Categories of Empirical Models

TL;DR

This paper develops a sheaf-theoretic framework for empirical models, introducing morphisms that formalize simulation and resource theories of contextuality, with implications for understanding non-contextuality, functoriality, and cloning limitations.

Contribution

It introduces a new morphism concept for empirical models using relations and stochastic mappings, formalizing simulation and resource theories of contextuality.

Findings

Non-contextual models are characterized by morphisms from the terminal object.

The contextual fraction is shown to be functorial.

Graham-reductions induce morphisms, providing a new perspective on Vorob'ev's theorem.

Abstract

A notion of morphism that is suitable for the sheaf-theoretic approach to contextuality is developed, resulting in a resource theory for contextuality. The key features involve using an underlying relation rather than a function between measurement scenarios, and allowing for stochastic mappings of outcomes to outcomes. This formalizes an intuitive idea of using one empirical model to simulate another one with the help of pre-shared classical randomness. This allows one to reinterpret concepts and earlier results in terms of morphisms. Most notably: non-contextual models are precisely those allowing a morphism from the terminal object; contextual fraction is functorial; Graham-reductions induce morphisms, reinterpreting Vorob'evs theorem; contextual models cannot be cloned.