From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept
From Problems to Structures: the Cousin Problems and the Emergence of the Sheaf Concept
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DOI：
10.1007/s00407-009-0052-3
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发表时间：
2009
影响因子：
0.5
通讯作者：
Renaud Chorlay
中科院分区：
文献类型：
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作者：
Renaud Chorlay
关于轴理论出现的历史工作主要集中在1945年至1950年期间轴同构的拓扑起源以及随后的发展。然而，在时间尺度和学科背景下的重点转移可以帮助我们对轴概念的出现获得新的见解。本文集中讨论了亨利·卡尔坦在复数变量解析函数理论方面的工作，以及他在堆理论出现的两个阶段中所起的显著不同的作用：1940-1944年新结构的定义和新研究计划的制定；1951-1952年意外地融入了轴上同调。为了将这两阶段的结构转变带入视野，我们将更具体地集中在一系列问题上，即所谓的表亲问题，从庞卡罗（1883）到卡尔坦。这种中期叙述提供了对当代数学历史上两个更普遍的问题的见解。一是注重问题在理论建构中的运用。其次，在几何风味的环境中设计结构的历史——比如束状结构和纤维束结构——这将有助于提供一个更全面的结构主义时刻的观点，到目前为止，它的代数成分一直是历史工作的主要焦点。
Historical work on the emergence of sheaf theory has mainly concentrated on the topological origins of sheafcohomologyin the period from 1945 to 1950 and on subsequent developments. However, a shift of emphasis both in time-scale and disciplinary context can help gain new insight into the emergence of the sheafconcept. This paper concentrates on Henri Cartan’s work in the theory of analytic functions of several complex variables and the strikingly different roles it played at two stages of the emergence of sheaf theory: the definition of a new structure and formulation of a new research programme in 1940–1944; the unexpected integration into sheaf cohomology in 1951–1952. In order to bring this two-stage structural transition into perspective, we will concentrate more specifically on a family of problems, the so-called Cousin problems, from Poincaré (1883) to Cartan. This medium-term narrative provides insight into two more general issues in the history of contemporary mathematics. First, we will focus on the use of problems in theory-making. Second, the history of the design of structures in geometrically flavoured contexts—such as for the sheaf and fibre-bundle structures—which will help provide a more comprehensive view of thestructuralist moment, a moment whose algebraic component has so far been the main focus for historical work.