A topos-theorist looks at dilators
type: Item
instance of: scholarly article
title: A topos-theorist looks at dilators
zbMATH Open document ID: 0675.18005
DOI: 10.1016/0022-4049(89)90039-X
published in: Journal of Pure and Applied Algebra
publication date: 1989-01-01T00:00:00Z
review text: The concept of a dilator is central to the development of \(\Pi^ 1_ 2\)-logic, which was investigated extensively by \textit{J.-Y. Girard} [Ann. Math. Logic 21, 75-219 (1981; Zbl 0496.03037)]. In this clearly written and interesting article, the author indicates how dilators can be looked at topos theoretically. The reader with the necessary category theoretical background will find this description clear and helpful. The stage is set by looking at the category \(\underline{\text{Lord}}\) of linearly ordered sets and strictly order preserving mappings, its full subcategory \(\underline{\text{Word}}\) of well-ordered sets and also the subcategory \underbar{Ford} of finite orders. Following Girard, a dilator is a functor \(\underline{\text{Word}}\to\) Word which preserves pullbacks and filtered colimits. The author proceeds to show that in fact dilators live in the atomic topos \({\mathcal E}=sh(\underline{\text{Ford}}^{op})\), where \(\underline{\text{Ford}}^{op}\) is an atomic site with its canonical Grothendieck topology, where every non-empty sieve is a cover. \({\mathcal E}\) is the classifying topos for the theory of dense linear orders without end points. Dilators become a full subcategory of \(\underline{\text{Lord}}({\mathcal E})\), the category of linearly ordered objects in \({\mathcal E}\). The article proceeds to analyze linearly ordered objects in an atomic topos and applies these ideas to the specific case of \({\mathcal E}\), and some of Girard's results about dilators are derived in this topos theoretic framework.
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03G30
zbMATH DE Number: 4105187
zbMATH Keywords: dilators
zbMATH Keywords: atomic topos
zbMATH Keywords: Grothendieck topology
zbMATH Keywords: classifying topos for the theory of dense linear orders without end points
zbMATH Keywords: linearly ordered objects in an atomic topos
author: Peter Johnstone
reviewed by: K. I. Rosenthal
MaRDI profile type: Publication
cites work: Atomic toposes
cites work: Molecular toposes
cites work: Q3682648
cites work: Boolean classifying topoi
cites work: Categories localement multipresentables
cites work: Π12-logic, Part 1: Dilators
cites work: Q5670775
cites work: Q3853627
cites work: Q3711309
cites work: Q3220729
cites work: An extension of the Galois theory of Grothendieck
cites work: Full Continuous Embeddings of Toposes
cites work: Forcing and classifying topoi
full work available at URL: 0022-4049(89)90039-x
OpenAlex ID: W1999447354
description: scientific article; zbMATH DE number 4105187