On the category of rough sets

We consider the class of approximation spaces in the present paper. In this class we define the concept of lower natural transformations, upper natural transformations and natural transformations. We prove that the class of approximation spaces with the lower natural transformations, upper natural transformations and natural transformations form categories which are denoted by Apr̲S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\mathbf{Apr }}{} \mathbf S $$\end{document}, Apr¯S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\overline{\mathbf{Apr }}{} \mathbf S $$\end{document} and AprS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf Apr {} \mathbf S $$\end{document}, respectively. We characterize a lower (upper) natural transformation through equivalence classes in an approximation space. We prove that two categories AprS\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\mathbf Apr {} \mathbf S $$\end{document} and Apr̲S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\mathbf{Apr }}{} \mathbf S $$\end{document} are the same. We characterize several kinds of epimorphisms and monomorphisms. In addition, we show that Apr̲S\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\underline{\mathbf{Apr }}{} \mathbf S $$\end{document} is a (ExtrEpi, ExtrMono)-structured.