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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

Sažetak

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 ${\underset{―}{\mathcal{A}}}_n$ 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 ${\underset{―}{\mathcal{A}}}_{n^{\prime }}$ to
${\underset{―}{\mathcal{A}}}_n$, $n\le n^{\prime }$, keeping the objects fixed and preserving the homotopy relation. Finally, the $S$-equivalence admits a category characterization in the corresponding sequential category $\underset{―}{\mathcal{A}}$.

Ključne riječi

compactum; ANR; shape; $S$-equivalence; $S_n$-equivalence; category

Hrčak ID:

23552

URI

https://hrcak.srce.hr/23552

Datum izdavanja:

28.5.2008.

Posjeta: 1.266 *