2021 Fiscal Year Final Research Report

Categorical and Higher-Categorical Approaches to Duality and Semantics across Mathematics, Physics, and Information

Research Project

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Project/Area Number	17K14231
Research Category	Grant-in-Aid for Young Scientists (B)
Allocation Type	Multi-year Fund
Research Field	Foundations of mathematics/Applied mathematics
Research Institution	Advanced Telecommunications Research Institute International (2020-2021) Kyoto University (2017-2018)
Principal Investigator	Maruyama Yoshihiro 株式会社国際電気通信基礎技術研究所, 佐藤匠徳特別研究所, 研究員 (20761290)
Project Period (FY)	2021-03-01 – 2022-03-31
Keywords	圏論的双対性 / 圏論的論理 / 圏論的人工知能 / 圏論的機械学習 / 圏論的認知科学 / 圏論的言語学 / 圏論的量子力学 / 圏論的統一科学
Outline of Final Research Achievements	We have developed a universal categorical duality theorem encompassing all Stone dualities for coherent spaces and applied it to computer science, by which we won a paper prize at an international conference on computational intelligence. We have discovered a novel infinitary Stone duality theorem which replaces the known infinitary Stone duality, which was published in a major journal in algebra, and further developed a general theory of categorical dualities which encompasses those infinitary Stone dualities, which was published in a top journal in category theory. To extend topos theory for arbitrary logical systems, we have generalised and applied tripos theory, thus having discovered a general theory of categorical logics, which was published in a top journal in mathematical logic.
Free Research Field	圏論的双対性、圏論的論理、圏論的人工知能、圏論的機械学習、圏論的量子力学、圏論的統一科学
Academic Significance and Societal Importance of the Research Achievements	圏論的双対性の普遍理論を構築し、圏論のトップジャーナル等において出版すると共に、計算知能に関する国際会議で論文賞を受賞した。圏論的論理の普遍理論を構築し、数理論理学のトップジャーナル等において出版し、トライポス理論を通じたトポス理論の拡張に成功した。これらは、純粋数学としての革新と同時に、人工知能・機械学習や量子物理・量子情報への応用を企図したものであり、従来の枠組みとは異なる全く新しい角度から、圏論に基づく人工知能と量子情報の新地平を切り拓くものとして非常に大きな可能性を秘めたものである。専門的な国際誌と同時に、成果の社会への普及のため数学セミナーや現代思想など一般向け雑誌にも多数寄稿した。