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Dissertations / Theses on the topic 'Topological categories'
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1
O'Sullivan, David Robert. "Topological C*-categories." Thesis, University of Sheffield, 2017. http://etheses.whiterose.ac.uk/16775/.
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Tensor C*-categories are the result of work to recast the fundamental theory of operator algebras in the setting of category theory, in order to facilitate the study of higher-dimensional algebras that are expected to play an important role in a unified model of physics. Indeed, the application of category theory to mathematical physics is itself a highly active field of research. C*-categories are the analogue of C*-algebras in this context. They are defined as norm-closed self-adjoint subcategories of the category of Hilbert spaces and bounded linear operators between them. Much of the theory of C*-algebras and their invariants generalises to C*-categories. Often, when a C*-algebra is associated to a particular structure it is not completely natural because certain choices are involved in its definition. Using C*-categories instead can avoid such choices since the construction of the relevant C*-category amounts to choosing all suitable C*-algebras at once. In this thesis we introduce and study C*-categories for which the set of objects carries topological data, extending the present body of work, which exclusively considers C*-categories with discrete object sets. We provide a construction of K-theory for topological C*-categories, which will have applications in widening the scope of the Baum-Connes conjecture, in index theory, and in geometric quantisation. As examples of such applications, we construct the C*-categories of topological groupoids, extending the familiar constructions of Renault.
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Razafindrakoto, Ando Desire. "Neighbourhood operators on Categories." Thesis, Stellenbosch : Stellenbosch University, 2013. http://hdl.handle.net/10019.1/80169.
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Thesis (PhD)--Stellenbosch University, 2013.ENGLISH ABSTRACT: While the notions of open and closed subsets in a topological space are dual to each other, they take on another meaning when points and complements are no longer available. Closure operators have been extensively used to study topological notions on categories. Though this has recovered a fair amount of topological results and has brought an economy of e ort and insight into Topology, it is thought that certain properties, such as convergence, are naturally associated with neighbourhoods. On the other hand, it is interesting enough to investigate certain notions, such as that of closed maps, which in turn are naturally associated with closure by means of neighbourhoods. We propose in this thesis a set of axioms for neighbourhoods and test them with the properties of connectedness and compactness.
AFRIKAANSE OPSOMMING: Al is die twee konsepte van oop en geslote subversamelings in 'n topologiese ruimte teenoorgesteldes van mekaar, verander hul betekenis wanneer punte en komplemente nie meer ter sprake is nie. Die gebruik van afsluitingsoperatore is alreeds omvattend in die studie van topologiese konsepte in kategorieë, toegepas. Alhoewel 'n redelike aantal topologiese resultate, groeiende belangstelling en groter insig tot Topologie die gevolg was, word daar geglo dat seker eienskappe, soos konvergensie, op 'n natuurlike wyse aan omgewings verwant is. Nietemin is dit van belang om sekere eienskappe, soos geslote afbeeldings, wat natuurlik verwant is aan afsluiting, te bestudeer. In hierdie proefskrif stel ons 'n aantal aksiomas oor omgewings voor en toets dit gevolglik met die eienskappe van samehangendheid en kompaktheid.
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Wüthrich, Samuel. "I-adic towers in algebraic and topological derived categories /." [S.l.] : [s.n.], 2004. http://www.zb.unibe.ch/download/eldiss/04wuethrich_s.pdf.
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De, Renzi Marco. "Construction of extended topological quantum field theories." Thesis, Sorbonne Paris Cité, 2017. http://www.theses.fr/2017USPCC114/document.
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La position centrale occupée par les Théories Quantiques des Champs Topologiques (TQFTs) dans l’étude de la topologie en basse dimension est due à leur structure extraordinairement riche, qui permet différentes interactions et applications à des questions de nature géométrique. Depuis leur première apparition, un grand effort a été mis dans l’extension des invariants quantiques de 3-variétés en TQFTs et en TQFT Étendues (ETQFTs). Cette thèse s’attaque à ce problème dans deux cadres généraux différents. Le premier est l’étude des invariants quantiques semi-simples de Witten, Reshetikhin et Turaev issus de catégories modulaires. Bien que les ETQFTs correspondantes étaient connues depuis un certain temps, une réalisation explicite basée sur la construction universelle de Blanchet, Habegger, Masbaum et Vogel apparaît ici pour la première fois. L’objectif est de tracer la route à suivre dans la deuxième partie de la thèse, où la même procédure est appliquée à une nouvelle famille d’invariants quantiques non semi-simples due à Costantino, Geer et Patureau. Ces invariants avaient déjà été étendus en TQFTs graduées par Blanchet, Costantino, Geer and Patureau, mais seulement pour une famille explicite d’exemples. Nous posons la première pierre en introduisant la définition de catégorie modulaire relative, un analogue non semi-simple aux catégories modulaires. Ensuite, nous affinons la construction universelle pour obtenir des ETQFTs graduées étendant à la fois les invariants quantiques de Costantino, Geer et Patureau et les TQFTs graduées de Blanchet, Costantino, Geer et Patureau dans ce cadre généralThe central position held by Topological Quantum Field Theories (TQFTs) in the study of low dimensional topology is due to their extraordinarily rich structure, which allows for various interactions with and applications to questions of geometric nature. Ever since their first appearance, a great effort has been put into extending quantum invariants of 3-dimensional manifolds to TQFTs and Extended TQFTs (ETQFTs). This thesis tackles this problem in two different general frameworks. The first one is the study of the semisimple quantum invariants of Witten, Reshetikhin and Turaev issued from modular categories. Although the corresponding ETQFTs were known to exist for a while, an explicit realization based on the universal construction of Blanchet, Habegger, Masbaum and Vogel appears here for the first time. The aim is to set a golden standard for the second part of the thesis, where the same procedure is applied to a new family of non-semisimple quantum invariants due to Costantino, Geer and Patureau. These invariants had been previously extended to graded TQFTs by Blanchet, Costantino, Geer an Patureau, but only for an explicit family of examples. We lay the first stone by introducing the definition of relative modular category, a non-semisimple analogue to modular categories. Then, we refine the universal construction to obtain graded ETQFTs extending both the quantum invariants of Costantino, Geer and Patureau and the graded TQFTs of Blanchet, Costantino, Geer and Patureau in this general setting
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Juer, Rosalinda. "1 + 1 dimensional cobordism categories and invertible TQFT for Klein surfaces." Thesis, University of Oxford, 2012. http://ora.ox.ac.uk/objects/uuid:b9a8fc3b-4abd-49a1-b47c-c33f919a95ef.
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We discuss a method of classifying 2-dimensional invertible topological quantum field theories (TQFTs) whose domain surface categories allow non-orientable cobordisms. These are known as Klein TQFTs. To this end we study the 1+1 dimensional open-closed unoriented cobordism category K, whose objects are compact 1-manifolds and whose morphisms are compact (not necessarily orientable) cobordisms up to homeomorphism. We are able to compute the fundamental group of its classifying space BK and, by way of this result, derive an infinite loop splitting of BK, a classification of functors K → Z, and a classification of 2-dimensional open-closed invertible Klein TQFTs. Analogous results are obtained for the two subcategories of K whose objects are closed or have boundary respectively, including classifications of both closed and open invertible Klein TQFTs. The results obtained throughout the paper are generalisations of previous results by Tillmann [Til96] and Douglas [Dou00] regarding the 1+1 dimensional closed and open-closed oriented cobordism categories. Finally we consider how our results should be interpreted in terms of the known classification of 2-dimensional TQFTs in terms of Frobenius algebras.
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Wasserman, Thomas A. "A reduced tensor product of braided fusion categories over a symmetric fusion category." Thesis, University of Oxford, 2017. http://ora.ox.ac.uk/objects/uuid:58c6aae3-cb0e-4381-821f-f7291ff95657.
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The main goal of this thesis is to construct a tensor product on the 2-category BFC-A of braided fusion categories containing a symmetric fusion category A. We achieve this by introducing the new notion of Z(A)-crossed braided categories. These are categories enriched over the Drinfeld centre Z(A) of the symmetric fusion category. We show that Z(A) admits an additional symmetric tensor structure, which makes it into a 2-fold monoidal category. ByTannaka duality, A= Rep(G) (or Rep(G; w)) for a finite group G (or finite super-group (G,w)). Under this identication Z(A) = VectG[G], the category of G-equivariant vector bundles over G, and we show that the symmetric tensor product corresponds to (a super version of) to the brewise tensor product. We use the additional symmetric tensor product on Z(A) to define the composition in Z(A)-crossed braided categories, whereas the usual tensor product is used for the monoidal structure. We further require this monoidal structure to be braided for the switch map that uses the braiding in Z(A). We show that the 2-category Z(A)-XBF is equivalent to both BFC=A and the 2-category of (super)-G-crossed braided categories. Using the former equivalence, the reduced tensor product on BFC-A is dened in terms of the enriched Cartesian product of Z(A)-enriched categories on Z(A)-XBF. The reduced tensor product obtained in this way has as unit Z(A). It induces a pairing between minimal modular extensions of categories having A as their Mueger centre.
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Maier, Jennifer [Verfasser], and Christoph [Akademischer Betreuer] Schweigert. "A Study of Equivariant Hopf Algebras and Tensor Categories through Topological Field Theories / Jennifer Maier. Betreuer: Christoph Schweigert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2013. http://d-nb.info/1032313412/34.
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Maier, Jennifer Verfasser], and Christoph [Akademischer Betreuer] [Schweigert. "A Study of Equivariant Hopf Algebras and Tensor Categories through Topological Field Theories / Jennifer Maier. Betreuer: Christoph Schweigert." Hamburg : Staats- und Universitätsbibliothek Hamburg, 2013. http://d-nb.info/1032313412/34.
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Araújo, Manuel. "Coherence for 3-dualizable objects." Thesis, University of Oxford, 2017. https://ora.ox.ac.uk/objects/uuid:a4b8f8de-a8e3-48c3-a742-82316a7bd8eb.
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A fully extended framed topological field theory with target in a symmetric monoidal n-catgeory C is a symmetric monoidal functor Z from Bord(n) to C, where Bord(n) is the symmetric monoidal n-category of n-framed bordisms. The cobordism hypothesis says that such field theories are classified by fully dualizable objects in C. Given a fully dualizable object X in C, we are interested in computing the values of the corresponding field theory on specific framed bordisms. This leads to the question of finding a presentation for Bord(n). In view of the cobordism hypothesis, this can be rephrased in terms of finding coherence data for fully dualizable objects in a symmetric monoidal n-category. We prove a characterization of full dualizability of an object X in terms of existence of a dual of X and existence of adjoints for a finite number of higher morphisms. This reduces the problem of finding coherence data for fully dualizable objects to that of finding coherence data for duals and adjoints. For n=3, and in the setting of strict symmetric monoidal 3-categories, we find this coherence data, and we prove the corresponding coherence theorems. The proofs rely on extensive use of a graphical calculus for strict monoidal 3-categories.
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Moreira, Charles dos Anjos. "Linguagem de categorias e o Teorema de van Kampen /." Rio Claro, 2017. http://hdl.handle.net/11449/152195.
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Orientador: Elíris Cristina RizziolliBanca: Aldício José Miranda
Banca: João Peres Vieira
Resumo: Esse trabalho trata de elementos da Topologia Algébrica, a qual tem como fundamental aplicação abordar questões acerca de Espaços Topológicos sob o ponto de vista algébrico. Uma das questões é tentar responder se dois espaços topológicos X e Y são homeomorfos. Neste sentido, o grupo fundamental é uma ferramenta algébrica útil por se tratar de um invariante topológico. Além disso, apresentamos o Teorema de van Kampen do ponto de vista da Linguagem de Categorias e Funtores
Abstract: This work treats of elements of the Algebraic Topology, which has as fundamental application to approach subjects concerning Topological Spaces under the algebraic point of view. One of the subjects is to try to answer if two topological spaces X and Y are homeomorphics. In this sense, the fundamental group is an useful algebraic tool for treating of an topological invariant. In addition, we presented the van Kampen's Theorem of the point of view of the language of Categories and Functors
Mestre
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Moreira, Charles dos Anjos [UNESP]. "Linguagem de categorias e o Teorema de van Kampen." Universidade Estadual Paulista (UNESP), 2017. http://hdl.handle.net/11449/152195.
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Esse trabalho trata de elementos da Topologia Algébrica, a qual tem como fundamental aplicação abordar questões acerca de Espaços Topológicos sob o ponto de vista algébrico. Uma das questões é tentar responder se dois espaços topológicos X e Y são homeomorfos. Neste sentido, o grupo fundamental é uma ferramenta algébrica útil por se tratar de um invariante topológico. Além disso, apresentamos o Teorema de van Kampen do ponto de vista da Linguagem de Categorias e Funtores.
This work treats of elements of the Algebraic Topology, which has as fundamental application to approach subjects concerning Topological Spaces under the algebraic point of view. One of the subjects is to try to answer if two topological spaces X and Y are homeomorphics. In this sense, the fundamental group is an useful algebraic tool for treating of an topological invariant. In addition, we presented the van Kampen's Theorem of the point of view of the language of Categories and Functors.
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Reis, Carla David. "Topology via enriched categories." Doctoral thesis, Universidade de Aveiro, 2014. http://hdl.handle.net/10773/12878.
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Doutoramento conjunto em Matemática - Matemática e Aplicações (PDMA)Having as a starting point the characterization of probabilistic metric spaces as enriched categories over the quantale , conditions that allow the generalization of results relating Cauchy sequences, convergence of sequences, adjunctions of V-distributors and its representability are established. Equivalence between L-completeness and L-injectivity is also established. L-completeness is characterized via the Yoneda embedding, and injectivity is related with exponentiability. Another kind of completeness is considered and the formal ball model is analyzed.
Tendo como ponto de partida a caracterização de espaços métricos probabilísticos como categorias enriquecidas no quantal , estabelecemos condições que permitem a generalização de resultados que relacionam sucessões de Cauchy, convergência de sucessões, adjunções de Vdistribuidores e a sua representabilidade. Também estabelecemos a equivalência entre L-injectividade e L-completude. Caracteriza-se L-completude via a imersão de Yoneda, e injectividade é relacionada com exponenciabilidade. Considera-se outra forma de completude e analisa-se o modelo das bolas formais.
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Blanc, Anthony. "Invariants topologiques des espaces non-commutatifs." Phd thesis, Université Montpellier II - Sciences et Techniques du Languedoc, 2013. http://tel.archives-ouvertes.fr/tel-01012109.
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Dans cette thèse, on donne une définition de la K-théorie topologique des espaces non-commutatifs de Kontsevich (c'est-à-dire des dg-catégories) définis sur les nombres complexes. L'introduction de ce nouvel invariant initie la recherche des invariants de nature topologique des espaces non-commutatifs, comme "simplifications" des invariants algébriques (K-théorie algébrique, homologie cyclique, périodique comme étudiés dans les travaux de Tsygan, Keller). La motivation principale vient de la théorie de Hodge non-commutative au sens de Katzarkov--Kontsevich--Pantev. En géométrie algébrique, la partie rationnelle de la structure de Hodge est donnée par la cohomologie de Betti rationnelle, qui est la cohomologie rationnelle de l'espace des points complexes du schéma. La recherche d'un espace associé à une dg-catégorie trouve une première réponse avec le champ (défini par Toën--Vaquié) classifiant les dg-modules parfaits sur cette dg-catégorie. La définition de la K-théorie topologique a pour ingrédient essentiel le foncteur de réalisation topologique des préfaisceaux en spectres sur le site des schémas de type fini sur les complexes. La partie connective de la K-théorie semi-topologique peut être définie comme la réalisation topologique du champ en monoïdes commutatifs des dg-modules parfaits. Cependant pour atteindre la K-théorie négative, on réalise le préfaisceau donné par la K-théorie algébrique non-connective. Un de nos résultats principaux énonce l'existence d'une équivalence naturelle entre ces deux définitions dans le cas connectif. On montre que la réalisation topologique du préfaisceau de K-théorie algébrique connective pour la dg-catégorie unité donne le spectre de K-théorie topologique usuel. Puis que c'est aussi vrai pour la K-théorie algébrique non-connective, en utilisant la propriété de restriction aux lisses de la réalisation topologique. En outre, cette propriété de restriction aux schémas lisses nécessite de montrer une généralisation de la descente propre cohomologique de Deligne, dans le cadre homotopique non-abélien.La K-théorie topologique est alors définie en localisant par rapport à l'élément de Bott. Cette définition repose donc sur des résultats non-triviaux. On montre alors que le caractère de Chern de la K-théorie algébrique vers l'homologie périodique se factorise par la K-théorie topologique, donnant un candidat naturel pour la partie rationnelle d'une structure de Hodge non-commutative sur l'homologie périodique, ceci étant énoncé sous la forme de la conjecture du réseau. Notre premier résultat de comparaison concerne le cas d'un schéma lisse de type fini sur les complexes -- la conjecture du réseau est alors vraie pour de tels schémas. On montre ensuite que cette conjecture est vraie dans le cas des algèbres associatives de dimension finie.
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Pinto, Darllan Conceição. "Dualidade Generalizada de Esakia com Aplicações." Instituto de Matemática, 2012. http://repositorio.ufba.br/ri/handle/ri/19491.
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FAPESB
Neste trabalho, inicialmente, apresentamos a Dualidade de Esakia. Enfraquecendo os mor smos de reticulados e considerando os mor smo de Esakia como sendo Mor smos Parciais de Esakia, obtemos a Dualidade Generalizada de Esakia. Com esses resultados, estabelecemos uma dualidade entre as categorias de L ogicas Abstratas Distributivas e espa cos Priestley, e uma representa c~ao de L ogicas Abstratas Intuicionistas em Espa cos de Esakia. Por m, aplicamos os resultados na compara c~ao da Condi c~ao de Dom nios Fechados (CDF) com a Condi c~oes de Dom nios Fechados de Zakharyaschev (CDFZ).
In this work, initially, we present the Esakia duality. Weakening the morphisms of lattices and considering the Esakia morphism as Partial Esakia morphism, we obtain the Generalized Esakia Duality. With these results, we establish a duality between the categories of Distributive Abstract Logic and Priestley Spaces, and a representation of intuitionistic Abstract Logic and Esakia Spaces. Finally, we apply the results of the comparison closed domain condition (CDC) with the Zakharyaschev's closed domain condition (ZCDC).
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Bugs, Cristhian Augusto. "Produtos em homologia e cohomologia na categoria dos complexos simpliciais." Universidade Federal de São Carlos, 2004. https://repositorio.ufscar.br/handle/ufscar/5849.
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Previous issue date: 2004-03-31Financiadora de Estudos e Projetos
In this work we present fundamental theory to establish the coordinates of the Kronecker Index, Cup and Cap Products in the finite Simplicial Complexes category in terms of chain and cochain.
Neste trabalho nós apresentamos a teoria fundamental para estabelecer as coordenadas do Índice de Kronecker, Produtos Cup e Cap na categoria dos complexos simpliciais finitos em termos de cadeia e cocadeia.
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Potier, Joris. "A few things about hyperimaginaries and stable forking." Doctoral thesis, Universitat de Barcelona, 2015. http://hdl.handle.net/10803/394029.
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The core of this PhD dissertation is basically twofold :
On one hand, I get some new results on the relationship between compact groups and bounded hyperimaginaries, extending a little bit the classical results of Lascar and Pillay in Hyperimaginaries And Automorphism Groups.
On the other hand, I prove some new results around the so called "stable forking" property, more specifically that a simple theory T has stable forking if Teq has. Quite surprisingly, the proof is not so straigtforward.En este texto se trata, por una parte, de la relación entre grupos compactos e hiper-imaginarios acotados, y por otra parte se prueba que una teoría T tiene la propiedad de bifurcación estable si i solo si Teq la tiene.
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Mendonça, Hudson Kazuo Teramoto. "Teorias de 2-gauge e o invariante de Yetter na construção de modelos com ordem topológica em 3-dimensões." Universidade de São Paulo, 2017. http://www.teses.usp.br/teses/disponiveis/43/43134/tde-01082017-155641/.
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Ordem topológica descreve fases da matéria que não são caracterizadas apenas pelo esquema de quebra de simetria de Landau. Em 2-dimensões ordem topológica é caracterizada, entre outras propriedades, pela existência de uma degenerescência do estado fundamental que é robusta sobre perturbações locais arbitrarias. Com o proposito de entender o que caracteriza e classifica ordem topológica 3-dimensional o presente trabalho apresenta um modelo quântico exatamente solúvel em 3-dimensões que generaliza os modelos em 2-dimensões baseados em teorias de gauge. No modelo proposto o grupo de gauge é substituído por um 2-grupo. A Hamiltonia, que é dada por uma soma de operadores locais, é livre de frustrações. Provamos que a degenerescência do estado fundamental nesse modelo é dado pelo invariante de Yetter da variedade 4-dimensional Sigma × S¹, onde Sigma é a variedade 3-dimensional onde o modelo está definido.Topological order describes phases of matter that cannot be described only by the symmetry breaking theory of Landau. In 2-dimensions topological order is characterized, among other properties, by the presence of a ground state degeneracy that is robust to arbitrary local perturbations. With the purpose of understanding what characterizes and classify 3-dimensional topological order this works presents an exactly soluble quantum model in 3-dimensions that generalize 2-dimensional models constructed using gauge theories. In the model we propose the gauge group is replaced by a 2-group. The Hamiltonian, that is given by a sum of local commuting operators, is frustration free. We prove that the ground state degeneracy of this model is given by the Yetters invariant of the 4-dimensional manifold Sigma × S¹, where Sigma is the 3-dimensional manifold the model is defined.
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Martins, Rafaella de Souza. "Sobre a topologia das fibrações de Milnor." Universidade de São Paulo, 2018. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-25072018-104835/.
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Nesta tese abordaremos dois tipos de problemas relacionados aos célebres Teorema da Fibração de Milnor e Teorema da Fibração de Milnor-Lê para o caso real com valores críticos não isolados. Primeiramente, asseguramos fibrações do tipo Milnor-Lê para F : (Xm, 0) → (Yn, 0), germe de aplicação subanalítico com X e Y espaços subanalíticos sobre C \\ {0} uma curva subanalítica conexa em Y e sobre um subespaço analítico suave W ⊂ Y de dimensão p, n ≥ p ≥ 2, sob algumas condições. Em particular, mostramos a existência das fibrações sobre o discriminantes de germe de aplicações subanalíticos, caso esse ainda não estudado na literatura, normalmente o conjunto dos valores críticos são desconsiderados. Finalizando nossa análise da categoria subanalítica, certificamos que existe a fibração de Milnor-Lê para f : (X, 0) →(Rp, 0), com dimensão de X maior que p ≥ 2, subanalítica e X subanalítico com valores críticos não isolados, definindo d-regularidade. Abordamos estes problemas utilizando resultados de campos de vetores rugosos. Em uma segunda etapa apresentamos um novo critério necessário e suficiente para verificar a importante propriedade de transversalidade de um germe de aplicação real f de classe Cl, l ≥ 1. Fazendo uso também de uma recente ferramenta desenvolvida, a D-regularidade, verificamos condições para a existência das fibrações do germe de aplicação Ψ F, X : (Cn, 0) → (C, 0) não holomorfo, dado por Ψ (z, z̄) = Σnj=1 kjtjzj a
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Nemer, Rodrigo Cohen Mota. "Resultados de multiplicidade para equações de Schrödinger com campo magnético via teoria de Morse e topologia do domínio." Universidade de São Paulo, 2013. http://www.teses.usp.br/teses/disponiveis/55/55135/tde-03012014-145233/.
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Neste trabalho, estudamos a existência de soluções não triviais para uma classe de equações de Schrödinger não lineares envolvendo um campo magnético com condição de Dirichlet ou condição de fronteira mista Dirichlet-Neumann. Nos dois primeiros capítulos, damos uma estimativa para o número de soluções não triviais para o problema de Dirichlet em termos da topologia do domínio. Nos dois capítulos restantes, consideramos o problema de fronteira mista e estimamos o número de soluções não triviais em termos da topologia da porção da fronteira onde é prescrita a condição de Neumann. Em ambos os casos, usamos a teoria de categoria de Ljusternik-Schnirelmann e a teoria de MorseWe study the existence of nontrivial solutions for a class of nonlinear Schrödinger equations involving a magnetic field with Dirichlet or mixed DirichletNeumann boundary condition. In the first two chapters we give an estimate for the number of nontrivial solutions for the Dirichlet boundary value problem in terms of topology of the domain. In the last two chapters we consider mixed DirichletNeumann boundary value problems and the estimation of the number of nontrivial solutions is given in terms of the topology of the part of the boundary where the Neumann condition is prescribed. In both cases, we use Lyusternik- Shnirelman category and the Morse theory
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Aasen, David. "Super Pivotal Categories, Fermion Condensation, and Fermionic Topological Phases." Thesis, 2018. https://thesis.library.caltech.edu/10982/7/aasen_dave_2018.pdf.
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We describe a systematic way of producing fermionic topological phases using the technique of fermion condensation. We give a prescription for performing fermion condensation in bosonic topological phases which contain an emergent fermion. Our approach to fermion condensation can roughly be understood as coupling the parent bosonic topological phase to a phase of physical fermions, and condensing pairs of physical and emergent fermions. There are two distinct types of objects in fermionic theories, which we call “m-type” and “q-type” particles. The endomorphism algebras of q-type particles are complex Clifford algebras, and they have no analogues in bosonic theories. We construct a fermionic generalization of the tube category, which allows us to compute the quasiparticle excitations in fermionic topological phases. We then prove a series of results relating data in condensed theories to data in their parent theories; for example, if C is a modular tensor category containing a fermion, then the tube category of the condensed theory satisfies Tube(C/ψ) ≅ C × C/ψ. We also study how modular transformations, fusion rules, and coherence relations are modified in the fermionic setting, prove a fermionic version of the Verlinde dimension formula, construct a commuting projector lattice Hamiltonian for fermionic theories, and write down a fermionic version of the Turaev-Viro-Barrett-Westbury state sum.
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Akhvlediani, Andrei. "Hausdorff and Gromov distances in quantale-enriched categories /." 2008. http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:MR45921.
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Thesis (M.A.)--York University, 2008. Graduate Programme in Mathematics and Statistics.Typescript. Includes bibliographical references (leaves 166-167). Also available on the Internet. MODE OF ACCESS via web browser by entering the following URL: http://gateway.proquest.com/openurl?url_ver=Z39.88-2004&res_dat=xri:pqdiss&rft_val_fmt=info:ofi/fmt:kev:mtx:dissertation&rft_dat=xri:pqdiss:MR45921
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Silva, Willian Ribeiro Valencia da. "Generalised enriched categories: exponentiation and injectivity." Doctoral thesis, 2019. http://hdl.handle.net/10316/88801.
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Tese no âmbito do Programa Interuniversitário de Doutoramento em Matemática e apresentada ao Departamento de Matemática da Faculdade de Ciências e Tecnologia da Universidade de Coimbra.Dentre as soluções clássicas para o problema da categoria Top dos espaços topológicos e aplicações contínuas não ser cartesiana fechada, nesta tese estamos interessados em espaços compactamente gerados, espaços equilógicos, e espaços quasi-topológicos; trabalhando com categorias enriquecidas generalizadas, que permitem um tratamento unificado de uma gama de categorias da Topologia e da Análise (e.g., espaços ordenados, métricos, topológicos e de aproximação), generalizamos estes três conceitos de Top para (T,V)-Cat. Para tal finalidade, começamos por estudar a relação entre os (T,V)-espaços injectivos e exponenciáveis, e por provar que (T,V)-Cat é uma categoria fracamente localmente cartesiana fechada. Em seguida, introduzimos a categoria (T,V)-Equ dos (T,V)-espaços equilógicos e seus morfismos, que provamos ser uma categoria cartesiana fechada. Ademais, estudamos uma relação generalizada entre os (T,V)-espaços equilógicos e os completamentos regular e exato de (T,V)-Cat, culminando no fato de que (T,V)-Equ é um quasitopos. Por fim, transportamos os conceitos de espaços C -gerados e espaços quasi-topológicos para (T,V)-Cat. Provamos que os (T,V)-espaços C -gerados formam uma subcategoria plena coreflectiva cartesiana fechada de (T,V)-Cat; exemplos de tais espaços incluem (T,V)-espaços compactamente gerados e (T,V)-espaços de Alexandroff. Para os últimos, fazemos algumas considerações que direcionam a uma generalização da equivalência entre os espaços topológicos de Alexandroff e os conjuntos ordenados. Quanto aos quasi-(T,V)-espaços, eles formam a categoria Qs(T,V)-Cat, a qual provamos ser cartesiana fechada e topológica sobre a categoria Set dos conjuntos e aplicações. Generalizamos também para (T,V)-Cat uma relação interessante entre espaços quasi-topológicos e espaços compactamente gerados.
Among the classical solutions to the problem of non-cartesian closedness of the category Top of topological spaces and continuous maps, in this thesis we are interested in compactly generated spaces, equilogical spaces, and quasi-topological spaces; working with generalised enriched categories, which allow for a unified treatment of a range of categories from Topology and Analysis (e.g., ordered, metric, topological, and approach spaces), we generalise these three concepts from Top to (T,V)-Cat. In order to do so, we start by studying the relation between injective and exponentiable (T,V)- spaces, and by proving that (T,V)-Cat is a weakly locally cartesian closed category. Then we introduce the category (T,V)-Equ of equilogical (T,V)-spaces and its morphisms, which we prove to be a cartesian closed category. Moreover, we study a generalised relation between equilogical (T,V)-spaces and the regular and exact completions of (T,V)-Cat, culminating in the fact that (T,V)-Equ is a quasitopos. We finish by carrying the concepts of C -generated spaces and quasi-topological spaces into (T,V)-Cat. We prove that C -generated (T,V)-spaces form a fully coreflective cartesian closed subcategory of (T,V)-Cat; examples of such spaces include compactly generated (T,V)-spaces and Alexandroff (T,V)-spaces. For the latter, we make some discussions towards a generalisation of the equivalence between Alexandroff topological spaces and ordered sets. Concerning quasi-(T,V)- spaces, they form the category Qs(T,V)-Cat which we prove to be cartesian closed and topological over the category Set of sets and maps. We also generalise to (T,V)-Cat an interesting relation between quasi-topological spaces and compactly generated spaces.