2018 Fiscal Year Final Research Report
Studies on stratifolds by a categorical approach
| Project/Area Number |
16K13753
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| Research Category |
Grant-in-Aid for Challenging Exploratory Research
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| Allocation Type | Multi-year Fund |
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| Research Field |
Geometry
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| Research Institution | Shinshu University |
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Principal Investigator |
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| Co-Investigator(Kenkyū-buntansha) |
岸本 大祐 京都大学, 理学研究科, 准教授 (60402765)
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| Research Collaborator |
AOKI Toshiki
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| Project Period (FY) |
2016-04-01 – 2019-03-31
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| Keywords | 階層体 / ディフェオロジカル空間 / Serre-Swanの定理 / 実スペクトラム |
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| Outline of Final Research Achievements |
The notion of a stratifold introduced by Kreck is a generalization of that of a smooth manifold. The aim of this research is to clarify categorical properties of stratifolds by comparing them with those of diffeological spaces whose notion also contains all smooth manifolds. As consequence, we have obtained a result (I) which asserts that the category of stratifolds fully and faithfully embeds into the category of R-algebras as does the category of smooth manifolds. (II) We also characterize morphisms of stratifolds in the category of diffeological spaces with an appropriate functor. The result (I) yields that (III) a stratifold recovers from the pullback along the inclusion from the real spectrum associated with the stratifold to the affine scheme as a ringed space. Moreover, we have proved the equivalence between vector bundles over a stratifold and finitely generated projective modules over the global sections; that is, (IV) the Serre-Swan theorem holds for stratifolds.
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| Free Research Field |
幾何学
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| Academic Significance and Societal Importance of the Research Achievements |
球面のように、局所的にユークリッド空間と同一視できる幾何学的対象は多様体と呼ばれ、その豊かな構造から微分幾何学やトポロジーにおける重要な研究対象となっている。次元の異なる幾つかの多様体を接着することで得られる対象が、階層体である。本研究では、階層体のつくる圏のR-代数の圏へのうめ込み定理や、階層体のR-代数上のアフィン・スキームの引き戻しにより表示する方法を確立した。さらに、多様体の概念の一般化であるディフェオロジカル空間がつくる圏の中で、階層体の間の射の特徴づけを与えた。階層体の圏論的立場から研究することで、階層体および多様体に潜む幾何学的な本質があぶり出されることになる。
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