Ω-sets and fuzzy prototypes

Ω-sets and M-valued sets provide frameworks for investigations into categorical and algebraic properties of fuzzy sets as well as for generalizations of fuzzy sets. For the most part, such investigations have not considered the question of whether or not (fuzzy) points need to be redefined or are, more generally, affected in any way when these approaches are used. This paper attempts to apply crisp approaches to the generalization of points in locales and powerobjects of locales to their fuzzy counterparts. The conclusion reached is that prototypical fuzzy points, which are the same as representable sieves of the powerlocale, reflect the nature of the corresponding subobject classifier: if the classifier must be weakened, the fuzzy prototype becomes larger (more inclusive)