On the \(n\)-movability of maps (Q2400872)

The notion of movability for metric compacta defined by \textit{K. Borsuk} [Fundam. Math. 66, 137--146 (1969; Zbl 0189.53802)] has been extended by the present authors to shape morphisms and maps [Bull. Pol. Acad. Sci., Math. 64, No. 1, 69--83 (2016; Zbl 1353.55005)]. A dual notion, co-movability (simple and uniform) is defined. In the present paper the authors continue their study of this type of properties for shape morphisms considering the notions of \(n\)-movability and \(n\)-co-movability. Applications of these properties for homology and homotopy groups are proved. They give a new class of movable morphisms and construct movable morphism between non-movable inverse systems. The notions of (uniform) movability and (uniform) co-movability of a morphism of inverse systems in a category with respect to a subcategory are defined and studied. They consider the new definitions for homotopical shape theory \((\text{\textbf{HTop}}, \text{\textbf{HPol}})\) with the subcategory \(\text{\textbf{HPol}}_{n}\) and obtain the notion of \(n\)-(uniform) movability and \(n\)-(uniform) co-movability of maps. As applications they study the implications of these properties on the induced morphisms between the homology and homotopy pro-groups/shape groups.