Fuzzy sets and geometric logic

Höhle has identified fuzzy sets, valued in a frame (complete Heyting algebra) Ω, with certain sheaves over Ω: the subsheaves of constant sheaves. More general sheaves can be got as quotients of the fuzzy sets. His principal approach to sheaves over Ω, and topos-theoretic constructions on them, is via complete Ω-valued sets. In this paper we show how the geometric fragment of those constructions can be described in a natural “stalkwise” manner, provided one works also with incomplete Ω-valued sets. Our exposition examines in detail the interactions between different technical expressions of the notion of sheaf, and highlights a conceptual view of sheaf as “continuous set-valued map”.