To the best of our knowledge, there are very few results on how Heyting-valued models are affected by the morphisms on the complete Heyting algebras that determine them: the only cases found in the literature are concerning automorphisms of complete Boolean algebras and complete embedding between them (i.e., injective Boolean algebra homomorphisms that preserves arbitrary suprema and arbitrary infima). In the present work, we consider and explore how more general kinds of morphisms between complete Heyting algebras H and H ′ induce arrows between V (H) and V (H ′ ) , and between their corresponding localic toposes Set (H) (≃ Sh (H)) and Set (H ′ ) (≃ Sh (H ′ )). In more details: any geometric morphism f * : Set (H) → Set (H ′ ) , (that automatically came from a unique locale morphism f : H → H ′ ), can be "lifted" to an arrow f : V (H) → V (H ′ ) . We also provide also some semantic preservation results concerning this arrow f : V (H) → V (H ′ ) .
This work is largely focused on extending D. Higgs' Ω-sets to the context of quantales, following the broad program of [7], we explore the rich category of Q-sets for strong, integral and commutative quantales, or other similar axioms. The focus of this work is to study the different notion of "completeness" a Q-set may enjoy and their relations, completion functors, resulting reflective subcategories, their relations to relational morphisms.We establish the general equivalence of singleton complete Q-sets with functional morphisms and the category of Q-sets with relational morphisms; we provide two characterizations of singleton completeness in categorical terms; we show that the singleton complete categorical inclusion creates limits.
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