Categorical Framework for the Study of Singular Spaces
Categorical Framework for the Study of Singular Spaces
复制标题
奇异空间研究的分类框架
DOI：
10.1090/memo/0243
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发表时间：
1981
期刊：
影响因子：
--
通讯作者：
R. Macpherson
中科院分区：
文献类型：
--
作者：
W. Fulton;R. Macpherson
在几何和拓扑的几个领域中，传统的协变和逆变函子显然是不够的，特别是对于处理有关奇异空间的几何问题。我们在这里发展了一种新的形式主义，称为二元理论。这些是协变群值“类同调”理论和逆变环值“类同调”理论的同时推广。大多数传统的协变和逆变理论对都可以扩展到双变理论。双变理论不将群分配给对象，而是分配给原始范畴的态射。它具有与态射组合兼容的产品。我们还将定义从一种二变理论到另一种二变理论的变换，称为格洛腾迪克变换，它概括了普通的自然变换。许多标准自然变换被证明可以扩展到格洛腾迪克变换，并且这种扩展具有深远的影响。
In several areas of geometry and topology it has become apparent that the traditional covariant and contravariant functors are insufficient, particularly for dealing with geometric questions about singular spaces. We develop here a new formalism called bivariant theories. These are simultaneous generalizations of covariant group valued" homology-like" theories and contravariant ring valued" cohomology-like" theories. Most traditional pairs of covariant and contravariant theories turn out to extend to bivariant theories. A bivariant theory assigns a group not to an object but to a morphism of the original category; it has products compatible with composition of morphisms. We will also define transformations from one bivariant theory to another, called Grothendieck transformations, which generalize ordinary natural transformations. A number of standard natural transformations turn out to extend to Grothendieck transformations, and this extension has deep consequences.