On sheaves on semicartesian quantales and their truth values
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 , , is canonically isomorphic to the quantale - 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.
Keywords
Cite
@article{arxiv.2204.08351,
title = {On sheaves on semicartesian quantales and their truth values},
author = {Ana Luiza Tenório and Caio de Andrade Mendes and Hugo Luiz Mariano},
journal= {arXiv preprint arXiv:2204.08351},
year = {2023}
}
Comments
43 pages. We have updated the paper to show that our sheaves on quantales do not form a topos and replaced a section about change of base by a section discussing a subobject classifier for $Sh(Q)$. The discussion about change of base will appear elsewhere with more details