In this work we study discrete analogues of an exact sequence of vector bundles introduced by M. Atiyah in 1957, associated to any smooth principal G-bundle $\pi :Q\to Q/G$. In the original setting, the splittings of the exact sequence correspond to connections on the principal bundle π. The discrete analogues that we consider here can be studied in two different categories: the category of fiber bundles with a (chosen) section, Fbs, and the category of local Lie groupoids, $lLGpd$. In Fbs we find a correspondence between a) (semi-local) splittings of the discrete Atiyah sequence (DAS) of π, b) discrete connections on the same bundle π, and c) isomorphisms of the DAS with certain fiber product extensions in Fbs. We see that the right splittings of the DAS (in Fbs) are not necessarily right splittings in $lLGpd$: we use this obstruction to define the discrete curvature of a discrete connection. Then, there is a correspondence between the right splittings of the DAS in $lLGpd$ and discrete connections with trivial discrete curvature, that is, flat discrete connections. We also introduce a semidirect product between (some) local Lie groupoids and prove that there is a correspondence between semidirect product extensions and right splittings of the DAS in $lLGpd$.

在这项工作中，我们研究由M.阿蒂亚在1957年引入的载体束的精确序列的离散的类似物，相关于任何光滑主要ģ -bundle$\pi ：问\to 问/G$. 在原始设置中，精确序列的分裂对应于主丛π上的连接。我们在这里考虑的离散类似物可以在两个不同的类别中研究：具有（选择的）截面的纤维丛类别，Fbs，以及局部李群群的类别，$升升Gpd$. 在Fbs 中，我们发现 a) π的离散 Atiyah 序列 (DAS) 的（半局部）分裂，b) 同一束π上的离散连接，以及 c) DAS 与Fbs 中某些纤维产品扩展的同构之间的对应关系. 我们看到 DAS 的正确拆分（在Fbs 中）不一定是在$升升Gpd$：我们使用这个障碍来定义离散连接的离散曲率。那么，在 DAS 的右分裂之间存在对应关系$升升Gpd$和具有平凡离散曲率的离散连接，即平坦的离散连接。我们还引入了（一些）局部 Lie groupoids 之间的半直积，并证明了半直积扩展和 DAS 的右分裂之间的对应关系$升升Gpd$.