Functoriality of the standard resolution of the Cartesian product of a compactum and a polyhedron

In a previous paper the author has associated with every inverse system of compact Hausdorff spaces X with limit X and every simplicial complex K (possibly infinite) with geometric realization P = vertical bar K vertical bar a resolution R-K(X) Of X X P, which consists of paracompact spaces. If X consists of compact polyhedra, then R-K (X) consists of spaces having the homotopy type of polyhedra. In the present paper it is proved that this construction is functorial. One of the consequences is the existence of a functor from the strong shape category of compact Hausdorff spaces X to the shape category of spaces, which maps X to the Cartesian product X x P. Another consequence is the theorem which asserts that, for compact Hausdorff spaces X, X', such that X is strong shape dominated by X' and the Cartesian product X' x P is a direct product in Sh(Top), then also X x P is a direct product in the shape category Sh(Top). (C) 2007 Elsevier B.V. All rights reserved.