras, the authors investigate two distinct dual adjunctions in which these algebras appear as objects on one side. The first adjunction uses the Boolean truth‑value set {0,1} as a dualising object. By considering characteristic functions χ : X → {0,1} of subsets of a convex set X, one obtains a preframe structure on the collection of such functions. A preframe is a poset with arbitrary directed joins and finite meets, generalising frames without requiring complete distributivity. The paper proves that the category of 𝔻‑algebras is dually equivalent to the category of preframes via the hom‑functor Hom(–,{0,1}). This establishes a logical, Boolean‑based duality: convex sets can be recovered from their Boolean‑valued predicates, and conversely every preframe determines a convex set of “states”.

The second adjunction is built on the real unit interval