Morphisms in categories of nonlocal games

TL;DR

This paper explores categories of nonlocal games with morphisms as correlations, characterizing their categorical properties like monomorphisms and epimorphisms across classical, quantum, and nonsignaling cases.

Contribution

It introduces a categorical framework for nonlocal games with various correlation types and characterizes morphism properties such as monic, epic, sections, and retractions.

Findings

Characterization of monic and epic morphisms in these categories

Identification of conditions for sections and retractions

Unified framework for classical, quantum, and nonsignaling correlations

Abstract

Synchronous correlations provide a class of nonlocal games that behave like functions between finite sets. In this work we examine categories whose morphisms are games with synchronous classical, quantum, or general nonsignaling correlations. In particular, we characterize when morphisms in these categories are monic, epic, sections, or retractions.