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

THE INDUCED HOMOLOGY AND HOMOTOPY FUNCTORS ON THE COARSE SHAPE CATEGORY

Nikola Koceić Bilan

University of Split, Faculty of Science and Mathematics, Teslina 12/III, 21000 Split, Croatia
e-mail: koceic@pmfst.hr


Abstract.   In this paper we consider some algebraic invariants of the coarse shape. We introduce functors pro*-Hn and pro*-πn 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*-Hn and pro*-πn extend standard functors pro-Hn and pro-πn 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.


Full text (PDF) (free access)

DOI: 10.3336/gm.45.2.18


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