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
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.
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