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.
翻译：我们从集体和个人的理论中为封闭的Petri网提供了明确的语义。 在这两种情况下,我们描述将网捆绑在内部的过程,仅仅通过使用comonads建造新的网络处决类别,而从外部,使用松绑的松绑的杀菌剂。我们的外部语义是非本地的,这意味着象征物具有描述网络全球状态的属性。 然后,在这两种情况下,我们用Grothendieck建筑顶端的机器来证明内部和外部建筑是等效的。 个人语则更难,因为它需要更明确地依靠抽象的方法。
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