Stratified categorical fixed-basis fuzzy topological spaces and their duality

For an abstract category C, a class MM of C-monomorphisms and a fixed C-object L, we introduce stratified C-MM-L-spaces to be categorical counterparts of stratified fixed-basis fuzzy topological spaces in C, and consider their category SC-MM-L-Top. As two main results of this paper, it is shown that SC-MM-L-Top is dually adjoint to the comma category L↓CL↓C, and this adjunction can be restricted to a dual equivalence between the full category of L↓CL↓C with comma-spatial objects and the full category of SC-MM-L-Top with comma-sober objects. The present paper also describes applications and relationships of these results to stratified fixed-basis fuzzy topological spaces. In this respect, a considerable part of this paper is devoted to stratified L-quasi-topological spaces and their duality.