Category-theoretic fuzzy topological spaces and their dualities

It is shown in this paper that various approaches to fuzzy topological spaces can be extended to an abstract category C. In this respect, variety-based generalized topological spaces, variable-basis fuzzy topological spaces, fixed-basis fuzzy topological spaces and Hutton fuzzy topological spaces are defined in C. We also construct their respective categories, as well as establish a generalized Papert–Papert–Isbell style adjunction for each of them. These adjunctions are later on refined to equivalences, involving appropriate definitions of spatiality and sobriety.