Categories and Sheaves
Categories and Sheaves
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DOI：
10.1007/3-540-27950-4
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发表时间：
2005-10
期刊：
影响因子：
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通讯作者：
柏原 正樹;P. Schapira
中科院分区：
文献类型：
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作者：
柏原 正樹;P. Schapira
自二十世纪中叶以来，尤其是在格罗森迪克的思想从代数几何传播到许多其他学科之后，数学的语言发生了巨大的变化。作为集合和函数概念的丰富，范畴和集合是当今几乎随处可见的新工具，有时只是作为一种有用的语言，但往往是加深对数学理解的自然途径。范畴理论由艾伦伯格和麦克·莱恩在40年代创立(见[19，20])，它可能被视为超越数学的更广泛运动的一部分，而知识各个领域的结构主义可能是其中的另一个方面。在范畴出现之前，人们习惯于处理具有给定结构的给定集合(例如拓扑空间)，并研究其性质。分类的观点本质上是不同的。重点不是放在对象上，而是放在范畴内对象之间的关系(态射)上。语言是自然的，它允许人们统一数学的不同分支，并在看似不同的学科之间建立意想不到的联系。范畴理论是基本的，因为它几乎没有学习的先决条件，尽管对许多人来说，它可能看起来抽象得令人望而生畏。的确，通常的数学教育课程并不利于这种概念性的思维方式。大多数数学家习惯于处理空间和函数、计算积分等，很少有人理解等式和同构之间区别的重要性，或者欣赏图的美和效率。另一个基本概念是一捆。许多数学(和物理)都围绕着这样的问题，提供了一个从局部到全局的工具。层允许我们研究局部存在但不是全局存在的对象，例如Riemann球面上的全纯函数或Möbius带上的方向，而层的上同调在某种意义上度量了从局部到全局传递的障碍。
The language of Mathematics has changed drastically since the middle of the twentieth century, in particular after Grothendieck’s ideas spread from algebraic geometry to many other subjects. As an enrichment for the notions of sets and functions, categories and sheaves are new tools which appear almost everywhere nowadays, sometimes simply in the role of a useful language, but often as the natural approach to a deeper understanding of mathematics. Category theory, initiated by Eilenberg and Mac Lane in the forties (see [19, 20]), may be seen as part of a wider movement transcending mathematics, of which structuralism in various areas of knowledge is perhaps another facet. Before the advent of categories, people were used to working with a given set endowed with a given structure (a topological space for example) and to studying its properties. The categorical point of view is essentially different. The stress is placed not upon the objects, but on the relations (the morphisms) between objects within the category. The language is natural and allows one to unify various branches of mathematics and to make unexpected links between seemingly different subjects.Category theory is elementary in the sense that there are few prerequisites to its study, though it may appear forbiddingly abstract to many people. Indeed, the usual course of mathematical education is not conducive to such a conceptual way of thinking. Most mathematicians are used to manipulating spaces and functions, computing integrals and so on, fewer understand the importance of the difference between an equality and an isomorphism or appreciate the beauty and efficiency of diagrams. Another fundamental idea is that of a sheaf. Sheaves provide a tool for passing from local to global situations and a good deal of mathematics (and physics) revolves around such questions. Sheaves allow us to study objects that exist locally but not globally, such as the holomorphic functions on the Riemann sphere or the orientation on a Möbius strip, and the cohomology of sheaves measures in some sense the obstruction to passing from local to global.