# A Structural and Nominal Syntax for Diagrams

## TL;DR

This paper proposes a unified syntax for diagrams that combines the categorical combinator approach with conventional label-based languages, maintaining semantics and equational properties.

## Contribution

It introduces a novel framework that reconciles combinator and label-based diagram syntax without altering existing semantics or equational theories.

## Key findings

- Unified syntax preserves existing semantics.
- Provides sound and complete equational theories.
- Bridges categorical and conventional diagram languages.

## Abstract

The correspondence between monoidal categories and graphical languages of diagrams has been studied extensively, leading to applications in quantum computing and communication, systems theory, circuit design and more. From the categorical perspective, diagrams can be specified using (name-free) combinators which enjoy elegant equational properties. However, conventional notations for diagrammatic structures, such as hardware description languages (VHDL, Verilog) or graph languages (Dot), use a different style, which is flat, relational, and reliant on extensive use of names (labels). Such languages are not known to enjoy nice syntactic equational properties. However, since they make it relatively easy to specify (and modify) arbitrary diagrammatic structures they are more popular than the combinator style. In this paper we show how the two approaches to diagram syntax can be reconciled and unified in a way that does not change the semantics and the existing equational theory. Additionally, we give sound and complete equational theories for the combined syntax.