Applications of Category Theory to Fuzzy Subsets is the first major work to comprehensively describe the deeper mathematical aspects of fuzzy sets, particularly those aspects which are category-theoretic in nature, and is intimately related to the first eleven years of the renowned International Seminar on Fuzzy Set Theory. Though it brings the reader to the very frontier of the mathematics of fuzzy set theory, its extensive bibliography, indices, and the tutorial nature of its longer chapters also make it suitable as a text for advanced graduate students. Part I develops model-theoretic foundations for fuzzy set theory, and in doing so, comprises an extensive study of monoid-valued sets, sheaves over commutative cl-monoids, weak and quasi topoi, local existence in such settings, and categories with two closed structures, including the logic and inference rules in these latter categories for the unbalanced subobjects modeling fuzzy subsets. Part II refines and works within non-model-theoretic approaches to fuzzy sets, giving a full account of the use of categorical methods to describe fuzzy topology from the structure-theoretic, category-theoretic, and point-set lattice-theoretic viewpoints. Explored in detail are set functors, topological constructs, convergence, and the relationship between locales, fuzzy topologies, and functor categories. Part III addresses issues related to Part I and II, including pointless geometry, fuzzy paths and relational databases, fuzzy points and locales, sobriety and Urysohn Lemmas, and the relationship between the Helly space and the fuzzy unit interval. The fourth part consists of several appendices: round tables and open questions, a comprehensive bibliography, and two extensive indices.
I: Topos-like and Model-Theoretic Approaches.- 1: Classification of Extremal Subobjects of Algebras over SM-SET.- 2: M-valued Sets and Sheaves over Integral Commutative CL-Monoids.- 3: The Logic of Unbalanced Subobjects in a Category with Two Closed Structures.- II: Categorical Methods in Topology.- 4: Fuzzy Filter Functors and Convergence.- 5: Convenient Topological Constructs.- 6: A Topological Universe Extension of FTS.- 7: Categorical Frameworks for Stone Representation Theories.- III: Applications and Related Topics in Logic and Topology.- 8: Pointless Metric Spaces and Fuzzy Spaces.- 9: Fuzzy Unit Interval and Fuzzy Paths.- 10: Lattice Morphisms, Sobriety, and Urysohn Lemmas.- 11: The Topological Modification of the L-Fuzzy Unit Interval.- 12: A Categorical Approach to Fuzzy Relational Database Theory.- 13: Fuzzy Points and Membership.- Appendices.- Index of Categories.- Addenda et Corrigenda.