Induced morphisms between Heyting-valued models

TL;DR

This paper investigates how general morphisms between complete Heyting algebras induce corresponding mappings between their Heyting-valued models and localic toposes, extending known results beyond automorphisms and embeddings.

Contribution

It introduces a framework for lifting geometric morphisms between localic toposes to arrows between Heyting-valued models, broadening the understanding of their interrelations.

Findings

Any geometric morphism between toposes induces a corresponding arrow between models.

Semantic preservation results are established for these induced arrows.

The work generalizes previous results limited to automorphisms and embeddings.

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 $\tilde{f} : V^{(\mathbb{H})} \to V^{(\mathbb{H}')}$.