We provide a categorical semantics for bounded Petri nets, both in the collective- and individual-token philosophy. In both cases, we describe the process of bounding a net internally, by just constructing new categories of executions of a net using comonads, and externally, using lax-monoidal-lax functors. Our external semantics is non-local, meaning that tokens are endowed with properties that say something about the global state of the net. We then prove, in both cases, that the internal and external constructions are equivalent, by using machinery built on top of the Grothendieck construction. The individual-token case is harder, as it requires a more explicit reliance on abstract methods.

中文翻译：

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有界Petri网的分类语义学。

在集体和个人令牌哲学中，我们为有界Petri网提供了分类语义。在这两种情况下，我们都描述了内部限制网络的过程，方法是仅使用comonads构造网络执行的新类别，而外部使用lax-monoidal-lax函子构造外部。我们的外部语义是非本地的，这意味着令牌具有赋予某些属性的功能，这些属性可以说明网络的整体状态。然后，通过使用在Grothendieck结构之上构建的机械，我们在两种情况下都证明内部结构和外部结构是等效的。单个令牌的情况比较困难，因为它需要更明确地依赖抽象方法。