Finite Completeness of Categories of Petri Nets

The problem of finite completeness of categories of Petri nets is studied. Since Petri nets have finite products, the problem reduces to the issue of the existence of equalizers. We show that the categories of Petri nets with general and Winskel morphisms do not admit equalizers, and hence are not finitely complete. An important class of multiplicative Petri net morphisms is also identified. The main positive result of the paper states that reachable Petri nets and multiplicative morphisms form a finitely complete category. As an application of this result, some well-known categories turn out to be finitely complete. Since, for instance, all morphisms between reachable safe Petri nets are multiplicative it follows that the category of reachable safe Petri nets nets is finitely complete.