Ultrametrization of pro\(^\ast\)-morphism sets (Q2862098)

In this paper, the author reaches, using metric tools, a better understanding of coarse and weak shape type classification (especially for metrizable compacta). The starting point is a naturally existing countable decreasing family of equivalence relations on a set \(inv^*\)-\(A(X,Y)\) where \(A\) is an arbitrary category and \(X\) and \(Y\) inverse systems. A pseudoultrametric is induced when the codomain inverse system \(Y\) is cofinite. By passing to the quotient set \(pro^*\)-\(A(X,Y)\), a complete ultrametric space \((pro^*\)-\(A(X,Y), d^*)\) is obtained. This is an extension of the known one \(pro\)-\(A(X,Y)\), such that the canonical injection is an isometric closed embedding. Next, the author considers the corresponding hom-bifunctor and establishes the sufficient condition for the hom to be the internal and invariant Hom of a subcategory, containing \(tow^*\)-\(A\), in the category of complete metric spaces. The paper concludes with several applications of the new results and a technique to obtain a better view into classifications by shapes, the coarse shape in particular. The author also defines coarse equivalence and uniform coarse equivalence (as the analogues and improvements of Borsuk quasi-equivalence). The considered applications are: the canonical complete ultrametric structures on the shape and coarse shape and weak shape morphism sets \(Sh(X,Y), Sh^*(X,Y)\) and \(Sh_*(X,Y)\) respectively, such that the natural (functional) injections are isometric closed embeddings.