A category whose isomorphisms induce an equivalence relation coarser than shape

Let Sh denote the usual shape category of metric compacta. In the paper one defines a new category S∗, whose objects are all metric compacta, and one defines a functor S∗:Sh→S∗, which preserves objects. In shape fibrations over a metric continuum fibers need not have the same shape, but they are isomorphic objects of S∗. Various shape invariant classes of compacta, like FANR's and movable continua, are also S∗-invariant classes, i.e., if X and X′ are isomorphic objects in S∗ and X is an FANR (is movable), then so is X′. Compact ANR's are isomorphic in S∗ if an only if they have the same homotopy type.