Elements of Petri nets and processes

Abstract

We present a formalism for Petri nets based on polynomial-style finite-set configurations and etale maps. The formalism supports both a geometric semantics in the style of Goltz and Reisig (processes are etale maps from graphs) and an algebraic semantics in terms of free coloured props: the Segal space of P-processes is shown to be the free coloured prop-in-groupoids on P. There is also an unfolding semantics \`a la Winskel, which bypasses the classical symmetry problems. Since everything is encoded with explicit sets, Petri nets and their processes have elements. In particular, individual-token semantics is native, and the benefits of pre-nets in this respect can be obtained without the need of numberings. (Collective-token semantics emerges from rather drastic quotient constructions \`a la Best--Devillers, involving taking $\pi_0$ of the groupoids of states.)Comment: 44 pages. The math is intended to be in reasonably final form, but the paper may well contain some misconceptions regarding the place of this material in the theory of Petri nets. All feedback and help will be greatly appreciated. v2: fixed a mistake in Section