Some algebraic and logical aspects of C∞-Rings
| Autor(a) principal: | |
|---|---|
| Data de Publicação: | 2018 |
| Tipo de documento: | Tese |
| Idioma: | eng |
| Título da fonte: | Biblioteca Digital de Teses e Dissertações da USP |
| Texto Completo: | http://www.teses.usp.br/teses/disponiveis/45/45131/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. |
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Some algebraic and logical aspects of C∞-RingsAlguns aspectos algébricos e lógicos dos C∞-AnéisÁlgebra comutativa C∞C∞-AnéisC∞-RingsFeixes e lógicaSheaves and logicSmooth commutative algebraAs 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.Biblioteca Digitais de Teses e Dissertações da USPMariano, Hugo LuizBerni, Jean Cerqueira2018-11-09info:eu-repo/semantics/publishedVersioninfo:eu-repo/semantics/doctoralThesisapplication/pdfhttp://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/reponame:Biblioteca Digital de Teses e Dissertações da USPinstname:Universidade de São Paulo (USP)instacron:USPLiberar o conteúdo para acesso público.info:eu-repo/semantics/openAccesseng2019-04-09T23:21:59Zoai: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.none.fl_str_mv |
Some algebraic and logical aspects of C∞-Rings 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 Berni, Jean Cerqueira Álgebra comutativa C∞ C∞-Anéis C∞-Rings Feixes e lógica Sheaves and logic Smooth commutative algebra |
| 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 |
Berni, Jean Cerqueira |
| author_facet |
Berni, Jean Cerqueira |
| author_role |
author |
| dc.contributor.none.fl_str_mv |
Mariano, Hugo Luiz |
| dc.contributor.author.fl_str_mv |
Berni, Jean Cerqueira |
| dc.subject.por.fl_str_mv |
Álgebra comutativa C∞ C∞-Anéis C∞-Rings Feixes e lógica Sheaves and logic Smooth commutative algebra |
| topic |
Álgebra comutativa C∞ C∞-Anéis C∞-Rings Feixes e lógica Sheaves and logic Smooth commutative algebra |
| 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.none.fl_str_mv |
2018-11-09 |
| dc.type.status.fl_str_mv |
info:eu-repo/semantics/publishedVersion |
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info:eu-repo/semantics/doctoralThesis |
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doctoralThesis |
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publishedVersion |
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http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/ |
| url |
http://www.teses.usp.br/teses/disponiveis/45/45131/tde-14022019-203839/ |
| dc.language.iso.fl_str_mv |
eng |
| language |
eng |
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|
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Liberar o conteúdo para acesso público. info:eu-repo/semantics/openAccess |
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Liberar o conteúdo para acesso público. |
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openAccess |
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application/pdf |
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|
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Biblioteca Digitais de Teses e Dissertações da USP |
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Biblioteca Digitais de Teses e Dissertações da USP |
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reponame:Biblioteca Digital de Teses e Dissertações da USP instname:Universidade de São Paulo (USP) instacron:USP |
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Universidade de São Paulo (USP) |
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USP |
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USP |
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Biblioteca Digital de Teses e Dissertações da USP |
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Biblioteca Digital de Teses e Dissertações da USP - Universidade de São Paulo (USP) |
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virginia@if.usp.br|| atendimento@aguia.usp.br||virginia@if.usp.br |
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