Separation and Gluing of Explanations on Sites of Dynamical Systems
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arXiv
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| Format: | Preprint |
| Published: |
2026
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| _version_ | 1866917351144816640 |
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| author | Wang, Paul Z. |
| author_facet | Wang, Paul Z. |
| contents | We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2603_17141 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Separation and Gluing of Explanations on Sites of Dynamical Systems Wang, Paul Z. Category Theory We construct a Grothendieck site whose objects are Mealy machines over definable sets in an o-minimal structure and whose coverings are jointly surjective families of definable open immersions. On this site, we define presheaves of explanations -- systems equipped with an interpretable interface, parameterised by a ``judge.'' We prove that the behavioral presheaf (quotienting by observable output equivalence) is separated: a global explanation is determined by its local restrictions. We show that gluing fails in general -- locally consistent explanations need not assemble globally -- and give, for stateless explanatory systems of the restricted-interface presheaf, a necessary and sufficient topological condition for the sheaf property in terms of robust disconnection of fibers of the judge. |
| title | Separation and Gluing of Explanations on Sites of Dynamical Systems |
| topic | Category Theory |
| url | https://arxiv.org/abs/2603.17141 |