Continuous cohesion over sets
type: Item
instance of: scholarly article
zbMATH Open document ID: 1307.18002
publication date: 2014-11-04T00:00:00Z
full work available at URL: 29-20abs.html
Mathematics Subject Classification ID: 18B25
Mathematics Subject Classification ID: 03G30
zbMATH DE Number: 6365574
zbMATH Keywords: topos
zbMATH Keywords: axiomatic cohesion
zbMATH Keywords: presheaf topos
author: Matias Menni
MaRDI profile type: Publication
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title: Continuous cohesion over sets
published in: Theory and Applications of Categories
review text: Lawvere proposed to analyze the contrast between cohesion (``space'') andNEWLINEnon-cohesion (``set'') in axiomatic terms: the basic data consist in theNEWLINEtwo categories and a string of four adjoint functors between them: ``set''NEWLINEof components, discrete ``space'', ``set'' of points, codiscrete ``space'' --NEWLINErequired to satisfy certain properties.NEWLINENEWLINEIn terms of such a string, one defines notions like ``Nullstellensatz''NEWLINE(``every component has at last one point''), contractibility, homotopyNEWLINEclasses of maps.NEWLINENEWLINEModels of the axiomatics are best studied in the context of geometricNEWLINEmorphisms between toposes.NEWLINENEWLINEThe paper by Menni under review is a contribution to this analysis, forNEWLINEthe case where the non-cohesive topos is just the category of sets, andNEWLINEthe cohesive one is a presheaf topos, like simplicial sets. It is provedNEWLINEthat for presheaf toposes, certain of the desirable properties of theNEWLINEcohesive/non-cohesive contrast are incompatible. More specifically forNEWLINEpresheaf toposes, the property ``every object embeds in a contractibleNEWLINEobject'' is incompatible with the property ``infinite products ofNEWLINEconnected objects are connected''.NEWLINENEWLINEBut on the other hand, Menni constructs toposes (over sets) where theseNEWLINEtwo properties are compatible, thus the Lawvere axiomatics of cohesionNEWLINEis consistent.
reviewed by: Anders Kock
description: scientific article; zbMATH DE number 6365574