The concept of a weak shape type (CROSBI ID 136461)
Prilog u časopisu | izvorni znanstveni rad | međunarodna recenzija
Podaci o odgovornosti
Uglešić, Nikica ; Červar, Branko
engleski
The concept of a weak shape type
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.
inverse system ; expansion ; dense subcategory ; (abstract) shape
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Podaci o izdanju
39 (3)
2007.
363-428
objavljeno
1311-8080
1314-3395