Some algebraic and logical aspects of C∞-Rings
| Ano de defesa: | 2018 |
|---|---|
| Autor(a) principal: | |
| Orientador(a): | |
| Banca de defesa: | , , , |
| Tipo de documento: | Tese |
| Tipo de acesso: | Acesso aberto |
| Idioma: | eng |
| Instituição de defesa: |
Universidade de São Paulo
|
| Programa de Pós-Graduação: |
Matemática
|
| Departamento: |
Não Informado pela instituição
|
| País: |
BR
|
| Link de acesso: | https://doi.org/10.11606/T.45.2019.tde-14022019-203839 |
Resumo: | As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings. |
| id |
USP_0433249723e816a1233c9e6f4726ea54 |
|---|---|
| oai_identifier_str |
oai:teses.usp.br:tde-14022019-203839 |
| network_acronym_str |
USP |
| network_name_str |
Biblioteca Digital de Teses e Dissertações da USP |
| repository_id_str |
|
| spelling |
info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesis Some algebraic and logical aspects of C∞-Rings Alguns aspectos algébricos e lógicos dos C∞-Anéis 2018-11-09Hugo Luiz MarianoPeter ArndtRicardo BianconiMarcelo Esteban ConiglioVinicius Cifú LopesJean Cerqueira BerniUniversidade de São PauloMatemáticaUSPBR Álgebra comutativa C∞ C∞-Anéis C∞-Rings Feixes e lógica Sheaves and logic Smooth commutative algebra As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings. Conforme observado por I. Moerdijk e G. Reyes em [63], os anéis C∞ têm sido estudados especialmente tendo em vista suas aplicações em Teoria de Singularidades e para construir toposes que sirvam de modelos para a Geometria Diferencial Sintética. Neste trabalho, seguimos um caminho complementar, aprofundando nosso conhecimento sobre eles por um viés mais puro, fazendo uso da Teoria das Categorias e os analisando a partir de pontos de vista algébrico e lógico-categorial. Iniciamos o trabalho apresentando uma sistematização abrangente dos fatos fundamentais da teoria (equacional) dos anéis C∞, distribuídos aqui e ali na literatura atual - a maioria sem demonstrações - mas que servem de base para a teoria. Na sequência, desenvolvemos alguns tópicos do que denominamos Álgebra Comutativa C∞, expandindo resultados parciais de [66] e [67]. Realizamos um estudo sistemático dos anéis C∞ von Neumann-regulares - na linha do estudo algébrico realizado em [2]- e apresentamos alguns resultados interessantes a seu respeito, juntamente com sua relação (funtorial) com os espaços booleanos. Estudamos algumas noções pertinentes à Teoria de Feixes para anéis ∞, tais como espaços (localmente) ∞anelados e o sítio de Zariski liso. Finalmente, descrevemos toposes classicantes para a teoria (algébrica) dos anéis C∞, a teoria (coerente) dos anéis locais C∞ e a teoria (algébrica) dos anéis C∞ von Neumann regulares. https://doi.org/10.11606/T.45.2019.tde-14022019-203839info:eu-repo/semantics/openAccessengreponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USP2023-12-21T18:55:20Zoai:teses.usp.br:tde-14022019-203839Biblioteca Digital de Teses e Dissertaçõeshttp://www.teses.usp.br/PUBhttp://www.teses.usp.br/cgi-bin/mtd2br.plvirginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.bropendoar:27212019-04-09T23:21:59Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP)false |
| dc.title.en.fl_str_mv |
Some algebraic and logical aspects of C∞-Rings |
| dc.title.alternative.pt.fl_str_mv |
Alguns aspectos algébricos e lógicos dos C∞-Anéis |
| title |
Some algebraic and logical aspects of C∞-Rings |
| spellingShingle |
Some algebraic and logical aspects of C∞-Rings Jean Cerqueira Berni |
| title_short |
Some algebraic and logical aspects of C∞-Rings |
| title_full |
Some algebraic and logical aspects of C∞-Rings |
| title_fullStr |
Some algebraic and logical aspects of C∞-Rings |
| title_full_unstemmed |
Some algebraic and logical aspects of C∞-Rings |
| title_sort |
Some algebraic and logical aspects of C∞-Rings |
| author |
Jean Cerqueira Berni |
| author_facet |
Jean Cerqueira Berni |
| author_role |
author |
| dc.contributor.advisor1.fl_str_mv |
Hugo Luiz Mariano |
| dc.contributor.referee1.fl_str_mv |
Peter Arndt |
| dc.contributor.referee2.fl_str_mv |
Ricardo Bianconi |
| dc.contributor.referee3.fl_str_mv |
Marcelo Esteban Coniglio |
| dc.contributor.referee4.fl_str_mv |
Vinicius Cifú Lopes |
| dc.contributor.author.fl_str_mv |
Jean Cerqueira Berni |
| contributor_str_mv |
Hugo Luiz Mariano Peter Arndt Ricardo Bianconi Marcelo Esteban Coniglio Vinicius Cifú Lopes |
| description |
As pointed out by I. Moerdijk and G. Reyes in [63], C∞-rings have been studied specially for their use in Singularity Theory and in order to construct topos models for Synthetic Differential Geometry. In this work, we follow a complementary trail, deepening our knowledge about them through a more pure bias, making use of Category Theory and accounting them from a logical-categorial viewpoint. We begin by giving a comprehensive systematization of the fundamental facts of the (equational) theory of C∞-rings, widespread here and there in the current literature - mostly without proof - which underly the theory of C∞-rings. Next we develop some topics of what we call a ∞Commutative Algebra, expanding some partial results of [66] and [67]. We make a systematic study of von Neumann-regular C∞-rings (following [2]) and we present some interesting results about them, together with their (functorial) relationship with Boolean spaces. We study some sheaf theoretic notions on C∞-rings, such as ∞(locally)-ringed spaces and the smooth Zariski site. Finally we describe classifying toposes for the (algebraic) theory of ∞ rings, the (coherent) theory of local C∞-rings and the (algebraic) theory of von Neumann regular C∞-rings. |
| publishDate |
2018 |
| dc.date.issued.fl_str_mv |
2018-11-09 |
| dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
| dc.type.driver.fl_str_mv |
info:eu-repo/semantics/doctoralThesis |
| format |
doctoralThesis |
| status_str |
publishedVersion |
| dc.identifier.uri.fl_str_mv |
https://doi.org/10.11606/T.45.2019.tde-14022019-203839 |
| url |
https://doi.org/10.11606/T.45.2019.tde-14022019-203839 |
| dc.language.iso.fl_str_mv |
eng |
| language |
eng |
| dc.rights.driver.fl_str_mv |
info:eu-repo/semantics/openAccess |
| eu_rights_str_mv |
openAccess |
| dc.publisher.none.fl_str_mv |
Universidade de São Paulo |
| dc.publisher.program.fl_str_mv |
Matemática |
| dc.publisher.initials.fl_str_mv |
USP |
| dc.publisher.country.fl_str_mv |
BR |
| publisher.none.fl_str_mv |
Universidade de São Paulo |
| dc.source.none.fl_str_mv |
reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
| instname_str |
Universidade de São Paulo (USP) |
| instacron_str |
USP |
| institution |
USP |
| reponame_str |
Biblioteca Digital de Teses e Dissertações da USP |
| collection |
Biblioteca Digital de Teses e Dissertações da USP |
| repository.name.fl_str_mv |
Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
| repository.mail.fl_str_mv |
virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
| _version_ |
1786376815890137088 |