Glasnik Matematicki, Vol. 42, No.1 (2007), 145-187.

THE COARSE SHAPE

Nikola Koceić Bilan and Nikica Uglešić

23287 Veli Rat, Dugi Otok, Croatia
e-mail: nuglesic@unizd.hr

Abstract.   Given a category C, a certain category pro*-C on inverse systems in C is constructed, such that the usual pro-category pro-C may be considered as a subcategory of pro*-C. By simulating the (abstract) shape category construction, Sh_{(C, D)}, an (abstract) coarse shape category Sh*_{(C, D)} is obtained. An appropriate functor of the shape category to the coarse shape category exists. In the case of topological spaces, C = HTop and D = HPol or D = HANR, he corresponding realizing category for Sh* is pro*-HPol or pro*-HANR respectively. Concerning an operative characterization of a coarse shape isomorphism, a full analogue of the well known Morita lemma is proved, while in the case of inverse sequences, a useful sufficient condition is established. It is proved by examples that for C = Grp (groups) and C = HTop, the classification of inverse systems in pro*-C is strictly coarser than in pro-C. Therefore, the underlying coarse shape theory for topological spaces makes sense.

2000 Mathematics Subject Classification.   55P55, 18A32.

Key words and phrases.   Topological space, compactum, polyhedron, ANR, category, homotopy, shape, S*-equivalence.

DOI: 10.3336/gm.42.1.12

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