The construction of π₀ in Axiomatic Cohesion

We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of π0 as the image of a natural map to a product of discrete spaces. A particular case of this construction produces, out of a local and hyperconnected geometric mor...

Descripción completa

Guardado en:
Detalles Bibliográficos
Autor principal: Menni, Matías
Formato: Articulo
Lenguaje:Inglés
Publicado: 2017
Materias:
Matemática
Axiomatic cohesion
Topology
Acceso en línea:http://sedici.unlp.edu.ar/handle/10915/98344
https://ri.conicet.gov.ar/11336/57061
http://tcms.org.ge/Journals/TMJ/Volume10/Volume10_3/Abstract/abstract10_3_9.html
Aporte de:
SEDICI (UNLP) de Universidad Nacional de La Plata
id I19-R120-10915-98344
record_format dspace
institution Universidad Nacional de La Plata
institution_str I-19
repository_str R-120
collection SEDICI (UNLP)
language Inglés
topic Matemática
Axiomatic cohesion
Topology
spellingShingle Matemática
Axiomatic cohesion
Topology
Menni, Matías
The construction of π₀ in Axiomatic Cohesion
topic_facet Matemática
Axiomatic cohesion
Topology
description We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of π0 as the image of a natural map to a product of discrete spaces. A particular case of this construction produces, out of a local and hyperconnected geometric morphism p : E → S, an idempotent monad π0 : E → E such that, for every X in E, π X = 1 if and only if (p* Ω)! : (p* Ω)1 → (p* Ω)X is an isomorphism. For instance, if E is the topological topos (over S = Set), the π0-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the π0-algebras in the category of compactly generated Hausdorff spaces are exactly the totally separated ones. Also, in order to relate the construction above with the axioms for Cohesion we prove that, for a local and hyperconnected p : E → S, p is pre-cohesive if and only if p* : E → S is cartesian closed. In this case, p! = p* π0 : E → S and the category of π0-algebras coincides with the subcategory p* : E → S.
format Articulo
Articulo
author Menni, Matías
author_facet Menni, Matías
author_sort Menni, Matías
title The construction of π₀ in Axiomatic Cohesion
title_short The construction of π₀ in Axiomatic Cohesion
title_full The construction of π₀ in Axiomatic Cohesion
title_fullStr The construction of π₀ in Axiomatic Cohesion
title_full_unstemmed The construction of π₀ in Axiomatic Cohesion
title_sort construction of π₀ in axiomatic cohesion
publishDate 2017
url http://sedici.unlp.edu.ar/handle/10915/98344
https://ri.conicet.gov.ar/11336/57061
http://tcms.org.ge/Journals/TMJ/Volume10/Volume10_3/Abstract/abstract10_3_9.html
work_keys_str_mv AT mennimatias theconstructionofp0inaxiomaticcohesion
AT mennimatias constructionofp0inaxiomaticcohesion
bdutipo_str Repositorios
_version_ 1764820493006798848

Ejemplares similares