Fuzzy set theory and topos theory.

*(English)*Zbl 0563.03040The purpose of this paper is to work out the relation between the category of fuzzy sets based on a locale L (L-fuzzy sets) and the topos of sheaves in the same locales (L-sheaves). The answer is this. Let \(L^+\) be the locale derived from L by adding a new element, less than all elements of L. Then the category of L-fuzzy sets is a full subcategory of \(L^+\)-sheaves consisting of the subsheaves of constant sheaves. The category of \(L^+\)-sheaves is the effective equivalence relation closure of the L-fuzzy sets, it may be thought of as being derived from the latter by allowing equality (as well as membership) to be fuzzy. The reason that \(L^+\) appears instead of L is that in fuzzy set theory an element whose degree of membership is 0 is still considered as being ”in” the set.