Duality for Convexity

2009/11/19 by Jacobs, Bart
#Category Theory (math.CT) #F.4.1 #FOS: Mathematics #Logic (math.LO)

paper · doi:10.48550/arxiv.0911.3834

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.

Related

• Convex and Sequential Effect Algebras
• Finite-Dimensional Convex Effect Algebras
• The operadic theory of convexity
• Conjectures and questions in convex geometry (of interest for quantum theory and other physical statistical theories)
• Cancellative Convex Semilattices