Fuzzy Categories: Generalities and Examples Related to Topology
by
Alexander P. Šostak
University of Latvia, Riga

There are known numerous examples of categories, whose objects or/and morphisms are endowed with some fuzzy structure (see e.g. [Ch], [Ek], [Go], [HoSo], [Ro], [Rs], [So1], [WG], etc.) We refer to such situations as "categories of fuzzily structured sets".

As oposed to categories of fuzzily structured sets we introduce here the concept of a fuzzy category   i.e. a category-like conglomerate, whose öbjects" and "morphisms" may be such only to a certain degree (cf [So2]). After briefly discussing the generalities of the theory of fuzzy categories, we shall concentrate attention on examples, mainly of topological nature, illustrating how fuzzy categories naturally arise from ordinary categories.

As most of these examples show, the appropriate context for our reasonings is the class of (L-)fuzzy sets taking their values in an GL-monoid L, i.e. L is a complete lattice endowed with an additional monoidal opeartion * satisfying certain axioms (see e.g. [Ho]).

Ch C.L. Chang, Fuzzy topological spaces, J. Math. Anal. Appl., 24 (1968), 182-190.
Ek P. Eklund, Category theoretic properties of fuzzy topological spaces, Fuzzy Sets and Syst. 19 (1984) 303-310. J.A. Goguen, -fuzzy sets, J. Math. Anal. Appl., 18 (1967) 145-174.
Ho U. Höhle, Commutative residuated l-monoids, In: Non-classical Logics and their Applications to Fuzzy Subsets - A Handbook of the Mathematical Foundations of Fuzzy Set Theory, U. Höhle and E.P. Klement eds., Kluwer, Dodrecht, Boston, 1994, pp. 53-106.
HoSo U. Höhle, A. Sostak, A general theory of fuzzy topological spaces Fuzzy Sets and Syst., 73 (1995), 131-149.
Ro S.E. Rodabaugh, A categorical accomodation of various notions of fuzzy topology, Fuzzy Sets and Syst., 9 (1983), 241-265.
Rs A. Rosenfeld, Fuzzy groups, J. Math. Anal. Appl. 35 (1971), 512-517. (1965), 338-365.
So1 A. Sostak, Two decades of fuzzy topology: basic ideas, notions and results, Russian Mathematical Surveys 44:6 (1989), 125-186.
So2 A. Sostak, Towards the concept of a fuzzy category, In: 14th Linz seminar on Fuzzy Set Theory: Non-Classical Logics and Their Applications, Linz, Austria, 1992, pp. 63-66.
WG Wang Guojun, Topological molecular lattices, Fuzzy Sets and Syst., 47 (1992), 351-376.

Date received: July 1, 1996

Copyright © 1996 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # caao-67.