Principal bundles, groupoids, and connections
Principal bundles, groupoids, and connections
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主束、群群和连接
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发表时间：
2007
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通讯作者：
A. Kock
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作者：
A. Kock
我们澄清了主丛理论和群胚理论在何种精确意义上是等价的;以及在可微情况下，理论的这种等价性如何在联络理论中反映出来。所用的方法是综合微分几何。导论.在本注记中，我们明确了主纤维丛理论等价于群胚理论的意义;特别是，联系的微分几何概念如何出现在这种等价中。对于后者，我们将利用综合微分几何的方法，它的基础是流形的“对角线的第一邻域”的概念。基本上，“主丛”和“广群”的概念，它们的本质等价性，以及在此上下文中的联络的概念，由Ehresmann，[4]，[5]等描述。经典的是用主G-丛中的联络来表达电磁场的概念，G是（交换）群U（1）。最近的粒子物理学正在将这种“规范理论”的观点扩展到非对易G，以及更高的“连接结构”，参见例如[3]和那里的参考文献。为了科普在这个扩展过程中出现的数学复杂性，Breen和Messing [2]，[3]发现利用过去几十年中阐述的一些形式或“合成”方法是有帮助的（例如[7]）。本说明希望也提供了一个贡献，这样的“综合规范理论”，更牢固地结合在一起的理论群胚。主丛和群胚之间关系的主要载体是一个函子，它把一个传递群胚PP与主丛P联系起来;然而，这个“Ehresmann函子”（普拉迪内斯[20]的术语），正如在上述文献中指出的那样。前引，不是范畴论意义上的范畴等价，因为它不是2000数学主题分类：51 K10，53 C 05，58 H 05，20 L05。
We clarify in which precise sense the theory of principal bundles and the theory of groupoids are equivalent; and how this equivalence of theories, in the differentiable case, reflects itself in the theory of connections. The method used is that of synthetic differential geometry. Introduction. In this note, we make explicit a sense in which the theory of principal fibre bundles is equivalent to the theory of groupoids; and in particular, how the differential geometric notion of connection appears in this equivalence. For the latter, we shall utilize the method of Synthetic Differential Geometry, which has for its base the notion of “first neighbourhood of the diagonal” of a manifold. Basically, the notions of “principal bundle” and “groupoid”, their essential equivalence, and the notion of connection in this context, were described by Ehresmann, [4], [5] etc. It is classical to formulate the notion of electromagnetic field in terms of a connection in a principal G-bundle, with G the (abelian) group U(1). Recent particle physics is extending this “gauge theory” viewpoint to non-commutative G, and also to higher “connective structures”, see e.g. [3] and references there. To cope with the mathematical complications arising in this extension process, Breen and Messing [2], [3] found it helpful to utilize some of the formal or “synthetic” method elaborated in the last decades (as in e.g. [7]). The present note hopefully also provides a contribution to such “synthetic gauge theory”, by combining it more firmly with the theory of groupoids. The main vehicle for the relationship between principal bundles and groupoids is a functor which to a principal bundle P associates a transitive groupoid PP; this “Ehresmann functor” (terminology of Pradines [20]) is however, as pointed out in loc. cit., not an equivalence of categories in the sense of category theory, since it is not 2000 Mathematics Subject Classification: 51K10, 53C05, 58H05, 20L05.