Categories of commutative semicartesian quantales valued sets

• Autor(a) principal: Mendes, Caio de Andrade
• Data de Publicação: 2024
• Tipo de documento: Tese
• Idioma: eng
• Título da fonte: Biblioteca Digital de Teses e Dissertações da USP
• Texto Completo: https://www.teses.usp.br/teses/disponiveis/45/45131/tde-16082024-205616/
• topic: Categorias monoidais; Conjuntos a valores em quantales; Lógicas não-clássicas; Monoidal categories.; Non-classical logics; Quantale valued sets; Quantales; Quantales

The main objective of this work is to provide descriptions of categories that have properties to some extent analogous to local topos. At the same time, these categories have a more comprehensive logical counterpart than intuitionistic logic algebraized by Heyting algebras and categorized in higher order in topos and also ukasiewicz logic algebraized by MV-algebras , these being a categorization of BCK- lattices, algebraization of affine logic. Instead of using some kind of usual sheaf categories, that is, categories of contravariant functors satisfying certain gluing conditions, this work explored the realization through sets of values in semicartesian commutative quantales (Q-Sets), as well as the categories defined by them. In addition to considering appropriate versions of different types of Q-Sets already existing in the literature for other classes of quantales, such as separable ones, with gluing property, and complete singletons; a new approach to the notion of Q-sets with restriction property, more suitable for the non-idempotent case, was also presented. Two well-known notions of morphism, functional and relational, are combined with the different types of proposed Q-sets, to generate a range of related categories with good properties. They are complete, cocomplete, locally presentable categories, have an extreme subobject classifier, and generators, and are closed monoidal. Part of these properties is effective, meaning that the precise description of the categorical constructions of limits, colimits, and other objects has provided. The precise descriptions characterizing various types of morphisms from these different categories were also provided.