arXiv subjects

José Goudet Alvim

Publications and source records attributed to José Goudet Alvim.

3 recordsLinked to original sources

arXiv2023

$\mathscr Q$-Sets and Friends: Categorical Constructions and Categorical Properties

This work mainly concerns the -- here introduced -- category of $\mathscr Q$-sets and functional morphisms, where $\mathscr Q$ is a commutative semicartesian quantale. We describe, in detail, the limits and colimits of this complete and cocomplete category and prove that it has a classifier for regular subobjects. Moreover, we prove that it is $κ^+$-locally presentable category, where $κ=max\{|\mathscr Q|, \aleph_0)\}$ and describe a hierarchy of semicartesian monoidal closed structures in this category. Finally, we discuss the issue of 'change of basis' induced by appropriate morphisms between the parametrizing quantales involved in the definition of $\mathscr Q$-sets. In a future work we will address such questions in the full subcategory given by all Scott-complete $\mathscr Q$-sets

math.CT↗

arXiv2023

$\mathscr Q$-Sets and Friends: Regarding Singleton and Gluing Completeness

This work is largely focused on extending D. Higgs' $Ω$-sets to the context of quantales, following the broad program of U. Höhle, we explore the rich category of $\mathscr Q$-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 $\mathscr Q$-set may enjoy and their relations, completion functors, resulting reflective subcategories, their relations to relational morphisms. We establish the general equivalence of singleton complete $\mathscr Q$-sets with functional morphisms and the category of $\mathscr Q$-sets with relational morphisms; we provide two characterizations of singleton completeness in categorical terms; we show that the singleton complete categorical inclusion creates limits.

math.CT↗

arXiv2021

Induced morphisms between Heyting-valued models

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

math.CT↗