M-valued Sets and Sheaves over Integral Commutative CL-Monoids

Abstract

The concept of Heyting algebra valued sets is intrinsically linked to intuitionistic models, sheaves and topos theory. Precursors of these ideas appear already in D. S. Scott’s and R. Solovay’s work on Boolean-valued models from the mid-sixties [Scott 1967]. Subsequently D. Higgs uses Boolean-valued sets in his unpublished, but widely circulated paper [Higgs 1973] and demonstrates that the category of sheaves over a complete Boolean algebra B is equivalent to the category of B-valued sets and maps in the original Scott-Solovay sense. The concept of Heyting algebra valued sets and its relationship to intuitionistic logic has been extensively studied by M. Fourman and D. S. Scott in their contributions to the Research Symposium on Applications of Sheaf Theory Durham, NC, 1977 [Fourman and Scott 1979; Scott 1979]. We can summarize the situation as follows: let Ω be a complete Heyting algebra; then 1° Ω-valued sets are intuitionistic models of the formalized mathematical theory of identity with existence predicate, and 2° the category sh(Ω) of sheaves over Ω forms a topos. In this context we mention explicitly the fact that the unique classification of subobjects in sh (Ω) depends essentially on the existence of non-trivial objects with local support.