Formal Relationships Between Geometrical and Classical Models for Concurrency

A wide variety of models for concurrent programs has been proposed during the past decades, each one focusing on various aspects of computations: trace equivalence, causality between events, conflicts and schedules due to resource accesses, etc. More recently, models with a geometrical flavor have been introduced, based on the notion of cubical set. These models are very rich and expressive since they can represent commutation between any bunch of events, thus generalizing the principle of true concurrency. While they seem to be very promising - because they make possible the use of techniques from algebraic topology in order to study concurrent computations - they have not yet been precisely related to the previous models, and the purpose of this paper is to fill this gap. In particular, we describe an adjunction between Petri nets and cubical sets which extends the previously known adjunction between Petri nets and asynchronous transition systems by Nielsen and Winskel.

💡 Research Summary

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The paper “Formal Relationships Between Geometrical and Classical Models for Concurrency” establishes precise categorical connections between a modern geometric model of concurrency—cubical sets (also known as higher‑dimensional automata, HDA)—and several classical models: transition systems (TS), asynchronous transition systems (ATS), event structures (ES), and Petri nets (PN). The authors begin by reviewing each classical model, emphasizing the aspects each captures: TS model nondeterministic interleaving, ATS adds pairwise independence, ES encodes binary conflict and conjunctive causality, and PN captures token flow and a built‑in notion of parallelism. They then introduce cubical sets, describing n‑cells as n‑dimensional cubes equipped with source, target, and degeneracy maps (∂⁻, ∂⁺, ι). Precubical sets allow “partial” cells, while cubical sets are the total, fully defined version. By viewing the cubical category 𝔾 as the free monoidal category generated by a co‑cubical object, they obtain a compact algebraic description of the face and degeneracy operators.

The core contribution is a family of adjunctions between the categories of the classical models and the category of cubical sets (CSet). For each classical model a left adjoint F (free functor) embeds the model into a cubical set by “completing” partial morphisms into total ones, while the right adjoint G extracts a classical structure from a cubical set by interpreting cells as states, transitions, or events. These adjunctions preserve limits, ensuring that pull‑back based parallel composition and other standard constructions survive the translation.

The most notable result is a new adjunction between Petri nets and cubical sets, which extends the well‑known Nielsen‑Winskel adjunction between Petri nets and asynchronous transition systems. In this construction, the degree of parallelism in a Petri net (the number of tokens that may fire simultaneously) corresponds to the dimension of a cube in the HDA; the source/target maps of a cell encode the pre‑ and post‑conditions of a transition, and the degeneracy maps capture the ability to “add idle dimensions”. This adjunction shows that every Petri net can be faithfully represented as a cubical set, and conversely, any cubical set satisfying certain coherence conditions yields a Petri net. Consequently, analysis techniques developed for one model (e.g., deadlock detection algorithms for HDA, invariant generation for Petri nets, sleep‑set reduction for ATS) can be transferred to the other without loss of semantic information.

Technical details include a monoidal reformulation of the cubical category, a Kleisli‑category description of the passage from precubical to cubical sets (mirroring the relationship between partial and total functions), and an explicit construction of the event equivalence relation ≈ on 1‑cells that recovers the notion of an “event” from a precubical set. The authors also discuss variations of each model (labelled vs. unlabelled, strict vs. partial morphisms, multiplicities) and argue that strongly labelled HDA provide the most suitable target for comparison.

In the concluding section the authors highlight that their work not only clarifies the subtle differences among the models but also opens avenues for further research: establishing direct adjunctions between event structures and Petri nets, integrating directed algebraic topology tools (e.g., homotopy invariants) into concurrency analysis, and building unified toolchains that exploit the categorical bridges. Overall, the paper delivers a rigorous, category‑theoretic framework that unifies geometric and classical concurrency semantics, enabling cross‑model reasoning and the reuse of verification techniques across disparate formalisms.