On the geometry of metric sites
type: Item
instance of: scholarly article
title: On the geometry of metric sites
zbMATH Open document ID: 0851.18008
author: Zhaohua Luo
published in: Journal of Algebra
publication date: 1995-11-09T00:00:00Z
review text: A metric site is a pair \(({\mathcal C}, \tau)\) where \({\mathcal C}\) is a category and \(\tau \) a covariant functor into a category of topological spaces supplying \({\mathcal C}\) with a geometric structure in terms of a Grothendieck topology; the covers of the latter are universal liftings of open covers of the ``underlying'' spaces. The author uses this notion to describe a broad class of categories (including those of topological spaces, differential manifolds, (affine) schemes, algebraic spaces, etc.) arising in modern geometry. There is a notion of complete metric site; any so-called strict metric site has a fibre product-preserving completion as certain sheaves on \({\mathcal C}\) called ``Dedekind cuts''. Example: the dual of the category of commutative rings is a strict metric site of which the category of schemes is a completion.
Mathematics Subject Classification ID: 18F10
Mathematics Subject Classification ID: 14A20
Mathematics Subject Classification ID: 14A15
Mathematics Subject Classification ID: 18F20
Mathematics Subject Classification ID: 18A35
zbMATH DE Number: 802376
zbMATH Keywords: metric site
zbMATH Keywords: Grothendieck topology
zbMATH Keywords: algebraic spaces
zbMATH Keywords: fibre product-preserving completion
zbMATH Keywords: sheaves
zbMATH Keywords: category of schemes
reviewed by: Paul Cherenack
MaRDI profile type: Publication
full work available at URL: jabr.1995.1240
OpenAlex ID: W1983619058
DOI: 10.1006/JABR.1995.1240
description: scientific article; zbMATH DE number 802376