# Categorical Equivalences from State-Effect Adjunctions

@inproceedings{Furber2018CategoricalEF, title={Categorical Equivalences from State-Effect Adjunctions}, author={Robert Furber}, booktitle={QPL}, year={2018} }

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…

## 4 Citations

### Scott Continuity in Generalized Probabilistic Theories

- MathematicsQPL
- 2019

It is shown that there are infinite-dimensional convex sets for which the set of Scott-continuous states on the corresponding set of 2-valued POVMs does not recover the original convex set, but is strictly larger, which shows the necessity of the use of topologies for state-effect duality in the general case.

### Edinburgh Explorer Scott Continuity in Generalized Probabilistic Theories

- Mathematics
- 2020

Scott continuity is a concept from domain theory that had an unexpected previous life in the theory of von Neumann algebras. Scott-continuous states are known as normal states, and normal states are…

### Dichotomy between deterministic and probabilistic models in countably additive effectus theory

- MathematicsQPL
- 2020

It is shown that a non-trivial $\sigma$-effectus with normalization has as scalars either the two-element effect monoid $\{0,1\}$ or the real unit interval $[ 0,1]$.

### Scott Continuity in Generalized Probabilistic Theories

- Physics
- 2020

Citation for published version: Furber, R 2020, Scott Continuity in Generalized Probabilistic Theories. in B Coecke & M Leifer (eds), Proceedings 16th International Conference on Quantum Physics and…

## References

SHOWING 1-10 OF 35 REFERENCES

### The expectation monad in quantum foundations

- MathematicsInf. Comput.
- 2016

The expectation monad is introduced abstractly via two composable adjunctions, but concretely captures measures, and leads to a new re-formulation of Gleason's theorem, expressing that effects on a Hilbert space are free effect modules on projections, obtained via tensoring with the unit interval.

### Convexity, Duality and Effects

- MathematicsIFIP TCS
- 2010

This paper describes some basic relationships between mathematical structures that are relevant in quantum logic and probability, namely convex sets, effect algebras, and a new class of functors that…

### Convex Spaces I: Definition and Examples

- Mathematics
- 2009

We propose an abstract definition of convex spaces as sets where one can take convex combinations in a consistent way. A priori, a convex space is an algebra over a finitary version of the Giry…

### Convex structures and operational quantum mechanics

- Mathematics
- 1973

A general mathematical framework called a convex structure is introduced. This framework generalizes the usual concept of a convex set in a real linear space. A metric is constructed on a convex…

### Effect algebras and unsharp quantum logics

- Mathematics
- 1994

The effects in a quantum-mechanical system form a partial algebra and a partially ordered set which is the prototypical example of the effect algebras discussed in this paper. The relationships among…

### On an Algebraic generalization of the quantum mechanical formalism

- Mathematics
- 1934

One of us has shown that the statistical properties of the measurements of a quantum mechanical system assume their simplest form when expressed in terms of a certain hypercomplex algebra which is…

### Categories for the Working Mathematician

- Mathematics
- 1971

I. Categories, Functors and Natural Transformations.- 1. Axioms for Categories.- 2. Categories.- 3. Functors.- 4. Natural Transformations.- 5. Monics, Epis, and Zeros.- 6. Foundations.- 7. Large…

### A Characterization of State Spaces of Orthomodular Lattices

- MathematicsJ. Comb. Theory, Ser. A
- 1974

### Ordered linear spaces and categories as frameworks for information-processing characterizations of quantum and classical theory

- Computer Science
- 2009

Some ways of formulating this framework in terms of categories are sketched, and in this light the relation of the work to that of Abramsky, Coecke, Selinger, Baez and others on information processing and other aspects of theories formulated categorically is considered.

### New trends in quantum structures

- Mathematics
- 2000

Preface. Introduction. 1. D-posets and Effect Algebras. 2. MV-algebras and QMV-algebras. 3. Quotients of Partial Abelian Monoids. 4. Tensor Product of D-Posets and Effect Algebras. 5. BCK-algebras.…