IJPAM: Volume 39, No. 3 (2007)

THE CONCEPT OF A WEAK SHAPE TYPE

N. Uglešic$^1$, B. Cervar$^2$
$^1$Department of Mathematics
University of Zadar
F. Tudjmana 24 D, Zadar, 23000, CROATIA
e-mail: nuglesic@unizd.hr
$^2$Department of Mathematics
University of Split
Teslina 12/III, Split, 21000, CROATIA
e-mail: brankoch@pmfst.hr


Abstract.For every category pair $(\mathcal{C},\mathcal{D})$, where $\mathcal{D}%
\subseteq\mathcal{C}$ is a dense and full subcategory, an (abstract) weak shape category $Sh_{\ast(\mathcal{C},\mathcal{D})}$ is constructed. There exists a faithful functor, which keeps the objects fixed, of the (abstract) shape category $Sh_{(\mathcal{C},\mathcal{D})}$ to $Sh_{\ast(\mathcal{C}%
,\mathcal{D})}$. The main benefit is that one may expect existence of a pair of $\mathcal{C}$-objects (especially, topological spaces) having the same weak shape type and different shape types. Further, the weak shape type is coarser than the recently introduced coarse shape type, because there also exists a functor of the (abstract) coarse shape category $Sh_{(\mathcal{C}%
,\mathcal{D})}^{\ast}$ to $Sh_{\ast(\mathcal{C},\mathcal{D})}$. It is interesting that, for metric compacta, both (coarse and weak shape) types coincide with the $S^{\ast}$-equivalence, which is strictly coarser than the shape type classification. An operative characterization of a weak shape isomorphism is established. Finally, it is proved that several important well known shape invariants are, actually, weak shape invariant properties.

Received: July 9, 2007

AMS Subject Classification: 55P55, 18A32

Key Words and Phrases: inverse system, expansion, dense subcategory, (abstract) shape

Source: International Journal of Pure and Applied Mathematics
ISSN: 1311-8080
Year: 2007
Volume: 39
Issue: 3