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Universes of fuzzy sets and axiomatizations of fuzzy set theory. II: Category theoretic approaches. (English) Zbl 1124.03027
This is the second part of an extensive overview paper of most of the known attempts at precise mathematical definition of the notion of a fuzzy set (for Part I see [Stud. Log. 82, No. 2, 211–244 (2006; Zbl 1111.03047)]). The approaches are classified into four groups: “naive” constructions of cumulative universes of fuzzy sets, model-theoretical constructions, “pure” axiomatizations, and category-theoretic approaches. This paper is devoted to the latter approach.
The development of fuzzy set theory in this direction is motivated by category SET of classical sets and the Higgs topos SET\((H)\). Unfortunately, when introducing SET\(([0,1])\) we do not obtain a category-theoretical characterization of fuzzy sets since the latter does not internalize Łukasiewicz negation (\(\neg a = 1-a\)) and, moreover, the internal logic of the topos is intuitionistic logic, which does not cover non-idempotent conjunction. The latter, however, is crucial in fuzzy set theory.
The paper overviews the first approaches introduced by J. A. Goguen (the categories S\((L)\) and Set\((L)\)) and mentions also categories of Heyting-algebra-valued sets. The latter approaches, which include Eytan and Wyler categories, suffer from the impossibility to introduce non-idempotent conjunction and so are less interesting for fuzzy set theory.
The most significant step in this direction has been done by Höhle, who considers \(M\)-sets with \(M\) being an integral, divisible, residuated, commutative completely lattice-ordered monoid with zero (i.e., complete residuated lattice). He constructs the category sh\((M)\) of sheafs, which has the following properties:
\(\bullet\) it has a suboject classifier \(\Omega\) and a truth arrow \(t\),
\(\bullet\) it allows the unique classification of the \((\Omega, t)\)-classifiable subobjects,
\(\bullet\) it internalizes \(M\)-valued maps as sh\((M)\)-morphisms with codomain \(\Omega\),
\(\bullet\) sh\((M)\) is equivalent with the Higgs topos in the case that the underlying monoid is a complete Heyting algebra.
The paper is finished by a section on categories of quantale-valued sets and remarks on further convergence of all the mentioned approaches.

MSC:
03E72 Theory of fuzzy sets, etc.
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