The existence of vague objects

We consider a category whose objects are defined as triplets (S,e,c̲) where S is a set, e is a similarity and c̲∈S. An object is called crisp if the similarity is crisp, vague otherwise. We indicate different ways of obtaining a similarity and therefore different ways to obtain an object. The paper is mathematical in nature, nevertheless I hope that the proposed formalisms are in some ways connected with the very interesting ontological question about the existence of vague objects in real world.