Hale has identified fuzzy sets, valued in a frame (complete Heyting algebra) Omega, with certain sheaves over Omega: the subsheaves of constant sheaves. More general sheaves can be got as quotients of the fuzzy sets. His principal approach to sheaves over Omega, and topos-theoretic constructions on them, is via complete Omega-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 Omega-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". (C) 2009 Elsevier B.V. All rights reserved.
|Number of pages||30|
|Journal||Fuzzy Sets and Systems|
|Early online date||4 Jul 2009|
|Publication status||Published - 1 May 2010|
- Omega-valued set
- Non-classical logics
- Category theory