nLab pre-net

Idea

A pre-net is a Petri net equipped with an ordering on the input and output of each event. These give appropriate data from which to freely generate a strict monoidal category or a symmetric strict monoidal category, whereas a Petri net is appropriate data from which to freely generate a commutative monoidal category. Pre-nets are the same as the tensor schemes defined by Joyal and Street.

Definition

A pre-net is a pair of functions

where $E$ is the set of events, $P$ is the set of places, and $P^*$ is the free monoid on the set $P$.

A morphism of of pre-nets $(f \colon E \to E', g \colon P \to P')$ is a pair of functions between the sets of events and places making the following diagrams commute

This defines a category $PreNet$ of pre-nets and pre-net homomorphisms.

References

Pre-nets were introduced in

• Roberto Bruni?, José Meseguer, Ugo Montanari?, and Vladimiro Sassone. Functorial models for Petri nets Information and Computation 170, no. 2 (2001): 207-236. pdf

Tensor schemes were introduced in

• André Joyal, and Ross Street. The geometry of tensor calculus I Advances in mathematics 88, no. 1 (1991): 55-112. pdf