A Classifying Topos for the Spectrum of Equivalences
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arXiv
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| Formato: | Preprint |
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2026
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| _version_ | 1866910036902543360 |
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| author | Oggad, Kenan |
| author_facet | Oggad, Kenan |
| contents | 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 $\mathbb{T}_M$ 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 $J_{\mathrm{bisim}} \subsetneq J_{\mathrm{sim}} \subsetneq J_{\mathrm{trace}}$, 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2603_01056 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | A Classifying Topos for the Spectrum of Equivalences Oggad, Kenan Logic in Computer Science Category Theory Logic 18B25, 68Q85 (primary), 03G30, 06D22 (secondary) F.4.1; F.3.2 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 $\mathbb{T}_M$ 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 $J_{\mathrm{bisim}} \subsetneq J_{\mathrm{sim}} \subsetneq J_{\mathrm{trace}}$, 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. |
| title | A Classifying Topos for the Spectrum of Equivalences |
| topic | Logic in Computer Science Category Theory Logic 18B25, 68Q85 (primary), 03G30, 06D22 (secondary) F.4.1; F.3.2 |
| url | https://arxiv.org/abs/2603.01056 |