On sheaves on semicartesian quantales and their truth values

TL;DR

This paper introduces a new notion of sheaves on semicartesian quantales, explores their categorical properties, and investigates their internal truth values, contributing to the development of a monoidal topos-like framework.

Contribution

It defines sheaves on semicartesian quantales, analyzes their properties, and examines internal truth value objects, advancing the understanding of non-elementary topos structures.

Findings

Sheaves on semicartesian quantales are similar to locale sheaves but do not form an elementary topos.

The lattice of truth values in sheaves is isomorphic to the quantale itself.

Analysis of subobject classifiers for subclasses of quantales.

Abstract

In this paper, we introduce a new definition of sheaves on semicartesian quantales, providing first examples and categorical properties. We note that our sheaves are similar to the standard definition of a sheaf on a locale, however, we prove in that in general it is not an elementary topos - since the lattice of external truth values of $Sh(Q)$, $Sub(1)$, is canonically isomorphic to the quantale $Q$ - placing this paper as part of a greater project towards a monoidal (not necessarily cartesian) closed version of elementary topos. To start the study the logical aspects of the category of sheaves we are introducing, we explore the nature of the "internal truth value objects" in such sheaves categories. More precisely, we analyze two candidates for subobject classifier for different subclasses of commutative and semicartesian quantales.