Authors: John C. Baez ^1,2; John Foley ³; Joe Moeller ¹

• 1 Department of Mathematics, University of California, Riverside CA, 92521, USA
• 2 Centre for Quantum Technologies
• 3 Metron (United States)

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.

• 18M05 - Monoidal categories, symmetric monoidal categories
• 18N45 - Categories of fibrations, relations to \(K\)-theory, relations to type theory
• 68Q85 - Models and methods for concurrent and distributed computing (process algebras, bisimulation, transition nets, etc.)