# Categorifying the ZX-calculus

## TL;DR

This paper develops a bicategorical framework combining spans and cospans within a topos to study open networks and diagrammatic languages, exemplified by the ZX-calculus.

## Contribution

It introduces a new bicategory construction that integrates the ZX-calculus, providing a formal categorical setting for diagrammatic reasoning.

## Key findings

- Constructed a symmetric monoidal, compact closed bicategory
- Applied the framework to the ZX-calculus diagrams
- Provided a categorical foundation for diagrammatic languages

## Abstract

We build a symmetric monoidal and compact closed bicategory by combining spans and cospans inside a topos. This can be used as a framework in which to study open networks and diagrammatic languages. We illustrate this framework with Coecke and Duncan's zx-calculus by constructing a bicategory with the natural numbers for 0-cells, the zx-calculus diagrams for 1-cells, and rewrite rules for 2-cells.