Interval-value Fuzzy Sets Category

Content: Interval-value fuzzy sets category (namely category IFuz)is mainly discussed in the paper. there are four sections in this pa-per. The first section is forward. Firstly, we introduce the theory oftopos; Secondly, the investigation of category Fuz is reviewed, then weshow that the category Fuz is not a topos; Finally, the theory of weaktopos of category Fuz presented by professor Yuan xue-hai is in-troduced. At the same time, the investigation of this paper is sum-marized. The second section is preliminary of knowledge. In the firstplace, we show the definition and operation of interval number. On suchbasis we introduce the definition and operation of interval-value fuzzysets, and the relationship between interval-value fuzzy sets and intu-itionistic fuzzy sets. In the second place, we introduce some relativeconceptions of topos and weak topos, including category, topos, middleobject, weak topos, power object and pullback lemma. We investigatea category IFuz in section 3. Firstly, the object and morphism of thecategory IFuz are given; Secondly, we prove the category is CartesianClosed(namely it has terminal object, finite products, equalizer and ex-ponentials), but it has no SC, so it is not a topos. In section 4, we provethat the category IFuz has a middle object and consequently the cate-gory IFuz is a weak topos. At the same time, power object of a interval-value fuzzy set in the category IFuz is described.