Fuzzy sets and geometric logic

(This paper is an extended abstract for an invited talk at 29th Linz Seminar on Fuzzy Set Theory, February 2008.) Höhle has shown that fuzzy sets, valued in a frame (complete Heyting algebra) Ω, can be identified with certain sheaves over Ω: they are the subsheaves of constant sheaves, and more general sheaves can be got as quotients of the fuzzy sets. The technical development is complicated by the fact that there are four different, but equivalent, technical expressions of the notion of sheaf: local homeomorphisms, presheaves with the sheaf pasting condition, and Ω-valued sets with or without a completeness condition. Fuzzy sets relate most naturally to the Ω-valued sets, and in fact they first lead one to incomplete Ω-valued sets. However for technical reasons Höhle describes the sheaf constructions known from topos theory in terms of complete