Functorial reducing pro⁎-Grp category to pro-Grp

Abstract

We introduce a new functor R˜ from pro⁎-Grp to pro-Grp and use it to give a new characterisation of coarse shape groups. It is shown that lim←⁡∘R˜∘pro⁎-πk=πˇk⁎, thus R˜∘pro⁎-πk is a full analogue of pro-πk. Furthermore, the Hurewicz theorem in a new context is proposed. We complete with a corollary stating isomorphicity of the first nontrivial coarse shape group and coarse shape homology group of a pointed continuum, assertion that does not hold for shape groups.