No description available.

Doctor of Philosophy / Department of Mathematics / David Yetter / This dissertation begins the development of the deformation theorem of monoidal categories which accounts for the function that all arrow-valued operations, composition, the arrow part of the monoidal product, and structural natural transformation are deformed. The first chapter is review of algebra deformation theory. It includes the Hochschild complex of an algebra, Gerstenhaber's deformation theory of rings and algebras, Yetter's deformation theory of a monoidal category, Gerstenhaber and Schack's bialgebra deformation theory and Markl and Shnider's deformation theory for Drinfel'd algebras. The second chapter examines deformations of a small $k$-linear monoidal category. It examines deformations beginning with a naive computational approach to discover that as in Markl and Shnider's theory for Drinfel'd algebras, deformations of monoidal categories are governed by the cohomology of a multicomplex. The standard results concerning first order deformations are established. Obstructions are shown to be cocycles in the special case of strict monoidal categories when one of composition or tensor or the associator is left undeformed. At the end there is a brief conclusion with conjectures.

No description available.

In this dissertation we generalise the basic theory of Hopf algebras to the context of autonomous pseudomonoids in monoidal bicategories. Autonomous pseudomonoids were introduced in [13] as generalisations of both autonomous monoidal categories and Hopf algebras. Much of the theory of autonomous pseudomonoids developed in [13] was inspired by the example of autonomous (pro)monoidal enriched categories. The present thesis aims to further develop the theory with results inspired by Hopf algebra theory instead. We study three important results in Hopf algebra theory: the so-called 'fundamental theorem of Hopf modules', the 'Drinfel'd quantum double' and its relation with the centre of monoidal categories, and 'Radford's formula'. The basic result of this work is a general fundamental theorem of Hopf modules that establishes conditions equivalent to the existence of a left dualization. With this result as a base, we are able to construct the centre (defined in [83]) and the lax centre of an autonomous pseudomonoid as an Eilenberg-Moore construction for certain monad. As an application we show that the Drinfel'd double of a finite-dimensional Hopf algebra is equivalent to the centre of the associated pseudomonoid. The next piece of theory we develop is a general Radford's formula for autonomous map pseudomonoids formula in the case of a (coquasi) Hopf algebra. We also introduce 'unimodular' autonomous pseudomonoids. In the last part of the dissertation we apply the general theory to enriched categories with a (chosen) class of (co)limits, with emphasis in the case of finite (co)limits. We construct tensor products of such categories by means of pseudo-commutative enriched monads (a slight generalisation of the pseudo-commutative 2-monads of [37], and showing that lax-idempotent 2-monads are pseudo-commutative. Finally we apply the general theory developed for pseudomonoids to deduce the main results of [27].

Cette thèse se propose d’étudier des problèmes de classification des groupes quantiques via des invariants issus de leur théorie de représentation. Plus précisément, nous classifions les algèbres de Hopf possédant un semi-anneau de fusion isomorphe à un groupe algébrique réductif donné G. De tels groupes quantiques sont alors appelés G-déformations. Dans cette thèse, nous étudions les cas GL(2) et SO(3). Nous donnons une classification complète des GL(2)-déformations en construisant une famille d’algèbres de Hopf indexées par des matrices inversibles. Nous décrivons leurs catégories de comodules et donnons certains résultats de classification quant à leurs objets de Hopf-Galois. Ensuite, nous donnons une classification des SO(3)-déformations compactes tout en étudiant le cas non-compact. Finalement, la dernière partie de la thèse est une étude de l’algèbre sous-jacente à une certaine famille d’algèbres de Hopf, dont nous exhibons une base. Cette base nous permet de calculer le centre des ces algèbres ainsi que quelques groupes de (co)homologie. / The purpose of this dissertation is to classify quantum groups according to invariants coming from their representation theory. More precisely, we classify Hopf algebras having a fusion semiring isomorphic to that of a given reductive algebraic group G. Such a quantum group is called a G-deformation. We study the case of GL(2) and SO(3). We give a complete classification of GL(2)-deformations by building a family of Hopf algebras parametrized by invertible matrices. We describe their comodule category and we give some classification results about the Hopf-Galois objects. We also classify compact SO(3)-deformations and we study the noncompact case. Finally, the last part of this dissertation is a study of the underlying algebra of some Hopf algebras, for which we exhibit a linear basis. This basis allows us to compute the centre and some (co)homology groups of those algebras.

Thèse par articles. / La présente thèse fait état des avancées en logique déontique et propose des outils formels pertinents à l'analyse de la validité des inférences légales. D'emblée, la logique vise l'abstraction de différentes structures. Lorsqu'appliquée en argumentation, la logique permet de déterminer les conditions de validité des inférences, fournissant ainsi un critère afin de distinguer entre les bons et les mauvais raisonnements. Comme le montre la multitude de paradoxes en logique déontique, la modélisation des inférences normatives fait cependant face à divers problèmes. D'un point de vue historique, ces difficultés ont donné lieu à différents courants au sein de la littérature, dont les plus importants à ce jour sont ceux qui traitent de l'action et ceux qui visent la modélisation des obligations conditionnelles. La présente thèse de doctorat, qui a été rédigée par articles, vise le développement d'outils formels pertinents à l'analyse du discours juridique. En première partie, nous proposons une revue de la littérature complémentaire à ce qui a été entamé dans Peterson (2011). La seconde partie comprend la contribution théorique proposée. Dans un premier temps, il s'agit d'introduire une logique déontique alternative au système standard. Sans prétendre aller au-delà de ses limites, le système standard de logique déontique possède plusieurs lacunes. La première contribution de cette thèse est d'offrir un système comparable répondant au différentes objections pouvant être formulées contre ce dernier. Cela fait l'objet de deux articles, dont le premier introduit le formalisme nécessaire et le second vulgarise les résultats et les adapte aux fins de l'étude des raisonnements normatifs. En second lieu, les différents problèmes auxquels la logique déontique fait face sont abordés selon la perspective de la théorie des catégories. En analysant la syntaxe des différents systèmes à l'aide des catégories monoïdales, il est possible de lier certains de ces problèmes avec des propriétés structurelles spécifiques des logiques utilisées. Ainsi, une lecture catégorique de la logique déontique permet de motiver l'introduction d'une nouvelle approche syntaxique, définie dans le cadre des catégories monoïdales, de façon à pallier les problèmes relatifs à la modélisation des inférences normatives. En plus de proposer une analyse des différentes logiques de l'action selon la théorie des catégories, la présente thèse étudie les problèmes relatifs aux inférences normatives conditionnelles et propose un système déductif typé. / The present thesis develops formal tools relevant to the analysis of legal discourse. When applied to legal reasoning, logic can be used to model the structure of legal inferences and, as such, it provides a criterion to discriminate between good and bad reasonings. But using logic to model normative reasoning comes with some problems, as shown by the various paradoxes one finds within the literature. From a historical point of view, these paradoxes lead to the introduction of different approaches, such as the ones that emphasize the notion of action and those that try to model conditional normative reasoning. In the first part of this thesis, we provide a review of the literature, which is complementary to the one we did in Peterson (2011). The second part of the thesis concerns our theoretical contribution. First, we propose a monadic deontic logic as an alternative to the standard system, answering many objections that can be made against it. This system is then adapted to model unconditional normative inferences and test their validity. Second, we propose to look at deontic logic from the proof-theoretical perspective of category theory. We begin by proposing a categorical analysis of action logics and then we show that many problems that arise when trying to model conditional normative reasoning come from the structural properties of the logic we use. As such, we show that modeling normative reasoning within the framework of monoidal categories enables us to answer many objections in favour of dyadic and non-monotonic foundations for deontic logic. Finally, we propose a proper typed deontic system to model legal inferences.

Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C*-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C*-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered.

Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C*-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C*-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered.

Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C*-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C*-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered.

Algebraic Quantum Field Theory (AQFT) is a mathematically rigorous framework that was developed to model the interaction of quantum mechanics and relativity. In AQFT, quantum mechanics is modelled by C*-algebras of observables and relativity is usually modelled in Minkowski space. In this thesis we will consider a generalization of AQFT which was inspired by the work of Abramsky and Coecke on abstract quantum mechanics [1, 2]. In their work, Abramsky and Coecke develop a categorical framework that captures many of the essential features of finite-dimensional quantum mechanics. In our setting we develop a categorified version of AQFT, which we call premonoidal C*-quantum field theory, and in the process we establish many analogues of classical results from AQFT. Along the way we also exhibit a number of new concepts, such as a von Neumann category, and prove several properties they possess. We also establish some results that could lead to proving a premonoidal version of the classical Doplicher-Roberts theorem, and conjecture a possible solution to constructing a fibre-functor. Lastly we look at two variations on AQFT in which a causal order on double cones in Minkowski space is considered.