Ultrametrization of pro*-morphism sets

Abstract

For every pair of inverse systems $\boldsymbol{X}$, $\boldsymbol{Y}$ in a
category $\mathcal{A}$, where $\boldsymbol{Y}$ is cofinite, there exists a
complete ultrametric structure on the set $pro^{\ast }\mbox{-}\mathcal{A}(
\boldsymbol{X},\boldsymbol{Y})$. The corresponding hom-bifunctor is the
internal and invariant $Hom$ of a subcategory, containing $tow^{\ast }\mbox{-}\mathcal{A}$, in the category of complete metric spaces. Several
applications to the shapes (ordinary, coarse and weak) are considered.