dc.contributor.author
Menni, Matías
dc.date.available
2018-08-24T19:15:54Z
dc.date.issued
2017-11
dc.identifier.citation
Menni, Matías; The construction of \pi_0 in Axiomatic Cohesion; De Gruyter; Tbilisi Mathematical Journal; 10; 3; 11-2017; 183-207
dc.identifier.issn
1512-0139
dc.identifier.uri
http://hdl.handle.net/11336/57061
dc.description.abstract
We study a construction suggested by Lawvere to rationalize, within a generalization of Axiomatic Cohesion, the classical construction of 0 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 pi_0 : E ightarrow E such that, for every X in E, pi_0 X = 1 if and only if (p^* Omega)^! : (p^* Omega)^1 ightarrow (p^* Omega)^X is an isomorphism. For instance, if E is the topological topos (over S = Set), the pi_0-algebras coincide with the totally separated (sequential) spaces. To illustrate the connection with classical topology we show that the pi_0-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 ightarrow S, p is pre-cohesive if and only if p^* : S ightarrow Eis cartesian closed. In this case, p_! = p_* pi_0 : E ightarrow S and the category of pi_0-algebras coincides with the subcategory p^* : S ightarrow E.
dc.format
application/pdf
dc.language.iso
eng
dc.publisher
De Gruyter
dc.rights
info:eu-repo/semantics/openAccess
dc.rights.uri
dc.subject
Axiomatic Cohesion
dc.subject
Topology
dc.subject.classification
Matemática Pura
dc.subject.classification
Matemáticas
dc.subject.classification
CIENCIAS NATURALES Y EXACTAS
dc.title
The construction of \pi_0 in Axiomatic Cohesion
dc.type
info:eu-repo/semantics/article
dc.type
info:ar-repo/semantics/artículo
dc.type
info:eu-repo/semantics/publishedVersion
dc.date.updated
2018-08-24T13:36:32Z
dc.journal.volume
10
dc.journal.number
3
dc.journal.pagination
183-207
dc.journal.pais
Polonia