Functorial models for Petri nets

We show that although the algebraic semantics of place/transition Petri nets under the collective token philosophy can be fully explained in terms of strictly symmetric monoidal categories, the analogous, construction under the individual token philosophy is not completely satisfactory, because it lacks universality and also functoriality. We introduce the notion of pre nets to overcome this, obtaining a fully satisfactory categorical treatment, where the operational semantics of nets yields an adjunction. This allows us to present a uniform logical description of net behaviors under both the collective and the individual token philosophies in terms of theories and theory morphisms in partial membership equational logic. Moreover, since the universal property of adjunctions guarantees that colimit constructions on nets are preserved in our algebraic models, the resulting semantic framework has good compositional properties.