Generalized logic and the representation of rings
type: Item
instance of: scholarly article
title: Generalized logic and the representation of rings
zbMATH Open document ID: 0856.03050
DOI: 10.1007/BF00122257
author: Simon J. Ambler
author: Dominic Verity
published in: Applied Categorical Structures
publication date: 1996-11-12T00:00:00Z
review text: Various authors considered weaker and weaker logics and, in particular, studied ``quantale-valued sets'' (equivalently, ``sheaves over a quantale''). The authors introduce ``quantic sheaves'' using some ideas from enriched category theory to motivate a quantic sheaf. If a localic subquantale of \(Q\) is given, it turns out that a quantic sheaf over \(Q\) gives a sheaf in the usual sense. As an application, the authors derive some sheaf representations for commutative rings including the Pierce representation.
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18D20
zbMATH DE Number: 934060
zbMATH Keywords: sheaves over a quantale
zbMATH Keywords: enriched category
zbMATH Keywords: quantic sheaf
zbMATH Keywords: localic subquantale
zbMATH Keywords: sheaf representations for commutative rings
zbMATH Keywords: Pierce representation
reviewed by: Boris M. Schein
MaRDI profile type: Publication
cites work: A theory of quantal sets
cites work: Algebra in a localic topos with applications to ring theory
cites work: Quantales and their sheaves
cites work: Q3978967
cites work: An analogue of the pierce sheaf for non-commutative rings
cites work: Q3050433
cites work: Q3965241
cites work: Metric spaces, generalized logic, and closed categories
cites work: Q4836222
cites work: Q3996704
cites work: ENRICHED CATEGORIES AND COHOMOLOGY
cites work: Sheaves on sites as Cauchy-complete categories
full work available at URL: https://doi.org/10.1007/bf00122257
OpenAlex ID: W2086288500
description: scientific article; zbMATH DE number 934060