Studies on association schemoids with insights gained from cohomology theory of small categories

2015 Fiscal Year Final Research Report

• Project/Area Number: 25610002
• Research Category: Grant-in-Aid for Challenging Exploratory Research
• Allocation Type: Multi-year Fund
• Research Field: Algebra
• Research Institution: Shinshu University
• Principal Investigator: KURIBAYASHI Katsuhiko 信州大学, 学術研究院理学系, 教授 (40249751)
• Research Collaborator: MATSUO Kentaro ; MOMOSE Yasuhiro ; HANAKI Akihide 信州大学, 学術研究院理学系, 教授 (50262647) ; NUMATA Yasuhide 信州大学, 学術研究院理学系, 准教授 (00455685)
• Project Period (FY): 2013-04-01 – 2016-03-31
• Keywords: スキーモイド / 2-圏 / 森田同値 / 強ホモトピー / モデル圏 / Mitchell埋め込み定理 / アソシエーションスキーム / 小圏のコホモロジー

Outline of Final Research Achievements

We have proposed the notion of (association) schemoids generalizing that of association schemes, which are widely used in algebraic combinatorics, from a small categorical point of view. In our study, the equivalence between the categories of groupoids and that of thin schemoids is established. Moreover, in order to develop homotopy theory for schemoids, we define a homotopy relation on the category of quasi-schemoids and study its fundamental properties. In consequence, the 2-category of small categories is embedded into the 2-category of quasi-schemoids. As for categorical representation theory for schemoids, we have proved Mitchell's embedding theorem for a tame schemoid. The result allows us to give a cofibrantly generated model category structure to the category of chain complexes over a functor category with a schemoid as the domain. We show that every Hamming scheme of binary codes is Morita equivalent to the association scheme arising from the cyclic group of order two.

Free Research Field

トポロジー