Metrizable shape and strong shape equivalences

Abstract

In this paper we construct a functor $\Phi : \rm{pro}\mathcal{T}op \to \rm{pro}\mathcal{ANR}$ which extends Mardešic correspondence that assigns to every metrizable space its canonical $\mathcal{ANR}$-resolution. Such a functor allows one to define the strong shape category of prospaces and, moreover, to define a class of spaces, called strongly fibered, that play for strong shape equivalences the role that $\mathcal{ANR}$-spaces play for ordinary shape equivalences. In the last section we characterize SSDR-promaps, as defined by Dydak and Nowak, in terms of the strong homotopy extension property considered by the author.