Petri net → free (commutative) SMC → elementary WBS & schedules (toy: cliffside mountain refuge)

In planning, one often wants not just “the list of actions” but the trace of state (which literals/facts carry causal responsibility for later steps). A Petri marking is exactly that kind of trace-friendly state, and a schedule can be seen as a path in the reachability graph: a sequence of markings connected by firings. Treating markings as interchangeable “for now” corresponds to working in the collective-token setting where only multiplicities matter (no token identities).

A Petri net freely generates a (very strict) symmetric/commutative monoidal category: objects are markings; morphisms are firing sequences; tensor is “doing in parallel”; composition is “doing in series”. This turns “project space” into a genuine algebra: schedule options are morphisms, WBS options are factorizations of morphisms, and “milestones” are just intermediate objects you choose to cut on.

“Free” means: you haven’t imposed any equations other than those required by the symmetric/monoidal axioms and your net’s incidence data. So the category has maximal explanatory power: if two schedules become equal in this semantics, it’s because they differ only by rearrangements the axioms declare irrelevant (e.g., commutations of independent parts). If you later care about “who did what token” (individual-token philosophy), you can refine the semantics (e.g. Σ-nets / symmetric strict monoidal rather than commutative) without changing the base net.

A WBS is, structurally, a typed tree of “sub-works composed into a whole.” That is exactly operadic syntax: operations compose subsystems; algebras interpret that syntax in a semantics (cost, duration, risk, etc.). In this page, the Petri net gives the generating operations; your chosen cuts/parenthesizations give the operadic decomposition tree.

One recurring complaint in practice is that WBS trees and schedule networks drift out of sync; categorically, you want them to be two views of the same underlying “atom” set of activities, related functorially rather than manually. Here, both are computed from the same Petri-derived causality, so coherence is enforced by construction.

Choosing a cut marking M is choosing an object through which your schedule morphism factors: (Initial ⟶ M) ; (M ⟶ Goal). Iterating cuts yields a WBS tree: a rooted factorization diagram whose leaves are “atomic” transitions (or bundles of commuting ones).

If you view the activity network as a category enriched in the tropical/“max-plus” style, a schedule is a (co)presheaf assigning start times to objects so that every dependency inequality holds. Earliest-start is the least such assignment; latest-start is the greatest assignment compatible with the project makespan.

If you later regret treating tokens as indistinguishable, the usual move is not to abandon Petri nets but to lift to a setting that remembers permutation symmetries. Σ-nets do exactly this: instead of collapsing all permutations, you keep track of them as structure. In project terms: “two planks” vs “plank A/B” is the same underlying feasibility logic, but a different granularity of explanation and resource accounting.

The unglamorous reason open nets matter: WBS modules are rarely disjoint; they share boundary conditions (“permits obtained”, “materials delivered”, “anchor grout cured”). Open Petri nets let you model a module as a net-with-ports and then compose modules by gluing interfaces—so your WBS decomposition becomes literally a composition in a cospan/double-category sense, not an informal grouping.

Open Petri net.pdf (open nets + compositional semantics),
Categories of sigma nets .pdf (token philosophies + Σ-nets),
Project Scheduling and Copresheaves | The n-Category Café.pdf (schedules as copresheaves / tropical enrichment),
Using categorical logic for AI planning – Topos Institute.pdf (rewrite-rule planning in copresheaves; rule limits; hierarchy+concurrency motivation),
Operads for complex system design specification, analysis and synthesis.pdf (operadic syntax/semantics separation).

(i) Tracelet theory (Behr, Kock, Kirvine) for compositional rewriting explanations, and
(ii) tile systems (Montanari, Bruni) relating double categories to proof search/logic programming—both feel extremely close to “project execution as proof search” once you squint.

This page is already a browser-only, dependency-light prototype that treats Petri nets as the execution core and schedules as derived semantics. The stack above extends that exact approach using mature open-source components before moving to heavier academic/CLI tooling.