Abstract
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.
Original language | English |
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Pages (from-to) | 1175-1204 |
Number of pages | 30 |
Journal | Fuzzy Sets and Systems |
Volume | 161 |
Issue number | 9 |
Early online date | 4 Jul 2009 |
DOIs | |
Publication status | Published - 1 May 2010 |
Keywords
- Topos
- Sheaf
- Omega-valued set
- Topology
- Frame
- Non-classical logics
- Locale
- Category theory