一般拓扑的基本前广群
The fundamental progroupoid of a general topos
摘要：
众所周知，局部连通拓扑的覆盖投影（即局部常数对象）范畴等价于严格前广群（或等价地，局部前离散广群）的分类拓扑，即\emph{fundamental progroupoid}，并且这个前广群表示一次上同调。本文将这些结果推广到任意拓扑。基本原广群现在是局部原广群，并且不能被局部广群替换。分类拓扑不再是伽罗瓦拓扑。不是所有的局部常数对象都可以被认为是覆盖投影。本文的主要贡献是对一般拓扑给出了一个新的覆盖投影定义，当拓扑是局部连通的时，该定义与通常的覆盖投影定义一致。本文的结果在2004年7月温哥华的范畴论会议上发表。
英文摘要：
It is well known that the category of covering projections (that is, locally constant objects) of a locally connected topos is equivalent to the classifying topos of a strict progroupoid (or, equivalently, a localic prodiscrete groupoid), the \emph{fundamental progroupoid}, and that this progroupoid represents first degree cohomology. In this paper we generalize these results to an arbitrary topos. The fundamental progroupoid is now a localic progroupoid, and can not be replaced by a localic groupoid. The classifying topos in not any more a Galois topos. Not all locally constant objects can be considered as covering projections. The key contribution of this paper is a novel definition of covering projection for a general topos, which coincides with the usual definition when the topos is locally connected. The results in this paper were presented in a talk at the Category Theory Conference, Vancouver July 2004.
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