跨度和单纯族
Spans and simplicial families
摘要：
在本文中，我们考虑单纯的家庭，也就是说，单纯对象索引的单纯集。我们开发了一种方法来构建家庭超覆盖加细的覆盖家庭的基础上的概念，我们在这里介绍的\emph{n-spans}。在[The fundamental progroupoid of a general topos，Journal of Pure and Applied Algebra 212（2008）]中，我们在topos中引入了\emph{covering projection}的概念。它们是局部常数对象，满足一个附加条件，当拓扑是局部连通的时，该条件在所有局部常数对象中都是有效的，并发展了一般拓扑的基本广群理论。在这里，我们表明，覆盖投影可以得到从一个家庭的集合上的单纯集的下降数据构造的对象。我们构造了一个广群$\nn{G}_\cc{H}$，使得被$\cc{H}$平凡化的覆盖投影范畴$\cc{G}_\cc{H}$是它的分类拓扑.这确定了一个原群$\{\cc{G} \cc{H}\}_\cc{H}$和一个progroupoid $\{\nn{G}_\cc{H}\}_\cc{H}$，适合于由超覆盖的过滤偏序集索引。然后，我们表明，这progroupoid分类torsors。这种构造在局部连通拓扑的情况下也是新颖的，表明局部连通拓扑中的局部常数对象是通过从集合族上的下降数据下降而构造的。区分局部连通topoi的显著特征是，progroupoid是\emph{strict}，也就是说，过渡态射在三角形上是满射的，或者等价地，底层的本征范畴中的过渡逆像函子是满的和忠实的。
英文摘要：
In this paper we consider simplicial families, that is, simplicial objects indexed by a simplicial set. We develop a method to construct family hypercover refinements of a cover family based on the notion of \emph{n-spans} that we introduce here. In [The fundamental progroupoid of a general topos, Journal of Pure and Applied Algebra 212 (2008)] we had introduced the notion of \emph{covering projection} in a topos. They are locally constant objects satisfying an additional condition which is valid in all locally constant objects when the topos is locally connected, and developed the theory of the fundamental groupoid of a general topos. Here we show that covering projections can be obtained as objects constructed from a descent datum of a simplicial set on a family of sets. We construct a groupoid $\nn{G}_\cc{H}$ such that the category $\cc{G}_\cc{H}$ of covering projections trivialized by $\cc{H}$ is its classifying topos. This determines a protopos $\{\cc{G} \cc{H}\}_\cc{H}$ and a progroupoid $\{\nn{G}_\cc{H}\}_\cc{H}$, suitable indexed by a filtered poset of hypercovers. Then we show that this progroupoid classifies torsors. This construction is novel also in the case of locally connected topoi, showing that locally constant object in a locally connected topos are constructed by descent from a descent datum on a family of sets. The salient feature that distinguishes locally connected topoi is that the progroupoid is \emph{strict}, that is, the transition morphisms are surjective on triangles, or, equivalently, the transition inverse image functors in the underlying indcategory are full and faithful.
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