Fundamental duality of abstract categories and its applications

Given an abstract category C, C-M-L-spaces that are categorical generalization of fixed-basis fuzzy topological spaces in C and their category C-M-L-Top are introduced. It is pointed out, as one of the main contributions of this paper, that C is dually adjoint to C-M-L-Top. By defining L-spatiality in C and L-sobriety in C-M-L-Top, this adjunction induces a dual equivalence between the full subcategory of C of all L-spatial objects and the full subcategory of C-M-L-Top of all L-sober objects. The present adjunction and duality are fruitful categorical extensions of the classical Top–Loc adjunction and SobTop–SpatLoc duality to abstract categories with a great deal of applications. In particular, their applications to Q-categories, quasivarieties and augmented posets are given.