Categories of fuzzy topology in the context of graded ditopologies on textures

This paper extends the notion of ditopology to the case where openness and closedness are given in terms of {\em a priori} unrelated grading functions. The resulting notion of graded ditopology is considered both in the setting of lattices and in that textures, the relation between the two approaches being discussed in detail. Interrelations between graded ditopologies and ditopologies on textures are also studied.