A survey of fuzzy set and topos theory

Abstract This paper is a comparison and contrast of approaches to many-valued mathematics offered by Fuzzy Set Theory and topos theory. It gives a survey of the categorical foundations of Fuzzy Set theory and related topoi. Topoi are not a basis for Fuzzy Set theory butthey do suggest appropriate directions to go and questions to ask for a synthesis which does provide a foundation. One possible structure which has a lopos-like internal logic and the rich variety of loggical connectives used in fuzzy sets is included.

[1]  G. C. Wraith.  Lectures on elementary topoi , 1975 .

[2]  M. Tierney.  Sheaf theory and the continuum hypothesis , 1972 .

[3]  D. Ralescu.  Fuzzy subobjects in a category and the theory of C-sets , 1978 .

[4]  M. Eytan,et al.  Fuzzy sets: A topos-logical point of view , 1981 .

[5]  Peter Freyd,et al.  Aspects of topoi , 1972, Bulletin of the Australian Mathematical Society.

[6]  M. Fourman.  The Logic of Topoi , 1977 .

[7]  Oswald Wyler,et al.  Are there topoi in topology , 1976 .

[8]  J. Lambek,et al.  Introduction to higher order categorical logic , 1986 .

[9]  Robert Paré,et al.  Colimits in Topoi , 1974 .

[10]  M. Barr.  FUZZY SET THEORY AND TOPOS THEORY , 1986 .

[11]  J. Goguen.  L-fuzzy sets , 1967 .

[12]  Denis Higgs,et al.  Injectivity in the Topos of Complete Heyting Algebra Valued Sets , 1984, Canadian Journal of Mathematics.

[13]  Review: Didier Dubois and Henri Prade, Fuzzy sets and systems: Theory and applications , 1982 .

[14]  M. Arbib,et al.  A Category-Theoretic Approach to Systems in a Fuzzy World , 1975 .

[15]  Didier Dubois,et al.  Fuzzy sets and systems ' . Theory and applications , 2007 .

[16]  E. Manes.  A class of fuzzy theories , 1982 .

[17]  F. William Lawvere,et al.  Metric spaces, generalized logic, and closed categories , 1973 .

[18]  Andrew M. Pitts,et al.  Fuzzy sets do not form a topos , 1982 .

[19]  Masaharu Mizumoto,et al.  Fuzzy conditional inference under max-odot composition , 1982, Inf. Sci..

[20]  M. Makkai,et al.  First order categorical logic , 1977 .

[21]  G. Monro.  Quasitopoi, logic and heyting-valued models , 1986 .

[22]  Christiane Rousseau,et al.  Topos theory and complex analysis , 1977 .

[23]  L. Kohout,et al.  FUZZY POWER SETS AND FUZZY IMPLICATION OPERATORS , 1980 .

[24]  R. Goldblatt.  Topoi, the Categorial Analysis of Logic , 1979 .

[25]  Lawrence Neff Stout,et al.  Topoi and categories of fuzzy sets , 1984 .

[26]  D. Scott,et al.  Sheaves and logic , 1979 .

[27]  J. Hyland.  The Effective Topos , 1982 .

[28]  J. L. Bell.  Some aspects of the category of subobjects of constant objects in a topos , 1982 .

[29]  Marta Bunge,et al.  Topos theory and souslin's hypothesis , 1974 .

[30]  A category-theoretic approach to boolean-valued models of set theory , 1986 .