TL;DR
This paper explores the categorical structure of convex sets, revealing dual adjunctions with preframes and effect algebras, which are relevant in quantum mechanics foundations.
Contribution
It introduces a categorical framework for convex sets as algebras of a distribution monad, connecting them to preframes and effect algebras.
Findings
Convex sets form dual adjunctions with preframes and effect algebras.
Effect algebras are linked to quantum mechanics foundations.
Convex sets are characterized as algebras of a distribution monad.
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.
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Taxonomy
TopicsAdvanced Algebra and Logic · Logic, Reasoning, and Knowledge · Advanced Topology and Set Theory