# Petri Nets Based on Lawvere Theories

## TL;DR

This paper introduces a generalized framework for Petri nets based on Lawvere theories, unifying various existing variants and providing a functorial, categorical semantics for these models.

## Contribution

It defines $ ext{Q}$-nets as a generalization of Petri nets using Lawvere theories, establishing their functoriality and categorical semantics.

## Key findings

- Unified framework for various Petri net variants
- Functorial semantics for $ ext{Q}$-nets
- Construction of semantics for multiple net types

## Abstract

We give a definition of $\mathsf{Q}$-net, a generalization of Petri nets based on a Lawvere theory $\mathsf{Q}$, for which many existing variants of Petri nets are a special case. This definition is functorial with respect to change in Lawvere theory, and we exploit this to explore the relationships between different kinds of $\mathsf{Q}$-nets. To justify our definition of $\mathsf{Q}$-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which $\mathsf{Q}$-nets present free models of $\mathsf{Q}$ in $\mathsf{Cat}$. This gives a functorial description of the operational semantics for an arbitrary category of $\mathsf{Q}$-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.