Glasnik Matematicki, Vol. 33, No.2 (1998), 287-297.

PROPER n-SHAPE CATEGORIES

Katsuro Sakai

Abstract.   In this paper, it is shown that the proper n-shape category of Ball-Sher type is isomorphic to a subcategory of the proper n-shape category defined by proper n-shaping. It is known that the latter is isomorphic to the shape category defined by the pair (H_pⁿ, H_pⁿ Pol), where H_pⁿ is the category whose objects are locally compact separable metrizable spaces and whose morphisms are proper n-homotopy classes of proper maps, and H_pⁿ Pol is the full subcategory of H_pⁿ whose objects are spaces having the proper n-homotopy type of polyhedra. In the case n = ∞, this shows the relation between the original Ball-Sher's category and the proper shape category defined by proper shapings. We also discuss the proper n-shape category of space of dimension ≤ n + 1.

1991 Mathematics Subject Classification.   54C56, 55P55.

Key words and phrases.   Proper n-shape category, proper n-homotopy, proper n-fundamental net, proper n-approximative net, proper n-shaping, expansion.