# Custom Hypergraph Categories via Generalized Relations

## TL;DR

This paper introduces a flexible, parameterized framework for constructing process theoretic models using generalized relations, enabling diverse applications across quantum computation, NLP, and network theory.

## Contribution

It generalizes relational models along four axes, creating a unifying, functorial approach for building process theories in various domains.

## Key findings

- Categories are preorder-enriched with relational analogues.
- Framework unifies existing models from literature.
- New models suggest avenues for further research.

## Abstract

Process theories combine a graphical language for compositional reasoning with an underlying categorical semantics. They have been successfully applied to fields such as quantum computation, natural language processing, linear dynamical systems and network theory. When investigating a new application, the question arises of how to identify a suitable process theoretic model. We present a conceptually motivated parameterized framework for the construction of models for process theories. Our framework generalizes the notion of binary relation along four axes of variation, the truth values, a choice of algebraic structure, the ambient mathematical universe and the choice of proof relevance or provability. The resulting categories are preorder-enriched and provide analogues of relational converse and taking graphs of maps. Our constructions are functorial in the parameter choices, establishing mathematical connections between different application domains. We illustrate our techniques by constructing many existing models from the literature, and new models that open up ground for further development.