For an abstract category C, a class M of C-monomorphisms and a fixed C-object L, we introduce stratified C-M-L-spaces to be categorical counterparts of stratified fixed-basis fuzzy topological spaces in C, and consider their category SC-M-L-Top. As two main results of this paper, it is shown that SC-M-L-Top is dually adjoint to the comma category L down arrow C, and this adjunction can be restricted to a dual equivalence between the full category of L down arrow C with comma-spatial objects and the full category of SC-M-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. (C) 2014 Elsevier B.V. All rights reserved.