There are several reasons to be interested in conformal field theories in two dimensions. Apart from arising in various physical applications, ranging from statistical mechanics to string theory, conformal field theory is a class of quantum field theories that is interesting on its own. First of all there is a large amount of symmetries. In addition, many of the interesting theories satisfy a finiteness condition, that together with the symmetries allows for a fully non-perturbative treatment, and even for a complete solution in a mathematically rigorous manner. One of the crucial tools which make such a treatment possible is provided by category theory. This thesis contains results relevant for two different classes of conformal field theory. We partly treat rational conformal field theory, but also derive results that aim at a better understanding of logarithmic conformal field theory. For rational conformal field theory, we generalize the proof that the construction of correlators, via three-dimensional topological field theory, satisfies the consistency conditions to oriented world sheets with defect lines. We also derive a classifying algebra for defects. This is a semisimple commutative associative algebra over the complex numbers whose one-dimensional representations are in bijection with the topological defect lines of the theory. Then we relax the semisimplicity condition of rational conformal field theory and consider a larger class of categories, containing non-semisimple ones, that is relevant for logarithmic conformal field theory. We obtain, for any finite-dimensional factorizable ribbon Hopf algebra H, a family of symmetric commutative Frobenius algebras in the category of bimodules over H. For any such Frobenius algebra, which can be constructed as a coend, we associate to any Riemann surface a morphism in the bimodule category. We prove that this morphism is invariant under a projective action of the mapping class group ofthe Riemann surface. This suggests to regard these morphisms as candidates for correlators of bulk fields of a full conformal field theories whose chiral data are described by the category of left-modules over H.

Autômatos não-seqüenciais constituem um domínio semântico categorial do tipo não-intercalação para sistemas reativos, comunicantes e concorrentes.É baseado em sistemas de transições etiquetados, inspirado em "Redes de Petri são Monóides" de Meseguer e Montanari, onde as operações de sincronização e encapsulação são funtoriais e as reificações constituem uma classe de morfismos especiais. Do que se tem conhecimento, é o primeiro modelo de concorrência a satisfazer a composicionalidade diagonal, ou seja, onde as reificações compõem (verticalmente) e distribuem-se sobre a composição paralela (verticalmente). Adjunções entre autômatos não-seqüenciais, redes de Petri e autômatos seqüenciais são introduzidas estendendo a abordagem de Winskel, Nielsen e Sassone onde é proposta uma classificação formal para modelos de concorrência. Dos passos que envolvem a passagem de um modelo para outro, pode-se inferir que os autômatos não-seqüenciais são mais concretos do que as redes de Petri e os autômatos seqüenciais. Para experimentar o domínio semântico proposto, é dada semântica a uma linguagem concorrente, baseada nos objetos, denominada Náutilus. Trata-se de uma versão simplificada e revisada da linguagem de especificação orientada aos objetos GNOME, onde são introduzidos algumas facilidades especiais, inspiradas no domínio semântico, como a reificação e a agregação. Neste contexto, a composicionalidade diagonal é uma propriedade essencial para dar a semântica. / Nonsequential automata constitute a non-interleaving categorial semantic domain for reactive, communicating and concurrent systems. It is based on labeled transition systems, inspired by Meseguer and Montanari's "Petri Nets are Monoids", where synchronization and encapsulation operations are functorial and a class of morphisms stands for reification. It is, for our knowledge, the first model for concurrency which satisfies the diagonal compositionality requirement, i. e., reifications compose (vertical) and distribute over the parallel composition (horizontal). Adjunctions between nonsequential automata, Petri nets and sequential automata are provided extending the approach of Winskel, Nielsen and Sassone where a scene for a formal classification of models for concurrency is set. The steps of abstraction involved in moving between models show that nonsequential automata are more concrete than Petri nets and sequential automata. To experiment with the proposed semantic domain, a semantics for a concurrent, object-based language named Nautilus is given. It is a simplified and revised version of the object-oriented specification language GNOME, introducing some special features inspired by the semantic domain such as reification and aggregation. The diagonal compositionality is an essential property to give semantics in this context.

Autômatos não-seqüenciais constituem um domínio semântico categorial do tipo não-intercalação para sistemas reativos, comunicantes e concorrentes.É baseado em sistemas de transições etiquetados, inspirado em "Redes de Petri são Monóides" de Meseguer e Montanari, onde as operações de sincronização e encapsulação são funtoriais e as reificações constituem uma classe de morfismos especiais. Do que se tem conhecimento, é o primeiro modelo de concorrência a satisfazer a composicionalidade diagonal, ou seja, onde as reificações compõem (verticalmente) e distribuem-se sobre a composição paralela (verticalmente). Adjunções entre autômatos não-seqüenciais, redes de Petri e autômatos seqüenciais são introduzidas estendendo a abordagem de Winskel, Nielsen e Sassone onde é proposta uma classificação formal para modelos de concorrência. Dos passos que envolvem a passagem de um modelo para outro, pode-se inferir que os autômatos não-seqüenciais são mais concretos do que as redes de Petri e os autômatos seqüenciais. Para experimentar o domínio semântico proposto, é dada semântica a uma linguagem concorrente, baseada nos objetos, denominada Náutilus. Trata-se de uma versão simplificada e revisada da linguagem de especificação orientada aos objetos GNOME, onde são introduzidos algumas facilidades especiais, inspiradas no domínio semântico, como a reificação e a agregação. Neste contexto, a composicionalidade diagonal é uma propriedade essencial para dar a semântica. / Nonsequential automata constitute a non-interleaving categorial semantic domain for reactive, communicating and concurrent systems. It is based on labeled transition systems, inspired by Meseguer and Montanari's "Petri Nets are Monoids", where synchronization and encapsulation operations are functorial and a class of morphisms stands for reification. It is, for our knowledge, the first model for concurrency which satisfies the diagonal compositionality requirement, i. e., reifications compose (vertical) and distribute over the parallel composition (horizontal). Adjunctions between nonsequential automata, Petri nets and sequential automata are provided extending the approach of Winskel, Nielsen and Sassone where a scene for a formal classification of models for concurrency is set. The steps of abstraction involved in moving between models show that nonsequential automata are more concrete than Petri nets and sequential automata. To experiment with the proposed semantic domain, a semantics for a concurrent, object-based language named Nautilus is given. It is a simplified and revised version of the object-oriented specification language GNOME, introducing some special features inspired by the semantic domain such as reification and aggregation. The diagonal compositionality is an essential property to give semantics in this context.

Made available in DSpace on 2015-05-14T12:11:59Z (GMT). No. of bitstreams: 1 arquivototal.pdf: 891353 bytes, checksum: 2d056c7f53fdfb7c20586b64874e848d (MD5) Previous issue date: 2010-12-01 / Coordenação de Aperfeiçoamento de Pessoal de Nível Superior / The basic concepts of what later became called category theory were introduced in 1945 by Samuel Eilenberg and Saunders Mac Lane. In 1940s, the main applications were originally in the fields of algebraic topology and algebraic abstract. During the 1950s and 1960s, this theory became an important conceptual framework in other many areas of mathematical research, especially in algrebraic homology and algebraic geometry, as shows the works of Daniel M. Kan (1958) and Alexander Grothendieck (1957). Late, questions mathematiclogics about the category theory appears, in particularly, with the publication of the Functorial Semantics of Algebraic Theories (1963) of Francis Willian Lawvere. After, other works are done in the category logic, such as the the current Makkai (1977), Borceux (1994), Goldblatt (2006), and others. As introduction of application of the category theory in logic, this work presents a study on the logic category propositional. The first section of this work, shows to the reader the important concepts to a better understanding of subject: (a) basic components of category theory: categorical constructions, definitions, axiomatic, applications, authors, etc.; (b) certain structures of abstract algebra: monoids, groups, Boolean algebras, etc.; (c) some concepts of mathematical logic: pre-order, partial orderind, equivalence relation, Lindenbaum algebra, etc. The second section, it talk about the properties, structures and relations of category propositional logic. In that section, we interpret the logical connectives of the negation, conjunction, disjunction and implication, as well the Boolean connectives of complement, intersection and union, in the categorical language. Finally, we define a categorical boolean propositional semantics through a Boolean category algebra. / Os conceitos básicos do que mais tarde seria chamado de teoria das categorias são introduzidos no artigo General Theory of Natural Equivalences (1945) de Samuel Eilenberg e Saunders Mac Lane. Já em meados da década de 1940, esta teoria é aplicada com sucesso ao campo da topologia. Ao longo das décadas de 1950 e 1960, a teoria das categorias ostenta importantes mudanças ao enfoque tradicional de diversas áreas da matemática, entre as quais, em especial, a álgebra geométrica e a álgebra homológica, como atestam os pioneiros trabalhos de Daniel M. Kan (1958) e Alexander Grothendieck (1957). Mais tarde, questões lógico-matemáticas emergem em meio a essa teoria, em particular, com a publica ção da Functorial Semantics of Algebraic Theories (1963) de Francis Willian Lawvere. Desde então, diversos outros trabalhos vêm sendo realizados em lógica categórica, como os mais recentes Makkai (1977), Borceux (1994), Goldblatt (2006), entre outros. Como inicialização à aplicação da teoria das categorias à lógica, a presente dissertação aduz um estudo introdutório à lógica proposicional categórica. Em linhas gerais, a primeira parte deste trabalho procura familiarizar o leitor com os conceitos básicos à pesquisa do tema: (a) elementos constitutivos da teoria das categorias : axiomática, construções, aplicações, autores, etc.; (b) algumas estruturas da álgebra abstrata: monóides, grupos, álgebra de Boole, etc.; (c) determinados conceitos da lógica matemática: pré-ordem; ordem parcial; equivalência, álgebra de Lindenbaum, etc. A segunda parte, trata da aproximação da teoria das categorias à lógica proposicional, isto é, investiga as propriedades, estruturas e relações próprias à lógica proposicional categórica. Nesta passagem, há uma reinterpreta ção dos conectivos lógicos da negação, conjunção, disjunção e implicação, bem como dos conectivos booleanos de complemento, interseção e união, em termos categóricos. Na seqüência, estas novas concepções permitem enunciar uma álgebra booleana categórica, por meio da qual, ao final, é construída uma semântica proposicional booleana categórica.

Autômatos não-seqüenciais constituem um domínio semântico categorial do tipo não-intercalação para sistemas reativos, comunicantes e concorrentes.É baseado em sistemas de transições etiquetados, inspirado em "Redes de Petri são Monóides" de Meseguer e Montanari, onde as operações de sincronização e encapsulação são funtoriais e as reificações constituem uma classe de morfismos especiais. Do que se tem conhecimento, é o primeiro modelo de concorrência a satisfazer a composicionalidade diagonal, ou seja, onde as reificações compõem (verticalmente) e distribuem-se sobre a composição paralela (verticalmente). Adjunções entre autômatos não-seqüenciais, redes de Petri e autômatos seqüenciais são introduzidas estendendo a abordagem de Winskel, Nielsen e Sassone onde é proposta uma classificação formal para modelos de concorrência. Dos passos que envolvem a passagem de um modelo para outro, pode-se inferir que os autômatos não-seqüenciais são mais concretos do que as redes de Petri e os autômatos seqüenciais. Para experimentar o domínio semântico proposto, é dada semântica a uma linguagem concorrente, baseada nos objetos, denominada Náutilus. Trata-se de uma versão simplificada e revisada da linguagem de especificação orientada aos objetos GNOME, onde são introduzidos algumas facilidades especiais, inspiradas no domínio semântico, como a reificação e a agregação. Neste contexto, a composicionalidade diagonal é uma propriedade essencial para dar a semântica. / Nonsequential automata constitute a non-interleaving categorial semantic domain for reactive, communicating and concurrent systems. It is based on labeled transition systems, inspired by Meseguer and Montanari's "Petri Nets are Monoids", where synchronization and encapsulation operations are functorial and a class of morphisms stands for reification. It is, for our knowledge, the first model for concurrency which satisfies the diagonal compositionality requirement, i. e., reifications compose (vertical) and distribute over the parallel composition (horizontal). Adjunctions between nonsequential automata, Petri nets and sequential automata are provided extending the approach of Winskel, Nielsen and Sassone where a scene for a formal classification of models for concurrency is set. The steps of abstraction involved in moving between models show that nonsequential automata are more concrete than Petri nets and sequential automata. To experiment with the proposed semantic domain, a semantics for a concurrent, object-based language named Nautilus is given. It is a simplified and revised version of the object-oriented specification language GNOME, introducing some special features inspired by the semantic domain such as reification and aggregation. The diagonal compositionality is an essential property to give semantics in this context.

Introduced by C.C. Chang in the 1950s, MV algebras are to many-valued (Łukasiewicz) logics what boolean algebras are to two-valued logic. More recently, effect algebras were introduced by physicists to describe quantum logic. In this thesis, we begin by investigating how these two structures, introduced decades apart for wildly different reasons, are intimately related in a mathematically precise way. We survey some connections between MV/effect algebras and more traditional algebraic structures. Then, we look at the categorical structure of effect algebras in depth, and in particular see how the partiality of their operations cause things to be vastly more complicated than their totally defined classical analogues. In the final chapter, we discuss coordinatization of MV algebras and prove some new theorems and construct some new concrete examples, connecting these structures up (requiring a detour through effect algebras!) to boolean inverse semigroups.

The goal of this paper is to extend Sanders Mac Lane's formulation of the torsion product as equivalence classes of projective chain complexes in the setting of modules over a ring to the setting of modules over small categories. The motivation to extend the definition was with a specific view to the orbit category. The main difficulty was in defining an appropriate dual for modules over small categories. During the course of our investigation it was discovered that modules over small categories can be formulated as modules over a matrix ring without losing any of the key features. / Thesis / Master of Science (MSc)

This thesis studies the categorical formalisation of quantum computing, through the prism of type theory, in a three-tier process. The first stage of our investigation involves the creation of the dagger lambda calculus, a lambda calculus for dagger compact categories. Our second contribution lifts the expressive power of the dagger lambda calculus, to that of a quantum programming language, by adding classical control in the form of complementary classical structures and dualisers. Finally, our third contribution demonstrates how our lambda calculus can be applied to various well known problems in quantum computation: Quantum Key Distribution, the quantum Fourier transform, and the teleportation protocol.

This work is about diagrammatic languages, how they can be represented, and what they in turn can be used to represent. More specifically, it focuses on representations and applications of string diagrams. String diagrams are used to represent a collection of processes, depicted as "boxes" with multiple (typed) inputs and outputs, depicted as "wires". If we allow plugging input and output wires together, we can intuitively represent complex compositions of processes, formalised as morphisms in a monoidal category. While string diagrams are very intuitive, existing methods for defining them rigorously rely on topological notions that do not extend naturally to automated computation. The first major contribution of this dissertation is the introduction of a discretised version of a string diagram called a string graph. String graphs form a partial adhesive category, so they can be manipulated using double-pushout graph rewriting. Furthermore, we show how string graphs modulo a rewrite system can be used to construct free symmetric traced and compact closed categories on a monoidal signature. The second contribution is in the application of graphical languages to quantum information theory. We use a mixture of diagrammatic and algebraic techniques to prove a new classification result for strongly complementary observables. Namely, maximal sets of strongly complementary observables of dimension D must be of size no larger than 2, and are in 1-to-1 correspondence with the Abelian groups of order D. We also introduce a graphical language for multipartite entanglement and illustrate a simple graphical axiom that distinguishes the two maximally-entangled tripartite qubit states: GHZ and W. Notably, we illustrate how the algebraic structures induced by these operations correspond to the (partial) arithmetic operations of addition and multiplication on the complex projective line. The third contribution is a description of two software tools developed in part by the author to implement much of the theoretical content described here. The first tool is Quantomatic, a desktop application for building string graphs and graphical theories, as well as performing automated graph rewriting visually. The second is QuantoCoSy, which performs fully automated, model-driven theory creation using a procedure called conjecture synthesis.

[pt] Este trabalho emprega conceitos e ferramentas de Teoria das Categorias e Teoria de Topoi para construir um modelo matemático de problemas, reduções entre problemas, espaços e estratégias de busca heurística. Mais precisamente, uma estratégia de construção de espaços de busca é representada por um funtor de uma certa categoria de problemas para uma certa categoria de florestas. A coleção de todos estes funtores forma um topos, um modelo específico equipado com uma lógica interna própria. Esta lógica interna é usada, então, para definir estratégias de busca e heurísticas em Teoria Local dos Conjuntos. Possíveis aplicações do trabalho incluem (1) a especificação lógica e a classificação de heurísticas e meta-heurísticas usadas na prática e (2) uma versão mais abstrata e geral de resultados específicos relacionando a estrutura de problemas com métodos de resolução adequados. / [en] This work employs concepts and tools from Category Theory and Topos Theory to construct a mathematical model for problems, reductions between problems, heuristic search spaces and strategies. More precisely, a search space construction strategy is represented by a functor from a certain category of problems to a certain category of forests. The collection of all such functors forms a topos, a specific model equipped with its own internal logic. This internal logic is then used to define search satrategies and heuristics in Local Set Theory. Possible applications of this work include (1) the logical specification and classification of heuristics and metaheuristics used in pratice and (2) a more abstract and general rendering of specific results relating the structure of problems to adequate problem-solving methods.