## Open Petri Nets and Their Categories of Processes

My student Jade Master will be talking about her work on open Petri nets at the online category theory seminar at UNAM on Wednesday October 21st at 18:00 UTC (11 am Pacific Time):

Open Petri Nets and Their Categories of Processes

Abstract. In this talk we will discuss Petri nets from a categorical perspective. A Petri net freely generates a symmetric monoidal category whose morphisms represent its executions. We will discuss how to make Petri nets ‘open’—i.e., equip them with input and output boundaries where resources can flow in and out. Open Petri nets freely generate open symmetric monoidal categories: symmetric monoidal categories which can be glued together along a shared boundary. The mapping from open Petri nets to their open symmetric monoidal categories is functorial and this gives a compositional framework for reasoning about the executions of Petri nets.

You can see the talk live, or later recorded, here:

Abstract. The reachability semantics for Petri nets can be studied using open Petri nets. For us an ‘open’ Petri net is one with certain places designated as inputs and outputs via a cospan of sets. We can compose open Petri nets by gluing the outputs of one to the inputs of another. Open Petri nets can be treated as morphisms of a category $\mathsf{Open}(\mathsf{Petri}),$ which becomes symmetric monoidal under disjoint union. However, since the composite of open Petri nets is defined only up to isomorphism, it is better to treat them as morphisms of a symmetric monoidal double category $\mathbb{O}\mathbf{pen}(\mathsf{Petri}).$ Various choices of semantics for open Petri nets can be described using symmetric monoidal double functors out of $\mathbb{O}\mathbf{pen}(\mathsf{Petri}).$ Here we describe the reachability semantics, which assigns to each open Petri net the relation saying which markings of the outputs can be obtained from a given marking of the inputs via a sequence of transitions. We show this semantics gives a symmetric monoidal lax double functor from $\mathbb{O}\mathbf{pen}(\mathsf{Petri})$ to the double category of relations. A key step in the proof is to treat Petri nets as presentations of symmetric monoidal categories; for this we use the work of Meseguer, Montanari, Sassone and others.