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Fuzzy set theory and topos theory. (English) Zbl 0563.03040
The 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.

MSC:
03G30 Categorical logic, topoi
03E72 Theory of fuzzy sets, etc.
18B25 Topoi
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