Towards the algebraic characterization of (coarse) shape path connectedness

One of the most interesting invariant in the coarse shape theory is the coarse shape path connectedness. We consider isomorphisms that coarse shape paths induce between coarse shape groups (and homotopy pro-groups, as well) at the different base points of topological space. We establish several interesting results for coarse shape path connected spaces such as the independence of n-shape connectedness from the base point. We also exhibit relationships between the (coarse) shape path connectedness and the triviality of low dimensional (coarse) shape groups and homotopy pro-groups.