Whole-grain Petri nets and processes

Petri nets are a useful model for concurrency, and also serve to
model processes in many other application areas in science and
engineering, such as chemistry, epidemiology, production and
business modelling, and so on. Their operational semantics come
in two main flavours: geometric and algebraic. People have
struggled for many years to reconcile the two viewpoints, the
problem being an issue with symmetries. In this talk I will
explain how the problem can be overcome with the help of some
elementary homotopy viewpoints, and a very slight adjustment to
the usual definition of Petri net. The new formalism for Petri
nets, called 'whole-grain', is based on polynomial-style
finite-set configurations and etale maps. The processes of a
whole-grain Petri net P are etale maps G -> P from graphs. The
main result I want to arrive at in the talk is that P-processes
(the geometric semantics) form a symmetric monoidal Segal space,
and that this is the free prop-in-groupoids on P (thus at the
same time the algebraic semantics). But most of the talk will be
spent just explaining Petri nets, markings, firings, and the
token game, and I will also spend some time on background in
simplicial homotopy theory.

Reference: "Whole-grain Petri nets and processes", J. ACM 70 (2023)
[https://doi.org/10.1145/3559103]