Yoneda representations of flat functors and classifying toposes
type: Item
instance of: scholarly article
zbMATH Open document ID: 1291.18003
author: Olivia Caramello
publication date: 2012-10-22T00:00:00Z
full work available at URL: https://arxiv.org/abs/0805.2187
full work available at URL: 26-21abs.html
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03G30
Mathematics Subject Classification ID: 18C10
Mathematics Subject Classification ID: 03C99
zbMATH DE Number: 6097027
zbMATH Keywords: classifying topos
zbMATH Keywords: Yoneda lemma
zbMATH Keywords: flat functor
MaRDI profile type: Publication
arXiv classification: math.CT
arXiv classification: math.LO
arXiv ID: 0805.2187
title: Yoneda representations of flat functors and classifying toposes
published in: Theory and Applications of Categories
review text: In this paper, the author obtains semantic characterizations, holding for any Grothendieck site \((C,J),\) for the models of a theory classified by a topos of the form \(\mathrm{Sh}(C,J)\) in terms of the models of a theory classified by a topos \([C^{\mathrm{op}},\mathrm{Set}].\)NEWLINENEWLINEThe author exploits the natural behavior of the notion of subtopos with respect to sites to achieve an entirely semantic characterization of the models of a quotient \(T^{\prime }\) of a theory \(T\) classified by a presheaf topos \([C^{\mathrm{op}},\mathrm{Set}]\) in terms of the models of \(T\) and of the Grotendieck topology \(J\) on \(C\) defined by saying that the subtopos \(\mathrm{Sh}(C,J) \rightsquigarrow \) \([C^{\mathrm{op}},\mathrm{Set}]\) corresponds to \(T^{\prime }.\)NEWLINENEWLINEThese characterizations arise from an appropriate representation of flat functors into Grotendieck toposes based on an application of the Yoneda lemma in conjunction with ideas from indexed category theory, and turn out to be relevant also in different contexts, in particular for addressing questions in classical model theory.NEWLINENEWLINEThis connection with classical model theory has already produced a number of useful insights.
reviewed by: Dimitru Buşneag
description: scientific article; zbMATH DE number 6097027