Fuzziness in a topos

Elementary topos can be seen as a categorical axiomatization of the classical set theory. They are basically categories in which each subobject can be uniquely described by reference to a subobject (named “true”) of a distinguished object, the subobject classifier. Morphisms from an object to the subobject classifier can then be seen as a membership morphism. Introducing fuzziness in a topos requires working not with points, which is a non-elementary notion, but with whole sets of subobjects. A comma construction with the category of *-autonomous lattices gives the desired category of fuzzy sets of subobjects