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Sheaf
Sheaf ฤ เอ๊ะเรื่องใด โปรดจารไข สักหน่อยนะ อย่ายาก ถึงเลิกละ "ฐาน"จะจะ มั่นคงจริง |
In mathematics, a sheaf is a tool for systematically tracking locally defined data attached to the open sets of a topological space. The data can be restricted to smaller open sets, and the data assigned to an open set is equivalent to all collections of compatible data assigned to collections of smaller open sets covering the original one. For example, such data can consist of the rings of continuous or smooth real-valued functions defined on each open set. Sheaves are by design quite general and abstract objects, and their correct definition is rather technical. They are variously defined, for example, as sheaves of sets or sheaves of rings, depending on the type of data assigned to open sets. |
Sheaf theory
2010 Mathematics Subject Classification: Primary: 14-XX [MSN][ZBL] A special mathematical tool which provides a unified approach to establishing connections between local and global properties of topological spaces (in particular geometric objects) and which is a powerful method for studying many problems in contemporary algebra, geometry, topology, and analysis. |
Sheaves have several applications in topology and especially in algebraic and differential geometry. First, geometric structures such as that of a differentiable manifold or a scheme can be expressed in terms of a sheaf of rings on the space. In such contexts several geometric constructions such as vector bundles or divisors are naturally specified in terms of sheaves. Second, sheaves provide the framework for a very general cohomology theory, which encompasses also the "usual" topological cohomology theories such as singular cohomology. |
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