Strong movable categories and strong movability of topological spaces

Abstract

The paper is devoted to one of the important notions of the shape theory: that of strong movability, which was primarily introduced by K. Borsuk for metrizable compacts. A strong movability criterion is proved for topological spaces, which in particular reveals a new, categorical approach to the strong movability.

Movability and Uniform Movability of Shape Morphisms

“…Let G = (G n , q nn+1 ) be an inverse sequence of groups defined as follows ( [11] , Ch.II, §6.2, p. 166). Let G n = Z 2 n , n = 1, 2, ..., and q nn+1 send the generator [1] of Z 2 n+1 to the generator [1] of Z 2 n . This system is not movable (see [11], p.166), but since q nn+1 are epimorphisms, G has the Mittag-Leffler property, and by Corollary 5 from [11], Ch.II, §6.2,this system is movable with respect to free groups.…”

“…Suppose that F ∈ Sh (T ,P) (X, Y ) is given of a morphism f = [(f µ , φ)] : X = (X λ , p λλ ′ , Λ) → Y = (Y µ , q µµ ′ , M), with p = (p λ ) : X → X a P-expansion of X. Then after an idea in [1] (Proposition 4) we can prove that the condition (2.10) from the definition of the strong movability can be written as…”