# Network Models from Petri Nets with Catalysts

## TL;DR

This paper introduces a method to combine Petri nets and network models using catalysts, enabling a structured categorical framework for modeling interacting systems with reusable entities.

## Contribution

It formalizes how catalysts in Petri nets can generate network models and constructs a fibered category capturing process compositions with catalysts.

## Key findings

- Defines catalysts as entities with equal in-degree and out-degree in Petri nets.
- Constructs a fibered category over catalyst lists representing enabled processes.
- Shows the symmetric monoidal and premonoidal structures for process composition.

## Abstract

Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.