Colored Petri Nets are Monoidal Double Functors
Jade Master, Joe Moeller
TL;DR
This work provides a category-theoretic account of colored Petri nets by modeling them as monoidal double functors from the underlying Petri net to Span(CMon), thereby turning syntax into semantics. It develops an operational semantics through the Grothendieck construction on a displayed category, yielding a category of executions that faithfully corresponds to firing sequences. A central result shows that unfolding colored Petri nets into ordinary Petri nets is free on unfolding, captured by a commuting diagram (up to isomorphism) between the semantics of the colored net and its unfolded form. The framework gives a canonical notion of morphisms as monoidal vertical transformations, connects to prior approaches with improved coherence by avoiding decategorification, and opens avenues for compositional open nets via structured cospans and monoidal displayed categories.
Abstract
We give a characterization of colored Petri nets as monoidal double functors. Framing colored Petri nets in terms of category theory allows for canonical definitions of various well-known constructions on colored Petri nets. In particular, we show how morphisms of colored Petri nets may be understood as natural transformations. The displayed category construction explains how lax double functors are equivalent to functors with codomain their former domain. We use this result to characterize the unfolding of colored Petri nets in terms of free symmetric monoidal categories.