On a Categorical Generalization of the Concept of Fuzzy Set

Abstract

The thesis considers a categorical approach to the concept of fuzzy set introduced by L. A. Zadeh in 1965. Given a concrete category (A,U) over X, we introduce the category X(A) of A-valued objects. The thesis is devoted to the study of the category X(A). We show the necessary and sufficient conditions for the category X(A) to be topological over its ground category, provide an insight into two its subcategories where the first one generalizes the category Set(L) of L-sets introduced by J. A. Goguen in 1967 and used later on by many researchers, present a fuzzification machinery for topological and algebraic structures based on the aforesaid subcategory of the category X(A) and show a particular realization of the category A as the category Q-Mod of (left) modules over a given quantale Q. The category Q-Mod is studied thoroughly in the last chapter.