We give a definition of Q-net, a generalization of Petri nets based on a Lawvere theory 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 Q-nets. To justify our definition of Q-net, we construct a family of adjunctions for each Lawvere theory explicating the way in which Q-nets present free models of Q in Cat. This gives a functorial description of the operational semantics for an arbitrary category of Q-nets. We show how this can be used to construct the semantics for Petri nets, pre-nets, integer nets, and elementary net systems.

中文翻译：

---

基于 Lawvere 理论的 Petri 网

我们给出了 Q-net 的定义，它是基于 Lawvere 理论 Q 的 Petri 网的泛化，其中许多 Petri 网的现有变体是一个特例。这个定义对于 Lawvere 理论的变化是函数式的，我们利用它来探索不同种类的 Q-nets 之间的关系。为了证明我们对 Q-net 的定义的合理性，我们为每个 Lawvere 理论构建了一个附加词族，以解释 Q-nets 在 Cat.1 中呈现 Q 的自由模型的方式。这给出了任意类别 Q-net 的操作语义的功能描述。我们展示了如何使用它来构建 Petri 网、前置网、整数网和基本网络系统的语义。