# Categorical Equivalences from State-Effect Adjunctions

## TL;DR

This paper explores how adjoint functors between effect algebras and convex sets induce categorical equivalences, revealing deep connections with reflexive spaces and dualities in probabilistic theories.

## Contribution

It demonstrates that restricting adjunctions yields equivalences between categories of reflexive spaces, linking effect algebras and convex sets in probabilistic frameworks.

## Key findings

- Reflexive order-unit and base-norm spaces form equivalent subcategories.
- Linearity of morphisms is automatic in these equivalences.
- Adding topology yields dualities for Banach spaces, but not simultaneously.

## Abstract

From every pair of adjoint functors it is possible to produce a (possibly trivial) equivalence of categories by restricting to the subcategories where the unit and counit are isomorphisms. If we do this for the adjunction between effect algebras and abstract convex sets, we get the surprising result that the equivalent subcategories consist of reflexive order-unit spaces and reflexive base-norm spaces, respectively. These are the convex sets that can occur as state spaces in generalized probabilistic theories satisfying both the no-restriction hypothesis and its dual. The linearity of the morphisms is automatic. If we add a compact topology to either the states or the effects, we can obtain a duality for all Banach order-unit spaces or all Banach base-norm spaces, but not both at the same time.