Gödel coding on fibrations and geminal categories
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arXiv
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| Format: | Preprint |
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2026
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| _version_ | 1866917551335800832 |
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| author | Ikeda, Yuto |
| author_facet | Ikeda, Yuto |
| contents | Ramesh's 2023 dissertation introduces the categorical notions of introspective theories and geminal categories, which formalize "self-internalizing" structures sharing the form of Löb's theorem ($\Box A \vdash A$ implies $\vdash A$). We reorganize the theory of geminal categories in a self-contained manner by introducing "code structures on fibrations," which serve as a categorical abstraction of Gödel coding. This framework leads to a significant simplification of the proof of Löb's theorem for geminal categories, as well as to a new categorical counterpart of the Gödel-Löb axiom ($\Box(\Box A \to A) \to \Box A$). This formulation offers an accessible framework for Ramesh's approach and suggests connections to modal type theories, where similar meta- and object-level interactions arise. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2606_01165 |
| institution | arXiv |
| publishDate | 2026 |
| record_format | arxiv |
| spellingShingle | Gödel coding on fibrations and geminal categories Ikeda, Yuto Logic Logic in Computer Science Category Theory 03G30 (Primary) 03F40, 03F45, 18D30, 18D40, 18C10 (Secondary) F.4.1 Ramesh's 2023 dissertation introduces the categorical notions of introspective theories and geminal categories, which formalize "self-internalizing" structures sharing the form of Löb's theorem ($\Box A \vdash A$ implies $\vdash A$). We reorganize the theory of geminal categories in a self-contained manner by introducing "code structures on fibrations," which serve as a categorical abstraction of Gödel coding. This framework leads to a significant simplification of the proof of Löb's theorem for geminal categories, as well as to a new categorical counterpart of the Gödel-Löb axiom ($\Box(\Box A \to A) \to \Box A$). This formulation offers an accessible framework for Ramesh's approach and suggests connections to modal type theories, where similar meta- and object-level interactions arise. |
| title | Gödel coding on fibrations and geminal categories |
| topic | Logic Logic in Computer Science Category Theory 03G30 (Primary) 03F40, 03F45, 18D30, 18D40, 18C10 (Secondary) F.4.1 |
| url | https://arxiv.org/abs/2606.01165 |