[Paper Review] A Universal Construction for (Co)Relations

Brendan Fong, Fabio Zanasi|DROPS (Schloss Dagstuhl – Leibniz Center for Informatics)|Mar 23, 2017

Natural Language Processing Techniques5 citations

TL;DR

This paper presents a universal construction for categories of (co)relations—such as relations, corelations, and linear subspaces—using pushouts of spans and cospans over a base category of maps. By leveraging distributive laws and categorical pushouts, it unifies diverse axiomatisations in quantum computation, control theory, and program semantics, showing that (co)relation categories arise as colimits, enabling complete, modular equational reasoning for string diagrams.

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).

Motivation & Objective

• To unify seemingly disparate constructions of (co)relation categories across different domains such as quantum processes, control systems, and program semantics.
• To provide a general categorical framework for constructing categories of relations and corelations as colimits of simpler diagrammatic components.
• To demonstrate that complete equational axiomatisations for string diagram semantics can be derived modularly from base categories of linear maps or functions.
• To extend the applicability of the pushout construction to algebras over monads and regular categories, broadening its scope beyond vector spaces.
• To establish a foundational method for constructing hypergraph categories from spans and cospans, supporting future work on decorated corelations and generalized network models.

Proposed method

• The paper constructs categories of (co)relations as pushouts in the category of props, using spans and cospans over a base category of maps (e.g., linear maps or functions).
• It applies distributive laws of props to combine the span and cospan constructions, enabling modular derivation of equational axiomatisations.
• The construction relies on the universal property of pushouts to derive a complete axiomatisation for the resulting (co)relation category from those of the base and dual categories.
• It generalises the construction to regular categories with split regular epimorphisms, allowing application to Eilenberg–Moore categories of monads.
• The method is illustrated through examples: linear subspaces (SV_k), equivalence relations (ER), and partial equivalence relations (PER), all derived from pushouts of spans and cospans.
• The framework extends to algebras over monads, such as algebras over a field, by verifying that the necessary categorical conditions (e.g., finite cocompleteness, split epimorphisms) are satisfied.

Experimental results

Research questions

• RQ1 Can categories of relations and corelations be universally constructed from simpler diagrammatic components such as spans and cospans?
• RQ2 How can a modular, complete equational axiomatisation for string diagram semantics be derived from base categories of maps?
• RQ3 What categorical conditions ensure that the pushout of spans and cospans yields a well-defined category of (co)relations?
• RQ4 To what extent does this construction unify examples from control theory, quantum computation, and program semantics?
• RQ5 Can the pushout construction be extended to algebras over monads, such as vector spaces with bilinear products?

Key findings

• The category of linear subspaces (SV_k) is constructed as the pushout of spans and cospans over the category of linear maps (Vect_k), providing a universal characterisation.
• The theory of interacting Hopf algebras arises as a complete equational axiomatisation of SV_k, derived from distributive laws over the span and cospan constructions.
• The construction generalises to equivalence relations (ER) and partial equivalence relations (PER), with analogous pushout decompositions using injections and functions.
• The method applies to Eilenberg–Moore categories of monads over regular categories with split regular epimorphisms, enabling construction of relations in algebraic structures.
• The resulting (co)relation categories are hypergraph categories, equipped with special commutative Frobenius structures, supporting network-style diagrammatic reasoning.
• The framework explains why field extensions are necessary in moving from FMod_R to Vect_k: the pushout construction requires formal inverses, which are only available in the field of fractions.