Existence of the Category $DTC_2 (K)$ Equivalent to the Given Category $KAC_2$

Abstract

For a given category $KAC_2$ , the present paper deals with the existence problem for the category $DTC_2 (K)$, which is equivalent to $KAC_2$ , where $DTC_2 (K)$ is the category whose objects are simple closed $K$-curves with even number $l$ of elements in $Z^n,\; l ≠ 6$, and morphisms are (digitally) $K$-continuous maps, and $KAC_2$ is a category whose objects are simple closed $A$-curves and morphisms are $A$-maps. To address this issue, the paper starts from the category denoted by $KAC_1$ whose objects are connected $nD$ Khalimsky topological subspaces with Khalimsky adjacency and morphisms are $A$-maps in [S. E. Han and A. Sostak, Comput. Appl. Math., 32, 521–536 (2013)]. Based on this approach, in $KAC_1$ the paper proposes the notions of $A$-homotopy and $A$-homotopy equivalence and classifies the spaces in $KAC_1$ or $KAC_2$ in terms of the $A$-homotopy equivalence. Finally, the paper proves that, for Sa given category $KAC_2$, there is $DTC_2 (K)$, which is equivalent to $KAC_2$.