{"title":"The category of Z−graded manifolds: What happens if you do not stay positive","authors":"Alexei Kotov , Vladimir Salnikov","doi":"10.1016/j.difgeo.2024.102109","DOIUrl":null,"url":null,"abstract":"<div><p>In this paper we discuss the categorical properties of <span><math><mi>Z</mi></math></span>-graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the <span><math><mi>N</mi></math></span>-graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make this construction intrinsic using a new type of filtrations. This sums up to proper definitions of objects and morphisms in the category. We also formulate the analog of the Borel's lemma for the functional spaces on <span><math><mi>Z</mi></math></span>-graded manifolds and the analogue of Batchelor's theorem for the global structure of them.</p></div>","PeriodicalId":51010,"journal":{"name":"Differential Geometry and its Applications","volume":"93 ","pages":"Article 102109"},"PeriodicalIF":0.6000,"publicationDate":"2024-02-01","publicationTypes":"Journal Article","fieldsOfStudy":null,"isOpenAccess":false,"openAccessPdf":"https://www.sciencedirect.com/science/article/pii/S0926224524000020/pdfft?md5=a5f043ddb99e39117ce84444d49aff31&pid=1-s2.0-S0926224524000020-main.pdf","citationCount":"0","resultStr":null,"platform":"Semanticscholar","paperid":null,"PeriodicalName":"Differential Geometry and its Applications","FirstCategoryId":"100","ListUrlMain":"https://www.sciencedirect.com/science/article/pii/S0926224524000020","RegionNum":4,"RegionCategory":"数学","ArticlePicture":[],"TitleCN":null,"AbstractTextCN":null,"PMCID":null,"EPubDate":"","PubModel":"","JCR":"Q3","JCRName":"MATHEMATICS","Score":null,"Total":0}
引用次数: 0
Abstract
In this paper we discuss the categorical properties of -graded manifolds. We start by describing the local model paying special attention to the differences in comparison to the -graded case. In particular we explain the origin of formality for the functional space and spell-out the structure of the power series. Then we make this construction intrinsic using a new type of filtrations. This sums up to proper definitions of objects and morphisms in the category. We also formulate the analog of the Borel's lemma for the functional spaces on -graded manifolds and the analogue of Batchelor's theorem for the global structure of them.
在本文中，我们将讨论 Z 级流形的分类属性。我们首先描述了局部模型，并特别关注了与 N 级流形的不同之处。我们特别解释了函数空间形式化的起源，并阐明了幂级数的结构。然后，我们利用一种新型滤波使这一结构内在化。这就总结出了该范畴中对象和变形的正确定义。我们还为 Z 级流形上的函数空间提出了类似的伯勒尔定理（Borel's lemma），并为它们的全局结构提出了类似的巴切洛定理（Batchelor's theorem）。
期刊介绍：
Differential Geometry and its Applications publishes original research papers and survey papers in differential geometry and in all interdisciplinary areas in mathematics which use differential geometric methods and investigate geometrical structures. The following main areas are covered: differential equations on manifolds, global analysis, Lie groups, local and global differential geometry, the calculus of variations on manifolds, topology of manifolds, and mathematical physics.
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