Network Models from Petri Nets with Catalysts
John C. Baez, John Foley, Joe Moeller
TL;DR
The paper develops a framework that fuses Petri nets with network models using catalysts. It constructs a network model $G:\mathsf{S}(C)\to\mathsf{Cat}$ from a Petri net $P$ with catalyst set $C$, and forms the symmetric monoidal category $\int G$ via the Grothendieck construction to capture parallel processes while preserving catalyst individuality in fibers. It further shows that catalyst-graded fibers $FP_i$ carry strict premonoidal structures, and lifts $G$ to $\hat{G}:\mathsf{S}(C)\to\mathsf{PreMonCat}$, connecting Petri nets, premonoidal categories, and network design. By highlighting the distinction between the individuality of catalyst tokens and the collective counting of species, the work suggests rich avenues for diagrammatic formalisms and functorial enrichments in compositional system modeling.
Abstract
Petri networks and network models are two frameworks for the compositional design of systems of interacting entities. Here we show how to combine them using the concept of a "catalyst": an entity that is neither destroyed nor created by any process it engages in. In a Petri net, a place is a catalyst if its in-degree equals its out-degree for every transition. We show how a Petri net with a chosen set of catalysts gives a network model. This network model maps any list of catalysts from the chosen set to the category whose morphisms are all the processes enabled by this list of catalysts. Applying the Grothendieck construction, we obtain a category fibered over the category whose objects are lists of catalysts. This category has as morphisms all processes enabled by some list of catalysts. While this category has a symmetric monoidal structure that describes doing processes in parallel, its fibers also have premonoidal structures that describe doing one process and then another while reusing the catalysts.