Original scientific paper
Category descriptions of the $S_{n}$- and $S$-equivalence
Branko Červar
; Department of Mathematics, University of Split, Split, Croatia
Nikica Uglešić
; University of Zadar, Zadar, Croatia
Abstract
By reducing the Mardešić S-equivalence to a finite
case, i.e. to each $n\in\{0\}\cup\mathbb{N}$ separately, the authors
recently derived the notion of $S_{n}$-equivalence of compacta. In
this paper an additional notion of $S_{n}^{+}$-equivalence is introduced such that $S_{n}^{+}$ implies $S_{n}$ and $S_{n}$ implies $S_{n-1}^{+}$. The implications $S_{1}^{+}\Rightarrow S_{1}\Rightarrow S_{0}^{+}\Rightarrow S_{0}$ as well as $Sh\Rightarrow S\Rightarrow S_{1}$ are strict. Further, for
every $n\in\mathbb{N}$, a category $\underline{\mathcal{A}}_{n}$ and a homotopy relation on its morphism sets are constructed such that the mentioned equivalence relations admit appropriate descriptions in the given settings.
There exist functors of $\underline{\mathcal{A}}_{n^{\prime}}$ to
$\underline{\mathcal{A}}_{n}$, $n\leq n^{\prime}$, keeping the objects fixed and preserving the homotopy relation. Finally, the $S$-equivalence admits a category characterization in the corresponding sequential category $\underline{\mathcal{A}}$.
Keywords
compactum; ANR; shape; $S$-equivalence; $S_{n}$-equivalence; category
Hrčak ID:
23552
URI
Publication date:
28.5.2008.
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