A Classifying Topos for the Spectrum of Equivalences

Abstract: What makes two computational systems equivalent? Topos theory answers with classifying toposes: a system's semantic content is encoded in the geometric theory it classifies, and two presentations are equivalent when their classifying toposes coincide. Process algebra answers with the linear time-branching time spectrum of van Glabbeek: a hierarchy of behavioral equivalences from trace equivalence to bisimilarity, each determined by which observations can distinguish processes. We show these are aspects of a single structure in which behavioral abstraction is localization. Each labeled transition system receives a geometric theory $\mathbb{T}<em>M$ whose classifying topos $\mathcal{E}[\mathbb{T}_M]$ determines its provable geometric sequents. Mutual simulation is strictly coarser than bisimulation, strictly coarser than topos equivalence; diamond-only Hennessy-Milner logic characterizes the bisimulation-invariant fragment of geometric logic -- a geometric van Benthem theorem. Grothendieck topologies yield $$J</em>{\mathrm{bisim}} \subsetneq J_{\mathrm{sim}} \subsetneq J_{\mathrm{trace}}$$, constructive for trace and bisimulation; a counterexample shows the observation-class approach inadequate for simulation, motivating Caramello's duality. Energy-topology extends this to all 13 named equivalences. Lattice closure yields 30 elements including 17 unnamed hybrids absent because the energy-game framework computes but does not close. $L_{30}$ is indecomposable with $S \to F = \mathrm{IF}$; a Geometric Closure Theorem computes presheaf Heyting implications at a single free extension. The hierarchy, bi-Heyting structure, and Closure Theorem are proved constructively with no known process-algebraic proof. The spectrum is a finite sub-poset of an infinite coframe whose operations (meets, implications, subtractions) yield structure inaccessible from process algebra. Formalized in Lean 4/Mathlib.

• The paper constructs a classifying-topos framework in which behavioral abstraction becomes localization, representing equivalences as Grothendieck topologies on labeled transition systems and proving the strict chain from bisimulation through simulation to trace semantics.
• The paper characterizes the bisimulation-invariant fragment of geometric logic as diamond-only Hennessy–Milner logic, with determinism, cyclic variable sharing, and repeated variables explaining how broader geometric logic distinguishes bisimilar systems.
• The paper closes 13 named equivalences under lattice operations to obtain a 30-element spectrum with 17 unnamed hybrids, while formalizing major results in Lean 4 and identifying open questions about a coordinatization-free topos embedding.

Overview

This paper constructs a topos-theoretic framework in which the van Glabbeek linear time–branching time spectrum of behavioral equivalences is realized as a finite sub-poset of the coframe of subtoposes of a classifying topos of labeled transition systems. The central claim is that behavioral abstraction is localization: each equivalence in the spectrum corresponds to a Grothendieck topology on a fixed site $\mathrm{f.p.LTS}_L$, and the passage between equivalences is a change of topology rather than an ad hoc operational definition. The paper develops three mutually reinforcing pillars—a geometric pillar of explicit topologies, an algebraic pillar of nuclei on Lindenbaum algebras, and a logical pillar identifying the bisimulation-invariant fragment of geometric logic—and formalizes all headline results in Lean 4 with Mathlib.

Geometric theories for transition systems and the separation hierarchy

Each rooted transition system $M$ receives a finitary geometric theory $T_M$ over a language with constants for states and predicates for transition, reachability, and path equivalence. The theory comprises six structural axioms plus system-specific existence, completeness, negative, and domain-closure axioms, totaling $O(|S_M|^2)$ sequents for finite systems. A soundness-and-completeness lemma establishes that geometric provability from $T_M$ coincides with semantic validity, using Deligne's theorem for enough Set-points. Provable geometric sequents serve as the invariant distinguishing classifying toposes.

On this foundation the paper proves a strict hierarchy by explicit finite witnesses:

• First Separation: the fork $F$ (which can halt) and the path $P$ (which cannot) mutually simulate but are not bisimilar; the totality sequent $\sigma_{\mathrm{tot}}$ is provable in $T_P$ but not $T_F$, so $M$0.
• Second Separation: the hub-spokes system $M$1 and two-cycle $M$2 are relationally bisimilar, yet the determinism sequent $M$3 separates their theories, so $M$4.
• Functional vs. relational bisimulation: the self-loop $M$5 and two-cycle $M$6 are relationally bisimilar but admit no functional bisimulation, yielding four strict levels: mutual simulation ⊋ relational bisimulation ⊋ functional bisimulation ⊋ topos equivalence.

The Second Separation has a pointed consequence: system-level bi-interpretation does not imply Morita equivalence of the associated geometric theories. A Quotient Bridge theorem partially repairs this—functional bisimilarity implies equivalent classifying toposes of the path-equivalence quotients—and the assignment $M$7 extends to a contravariant 2-functor into Grothendieck toposes, with each geometric morphism factoring as hyperconnected followed by localic.

The geometric van Benthem theorem

The paper identifies the exact boundary between what bisimulation can see and what geometric logic can express: the bisimulation-invariant fragment of geometric logic is precisely diamond-only Hennessy–Milner logic. The forward direction (HML invariance) is constructive; the converse is proved via bounded tree unraveling, using characteristic formulas on trees and a case split resolved by a decision procedure on the finite unraveling.

Three syntactic mechanisms account for all ways geometric logic exceeds HML, each witnessed by a bisimilar pair:

1. Consequent equality ($M$8): equality between universally bound variables, separating $M$9 from $T_M$0.
2. Cyclic variable sharing ($T_M$1, weak confluence): a join constraint creating a cycle in the variable dependency graph, separating the diamond graph from the confluence tree.
3. Repeated variables ($T_M$2, universal self-loop): source equals target, separating $T_M$3 from $T_M$4.

All three assert that two structural positions are filled by the same state—an equalizer condition in the syntactic category that bisimulation systematically breaks by duplicating states. Exhaustive verification at depths 0, 1, and 2 confirms the characterization constructively at low depth: at depth 1 only $T_M$5 survives among seven atoms; at depth 2 only $T_M$6 among eight. The paper notes honestly that the gap between the syntactic observation (three violations of equality-freeness, linearity, tree-shape) and the full formula-level decomposition is where the proof effort lies, and that scaling beyond depth 2 by brute enumeration would face coNP/PSPACE-complete characteristic-formula complexity.

The spectrum lattice $T_M$7

Parameterizing the 13 named equivalences by Bisping's 6-dimensional energy vectors, the paper closes the resulting set under componentwise min/max. The closure is not the original set: it stabilizes after three rounds into a 30-element lattice containing 17 unnamed hybrids absent from the classical spectrum. This is a substantive claim: the named spectrum is algebraically incomplete, and the missing elements are discovered purely by lattice closure.

$T_M$8 exhibits strong structure:

• It is directly indecomposable (its join-irreducible poset is connected, since traces lie below every join-irreducible), despite Bisping's framework treating the six energy dimensions as independent parameters.
• Its negation is maximally degenerate—$T_M$9 for every $O(|S_M|^2)$0, Boolean core $O(|S_M|^2)$1—yet its Heyting implication is fully informative: among 32 incomparable ordered pairs, every implication is nontrivial, and the striking identity

$O(|S_M|^2)$2

identifies impossible futures as the algebraic mediator of the simulation–failures divide.

• Co-Heyting subtractions yield operationally meaningful decompositions such as $O(|S_M|^2)$3 and $O(|S_M|^2)$4; notably, naive componentwise subtraction in the product frame never coincides with the exact Birkhoff computation.

An important caveat is stated plainly: $O(|S_M|^2)$5 depends on Bisping's coordinatization, not merely on the van Glabbeek poset—the free distributive lattice on the 13-element poset has 13 join-irreducibles versus $O(|S_M|^2)$6's 10, so the embedding introduces accidental join-relations. The topos-theoretic embedding offers a potential coordinatization-free alternative, contingent on the conjecture below.

Explicit Grothendieck topologies and the Energy–LT Bridge

On the fixed site $O(|S_M|^2)$7, the paper defines $O(|S_M|^2)$8 via path digraphs and $O(|S_M|^2)$9 via rooted trees, proving all three Grothendieck axioms constructively, and establishes the Spectrum Bracket: any observation class between paths and trees yields a topology sandwiched $T_M$0. Via the Comparison Lemma, the trace sheaf topos identifies with $T_M$1, recovering the Cattani–Winskel presheaf model.

Simulation is the structural outlier. A constructive counterexample shows the naive existence-based covering predicate fails stability under pullback, reflecting a truncation-level mismatch: trace and bisimulation operate at sieve level (0-truncated), while simulation requires only homomorphism existence ((−1)-truncated). Consequently $T_M$2 is obtained indirectly through Caramello's quotient-theory duality, completing the strict chain

$T_M$3

and this indirection is provably necessary for the observation-class approach—not an artifact of technique. An energy-topology pipeline extends the embedding to all 13 named equivalences (antitone, injective for $T_M$4); for $T_M$5 the spectrum collapses to two distinct topologies. The coframe meet $T_M$6 provably escapes the image of the spectrum, witnessing that the subtopos coframe strictly exceeds process algebra.

The Geometric Closure Theorem

At the presheaf level, positive existential HML formulas define subfunctors of the generic underlying-set functor. Negation collapses entirely—every satisfiable $T_M$7-definable subfunctor has $T_M$8—mirroring the degenerate negation of $T_M$9; both instances share the mechanism of a common lower bound forcing complementation to vanish. Yet Heyting implication remains informative, and the Geometric Closure Theorem reduces it to a single test:

$F$0

where $F$1 adjoins a minimal witness tree. This replaces Kripke–Joyal's universal quantification over all extensions with one canonical free extension, and yields four qualitatively distinct implication regimes at depth ≤ 2. The paper claims no known process-algebraic proof of this result.

Limitations and open questions

Several boundaries are conceded explicitly. The simulation topology lacks explicit covering sieves; Caramello–Lafforgue's recent generation techniques are proposed as a route. The full $F$2 embedding into the subtopos coframe is conjectural—only the three-element chain is established—and a positive answer would resolve the coordinatization dependence on Bisping's parameterization. Whether every topology between $F$3 and $F$4 arises from an observation class remains open. The bridge identity connecting Lindenbaum fixpoints to nuclei rests on van Glabbeek's logical characterizations, whose proofs use classical König lemma arguments, so full constructivity for all 13 levels is unverified. Deadlock-sensitive equivalences fall outside the current totality axiom convention, and quantitative models would require enrichment beyond geometric sequents. Finally, whether the $F$5-sheaf-theoretic equivalence is exactly trace equivalence or strictly finer (as the Cattani–Winskel correspondence suggests) is unresolved, as is the spatiality of the colimit locale for non-image-finite processes.

Conclusion

The paper demonstrates that the van Glabbeek spectrum embeds as a finite sub-poset of the coframe of subtoposes of a classifying topos, with localization as the uniform mechanism of behavioral abstraction. Its most concrete contributions are the strict three-topology chain with a proven obstruction for naive simulation coverings, the 30-element lattice closure exposing 17 unnamed equivalences and the identity $F$6, the geometric van Benthem characterization with its three-mechanism taxonomy, and the free-extension reduction of presheaf Heyting implication—all with substantial zero-axiom Lean 4 formalization. The central open question, whether the embedding extends to a full sublattice of the coframe, would determine if the spectrum lattice admits a coordinatization-free, intrinsically topos-theoretic characterization.