Discrete connections on principal bundles: The discrete atiyah sequence

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