Induced morphisms between Heyting-valued models

Abstract: 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 (\emph{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 $\mathbb{H}$ and $\mathbb{H}'$ induce arrows between $V^{(\mathbb{H})}$ and $V^{(\mathbb{H}')}$, and between their corresponding localic toposes $\mathbf{Set}^{(\mathbb{H})}$ ($\simeq \mathbf{Sh}(\mathbb{H})$) and $\mathbf{Set}^{(\mathbb{H}')}$ ($\simeq \mathbf{Sh}(\mathbb{H}')$). In more details: any {\em geometric morphism} $f^* : \mathbf{Set}^{(\mathbb{H})} \to \mathbf{Set}^{(\mathbb{H'})}$, (that automatically came from a unique locale morphism $f : \mathbb{H} \to \mathbb{H}'$), can be "lifted" to an arrow $\tilde{f} : V^{(\mathbb{H})} \to V^{(\mathbb{H}')}$. We also provide also some semantic preservation results concerning this arrow $\tilde{f} : V^{(\mathbb{H})} \to V^{(\mathbb{H}')}$.