Glasnik Matematicki, Vol. 45, No.2 (2010), 531-557.

THE INDUCED HOMOLOGY AND HOMOTOPY FUNCTORS ON THE COARSE SHAPE CATEGORY

Nikola Koceić Bilan

Abstract.   In this paper we consider some algebraic invariants of the coarse shape. We introduce functors pro^*-H_n and pro^*-π_n relating the (pointed) coarse shape category (Sh_*^*) Sh^* to the category pro^*-Grp. The category (Sh_*^*) Sh^*, which is recently constructed, is the supercategory of the (pointed) shape category (Sh_*) Sh^*, having all (pointed) topological spaces as objects. The category pro^* -Grp is the supercategory of the category of pro-groups pro-Grp, both having the same object class. The functors pro^*-H_n and pro^*-π_n extend standard functors pro-H_n and pro-π_n which operate on (Sh_*) Sh^*. The full analogue of the well known Hurewicz theorem holds also in Sh_*^*. We proved that the pro-homology (homotopy) sequence of every pair (X,A) of topological spaces, where A is normally embedded in X, is also exact in pro^*-Grp. Regarding this matter the following general result is obtained: for every category C with zero-objects and kernels, the category pro-C is also a category with zero-objects and kernels, while morphisms of pro^*-C generally don't have kernels.

2000 Mathematics Subject Classification.   55P55, 55Q05, 55N99.

Key words and phrases.   Topological space, polyhedron, inverse system, pro-category, pro^*-category, expansion, shape, coarse shape, homotopy pro-group, homology pro-group, n-shape connectedness, kernel, exact sequence.

DOI: 10.3336/gm.45.2.18

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