The induced homology and homotopy functors on the coarse shape category

In this paper we consider some algebraic invariants of the coarse shape. We introduce functors pro*-Hn and pro*-πn relating the (pointed) coarse shape category (Sh**) Sh* to the category pro*-Grp. The category (Sh**) Sh*, which is recently constructed, is the supercategory of the (pointed) shape category (Sh*) Sh*, having all (pointed) topological spaces as objects. The category pro* -Grp is the supercategory of the category of pro-groups pro-Grp, both having the same object class. The functors pro*-Hn and pro*-πn extend standard functors pro-Hn and pro-πn which operate on (Sh*) Sh*. The full analogue of the well known Hurewicz theorem holds also in Sh**. We proved that the pro-homology (homotopy) sequence of every pair (X,A) of topological spaces, where A is normally embedded in X, is also exact in pro*-Grp. Regarding this matter the following general result is obtained: for every category C with zero-objects and kernels, the category pro-C is also a category with zero-objects and kernels, while morphisms of pro*-C generally don\u27t have kernels