The Category Of Fuzzy Sets And Weak Topos

Firstly, by the use of the relations between fuzzy points and fuzzy sets, the definitions of (β|-,α|-)-fuzzy mapping, (β|-,α|-)-convex fuzzy cone and (β|-,α|-)-fuzzy topology are introduced. Secondly, the category of fuzzy sets, category of convex fuzzy cone and category of consensus set are studied respectively. The concepts of middle object and weak topos are acquired. Weak topos is a new kind of categorical theory which is stronger than Cartesian colsed category and weaker than topos theory, and a weak topos can serve a similiar function to a topos. Finally, the concept of fuzzy subobject is given and the concept of Zadeh's fuzzy subset is generalized to a topos. The main results obtained in this dissertation are as follows:1. In Chapter 2, first, the definition of (β|-,α|-)-fuzzy mapping is introduced, three fuzzy mappings such as (∈,∈)-fuzzy mapping, (∈,∈∨ q)-fuzzy mapping and (∈|-,∈|- ∨ q|-)-fuzzy mapping are obtained. By generalizing those three fuzzy mappings, (λ,μ]-fuzzy mapping is acquired and the relations between (λ,μ]-fuzzy mapping and HX-fuzzy mapping are discussed. Second, a category Fuz_λ^μ of fuzzy subsets and (λ,μ]-fuzzy mappings is built. It is proved that the category Fuz_λ^μ is a Cartesian closed category, but the category Fuz_λ^μ is not a topos. Third, the concept of middle object and weak topos are introduced. It is proved that the category Fuz_λ^μ, the category RVF of real valued fuzzy sets and the category Fuz^C of functors from small category C to the category Fuz are weak topos. The properties of a weak topos are studied. The relations between the smallest characteristic morphisms of two objects and the relation between the characteristic morphism X_f of monomorphism f : A' → A and the smallest characteristic morphism α_A' of object A' are described. A sufficient and necessary condition that A is a Terminal object for middle object m : Λ → Δ is acquired and power object of an object is described. Final, the concept of fuzzy subobject is given and the concept of Zadeh's fuzzy subset is generalized to a topos. The category FC of fuzzy subobject is built and it is proved that the category FC is finitely complete.2. In Chapter 3, first, the concept of (β|-,α|-)-convex fuzzy cone is given and (∈,∈)-convex fuzzy cone, (∈,∈∨ q)-convex fuzzy cone and (∈|-,∈|- ∨ q|-)-convex fuzzy cone are obtained. Second, by the use of consensus space, the definition of C-convex fuzzy cone is given. It is proved that a (∈,∈)-convex fuzzy cone is a C-convex fuzzy cone, and a C-convex fuzzy cone is isomorphic to the C-convex fuzzy cone generated by a cone S. Third, the category CFC of convex fuzzy cone is built. It is proved that the category CFC is finitely complete and has the similar properties to Exponential. Final, the category C(Ω,(?)) of consensus sets is built and it is proved that the category C(Ω,(?))is a topos.