A Universal Construction for (Co)Relations

TL;DR

This paper presents a universal, modular construction for (co)relations that simplifies the axiomatization of string diagram semantics across various domains like circuits and quantum processes.

Contribution

It introduces a unifying colimit-based framework for characterizing relations and corelations, unifying multiple existing theorems and simplifying semantic equivalence axiomatization.

Findings

Unified categorical framework for relations and corelations

Simplifies axiomatization of string diagram semantics

Connects various existing theorems in literature

Abstract

Calculi of string diagrams are increasingly used to present the syntax and algebraic structure of various families of circuits, including signal flow graphs, electrical circuits and quantum processes. In many such approaches, the semantic interpretation for diagrams is given in terms of relations or corelations (generalised equivalence relations) of some kind. In this paper we show how semantic categories of both relations and corelations can be characterised as colimits of simpler categories. This modular perspective is important as it simplifies the task of giving a complete axiomatisation for semantic equivalence of string diagrams. Moreover, our general result unifies various theorems that are independently found in literature, including the cases of linear corelations (relevant for the semantics of electrical circuits), of partial equivalence relations and of linear subspaces (semantics of signal flow graphs and of the phase-free ZX calculus).