Composable constraints

TL;DR

This paper introduces a categorical framework for compatible constraint encoding, enabling the construction of constrained categories and analyzing their properties in quantum and relational theories.

Contribution

It formalizes the concept of composable constraint encoding within category theory and explores its applications to quantum protocols and relational constraints.

Findings

Every composable constraint encoding yields an equivalent constrained category.

Compatibility of constraints with categorical structures like parallel composition is characterized.

Time-symmetric relational theories admit a faithful intersection of constraints.

Abstract

We introduce a notion of compatibility between constraint encoding and compositional structure. Phrased in the language of category theory, it is given by a "composable constraint encoding". We show that every composable constraint encoding can be used to construct an equivalent notion of a constrained category in which morphisms are supplemented with the constraints they satisfy. We further describe how to express the compatibility of constraints with additional categorical structures of their targets, such as parallel composition, compactness, and time-symmetry. We present a variety of concrete examples. Some are familiar in the study of quantum protocols and quantum foundations, such as signalling and sectorial constraints; others arise by construction from basic categorical notions. We use the language developed to discuss the notion of intersectability of constraints and the simplifications it allows for when present, and to show that any time-symmetric theory of relational constraints admits a faithful notion of intersection.