Drawing data from a variety of sources, this thesis compares functional evidence regarding lexical categories from a number of Salish and Wakashan languages, as well as from the Michif language. It then applies Prototype Theory to examine the structure of the lexicons of these languages. They are described in terms of prototype categories that overlap to varying extents, with each category and each area of overlap defined by a central set of prototypical features. A high degree of gradience appears to exist between categories in Salish and Wakashan languages, with no clear boundary between categories or areas of overlap, indicating that lexical categories in these languages, rather than being clearly demarcated, are instead fuzzy categories with very little distinguishing them. Categories in Michif, on the other hand, exhibit far less overlap. This variation is compared to variation in conceptual categories across languages, and challenges the notions of the universality of clearly demarcated lexical categories and the existence of separately stored language module in the human mind. In spite of the variation in lexical category demarcation observed across the languages studied, it is possible to demarcate the categories of Noun and Verb to at least some extent in all languages, as well as a category of Adjective in some languages. This supports the proposed universality of the categories of Noun and Verb, as well as the implicational universals proposed in the Amsterdam Model of Parts of Speech (Hengeveld 1992a, b). It is also possible to identify a number of defining characteristics for each lexical category that appear to hold across languages. Since similar characteristics can be identified across languages for all categories, but the categories themselves display varying degrees of overlap in individual languages, this research supports the proposal that language universals, rather than consisting of structures, rules, and categories that are identical in all languages, are rather collections of prototypical characteristics for grammatical categories that are similar across languages (Croft 2000).

String diagrams turn algebraic equations into topological moves that have recurring shapes, involving the sliding of one diagram past another. We individuate, at the root of this fact, the dual nature of polygraphs as presentations of higher algebraic theories, and as combinatorial descriptions of "directed spaces". Operations of polygraphs modelled on operations of topological spaces are used as the foundation of a compositional universal algebra, where sliding moves arise from tensor products of polygraphs. We reconstruct several higher algebraic theories in this framework. In this regard, the standard formalism of polygraphs has some technical problems. We propose a notion of regular polygraph, barring cell boundaries that are not homeomorphic to a disk of the appropriate dimension. We define a category of non-degenerate shapes, and show how to calculate their tensor products. Then, we introduce a notion of weak unit to recover weakly degenerate boundaries in low dimensions, and prove that the existence of weak units is equivalent to a representability property. We then turn to applications of diagrammatic algebra to quantum theory. We re-evaluate the category of Hilbert spaces from the perspective of categorical universal algebra, which leads to a bicategorical refinement. Then, we focus on the axiomatics of fragments of quantum theory, and present the ZW calculus, the first complete diagrammatic axiomatisation of the theory of qubits. The ZW calculus has several advantages over ZX calculi, including a computationally meaningful normal form, and a fragment whose diagrams can be read as setups of fermionic oscillators. Moreover, its generators reflect an operational classification of entangled states of 3 qubits. We conclude with generalisations of the ZW calculus to higher-dimensional systems, including the definition of a universal set of generators in each dimension.

Cette thèse porte sur l'étude des données $mathbb{Z}$-modulaires et leur catégorification, et particulièrement sur des données $mathbb{Z}$-modulaires reliées aux groupes de réflexions complexes, ainsi que sur la notion de caractère cellulaire pour ces derniers. Dans sa classification des caractères des groupes finis de type de Lie, Lusztig décrit une transformée de Fourier non abélienne et définit des données $mathbb{N}$-modulaires pour chaque famille de caractères unipotents. Dans des tentatives de généralisation aux Spetses, Broué, Malle et Michel introduisent des données $mathbb{Z}$-modulaires. On commence par donner une explication catégorique de certaines de ces données via la catégorie des représentations du double de Drinfeld d'un groupe fini, que l'on munit d'une structure pivotale non sphérique. Une étude approfondie de la notion de catégorie de fusion pivotale et légèrement dégénérée montre que l'on peut ainsi produire des données $mathbb{Z}$-modulaires. Afin de construire des exemples de telles catégories, on considère des extensions des catégories de fusion associées à $qgrroot{mathfrak{g}}$, où $mathfrak{g}$ est une algèbre de Lie simple, et $xi$ une racine de l'unité. Ces dernières sont construites comme des semi-simplifications de la catégorie des modules basculants de l'algèbre $qdblroot{mathfrak{g}}$, qui est une extension centrale de $qgrroot{mathfrak{g}}$. Dans le cas où $mathfrak{g}=mathfrak{sl}_{n+1}$, on relie cette catégorie à une des données $mathbb{Z}$-modulaires associée au groupe de réflexions complexes $Gleft(d,1,frac{n(n+1)}{2}right)$. Les groupes de réflexions exceptionnels sont également étudiés, et les catégorifications des données $mathbb{Z}$-modulaires associées font apparaître diverses catégories : des catégories de représentations de doubles de Drinfeld tordus ainsi que des sous-catégories des catégories de fusion des modules basculants en $qdblroot{mathfrak{g}}$ en type $A$ et $B$. / This work is a contribution to the categorification of $mathbb{Z}$-modular data and deals mainly with $mathbb{Z}$-modular data arising from complex reflection groups, as well as cellular characters for these groups. In his classification of representations of finite groups of Lie type, Lusztig defines a nonabelian Fourier transform, and associate a $mathbb{N}$-modular datum to each family of unipotent characters. In a generalization of Lusztig's theory to Spetses, Broué, Malle and Michel construct $mathbb{Z}$-modular data associated to some complex reflection groups. We first give a categorical explanation of some of these $mathbb{Z}$-modular data in terms of representation of the Drinfeld double of a finite group. We had to endow the category of representations with a non-spherical structure. The study of slightly degenerate categories shows that they naturally give rise to $mathbb{Z}$-modular data. In order to construct some examples, we consider an extension of the fusion categories associated to $qgrroot{mathfrak{g}}$, where $mathfrak{g}$ is a simple Lie algebra and $xi$ a root of unity. These categories are constructed as semisimplification of the category of tilting modules of $qdblroot{mathfrak{g}}$, which is a central extension of $qgrroot{mathfrak{g}}$. If $mathfrak{s}=mathfrak{sl}_{n+1}$, we show that this category is related to some $mathbb{Z}$-modular data associated to the complex reflection group $Gleft(d,1,frac{n(n+1)}{2}right)$. Exceptional complex reflection groups are also considered and many different categories appear in the categorification of the associated $mathbb{Z}$-modular data : modules categories over twisted Drinfeld doubles as well as some subcategories of fusion categories of tilting modules over $qdblroot{mathfrak{g}}$ in type $A$ and $B$.

The aim of this work is to study the ring-theoretic properties of the diagonals of a Rees algebra, which from a geometric point of view are the homogenous coordinate rings of embeddings of blow-ups of projective varieties along a subvariety. First we are going to introduce the subject and the main problems. After that we shall review the known results about these problems, and finally we will give a summary of the contents and results obtained in this work. / L’objectiu d’aquesta memòria és l’estudi de les propietats aritmètiques de les diagonals d’una àlgebra de Rees o, des d’un punt de vista geomètric, dels anells de coordenades homogenis d’immersions d’explosions de varietats projectives al llarg d’una subvarietat. En primer lloc, anem a introduir el tema i els principals problemes que tractarem. A continuació, exposarem els resultats coneguts sobre aquests problemes i finalment farem un resum dels resultats obtinguts en aquesta memòria.

Une théorie des champs quantique topologique (TQFT) en dimension 3 est un foncteur monoidal symétrique de la catégorie des cobordismes de dimension 3 vers celle des espaces vectoriels. Une TQFT fournit en particulier un invariant scalaire des variétés fermées de dimension 3 ainsi que des représentations du groupe de difféotopie des surfaces fermées.Turaev explique en 1994 comment construire à partir d'une catégorie modulaire une TQFT qui étend l'invariant scalaire de 3-variétés fermées introduit en 1991 par Reshetikhin et Turaev. Dans cette thèse, nous généralisons cette construction à l'aide d'une catégorie C en ruban avec coend. On représente un cobordisme par un enchevêtrement d'un type particulier (enchevêtrement de cobordisme) et on associe à celui-ci un morphisme défini entre puissances tensorielles de la coend comme décrit par Lyubashenko en 1995. A l'aide de l'extension du calcul de Kirby aux cobordismes de dimension 3, cette construction nous permet de produire un invariant de cobordismes puis une TQFT à valeurs dans la sous-catégorie monoïdale symétrique des objets transparents de C.Dans le cas où C est une catégorie modulaire, cette sous-catégorie s'identifie à celle des espaces vectoriels et on retrouve ainsi la TQFT de Turaev. Dans le cas où C est une catégorie prémodulaire modularisable, notre TQFT est un relèvement de la TQFT de Turaev associée à la modularisée de C. / A 3-dimensional topological quantum field theory (TQFT) is a symmetric monoidal functor from the category of 3-cobordisms to the category of vector spaces. Such TQFTs provide in particular numerical invariants of closed 3-manifolds and representations of the mapping class group of closed surfaces.In 1994, Turaev explains how to construct a TQFT from a modular category; the scalar invariant is then the Reshethikhin-Turaev invariant introduced in 1991. In this thesis, we describe a generalization of this construction starting from a ribbon category C with coend. We present a cobordism by a certain type of tangle (cobordism tangle) and we associate to such a tangle a morphism between tensor products of the coend as described by Lyubashenko in 1994. Extending the Kirby calculus to 3-cobordisms, we obtain in this way an invariant of cobordisms and a TQFT which takes values in the symmetric monoidal subcategory of transparent objects of C. If the category C is modular, this subcategory can be identified with the category of vector spaces, and we recover Turaev's TQFT. If the category C is modularizable, our TQFT is a lift of the Turaev TQFT for the modularization of C.

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 général / The 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

Antalet övervakningskameror i samhället ökar, samtidigt som deras miljöpåverkan är relativt okänd. För en produkt som tidigare bara utvärderats gällande prestanda, börjar kunderna nu efterfråga en kartläggning av miljöpåverkan. Kamerans miljöpåverkan studeras ur ett livscykelperspektiv: energi- och materialtillförsel samt utsläpp och avfall som berör allt ifrån råmaterialutvinning till slutåtervinning. Projektet är genomfört på begäran av, och för, Axis Communications AB (härmed refererade till som Axis) med huvudsyfte att öka Axis kunskap om deras produkters miljöpåverkan och utveckla en metod för genomförande av förenklade livscykelanalyser på Axis produkter. Livscykelanalysen, LCA, genomförs på en nätverkskamera utvecklad av Axis Communications AB; modell AXIS Q6032-E PTZ. Samtidigt togs metoden för förenklade livscykelanalyser fram för att möjliggöra jämförelse med andra kameramodeller i företagets produktportfolio. Även en plattform har skapats för att kunna användas i produktutvecklingens tidigare skede, för att redan där göra ett aktivt miljöval. Den metod som används för bedömning av miljöpåverkan är Eco-indicator 99. Simulering och beräkningar sker i LCA-programmet SimaPro 7.1.Resultatet visar att den största miljöpåverkan kommer ifrån användningsfasen och behovet av elektricitet. För scenariot där kameran används i Europa har tillverkningen näst störst påverkan, därefter materialanvändningen och sist transporterna. Återvinningen påverkar med ett negativt värde, d.v.s. den påverkar miljön på ett positivt sätt. Det alternativa scenariot (där kameran flygs till USA och installeras där) ger en totalt större miljöpåverkan och har transporterna som andra värsta kategorin. Vid beräkningar för Europa släpper kameran ut 663 kg CO2 under sin livstid. Den utvecklade modellen överensstämmer till 0,24% med resultatet ifrån simuleringsprogrammet. Modellen kan enligt den genomförda känslighetsanalysen anses stabil / The number of surveillance cameras installed for various purposes have increased substantially in society over the past decade. The environmental impacts from network cameras are relatively unknown and their rapid increase in number calls for studying the impacts from a life cycle perspective; from raw material extraction to decommissioning. The project is performed on request by Axis Communications AB (hereby referred to as Axis) with the main purposes to increase Axis's knowledge of the environmental impact from their products and establish a method for conducting simplified life cycle assessments (LCA) on Axis products. A case study LCA is conducted on a network camera developed by Axis; model AXIS Q6032-E PTZ. Concurrently a method for conducting simplified LCAs on other Axis cameras is developed as well as a platform to be used in product development processes to enhance life cycle thinking (LCT). The Eco-indicator 99 Method is used for the environmental impact assessment and for simulations and calculations the software program SimaPro 7.1 is used.The results emphasize the life stages and their particular activities having the largest potential environmental impacts; primarily utilization and the production of electricity. For the scenario where the camera is installed in Europe the manufacturing comes as second, then raw material extraction and processing, followed by transportations. Decommissioning impacts with a negative value, i.e. impacts the environment in a positive way. The alternative scenario (where the camera is transported by air to U.S. and installed there) gives a total higher score and has the transportation category as the second highest regarding the total environmental impact. During the whole lifetime the camera emits 663 kg CO2.The results from using the developed model to conduct simplified LCAs only differ by 0.24% from the results of the case study LCA. The LCA is considered stable based on the performed sensitivity analysis.

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This thesis concerns a range of aspects of theoretical physics. It is composed of two parts. In the first part we motivate our line of research, and introduce and discuss the relevant concepts. In the second part, four research papers are collected. The first paper deals with a possible extension of general relativity, namely the recently discovered classically consistent bimetric theory. In this paper we study the behavior of perturbations of the metric(s) around cosmologically viable background solutions. In the second paper, we explore possibilities for particle physics with low-scale supersymmetry. In particular we consider the addition of supersymmetric higher-dimensional operators to the minimal supersymmetric standard model, and study collider phenomenology in this class of models. The third paper deals with a possible extension of the notion of Lie algebras within category theory. Considering Lie algebras as objects in additive symmetric ribbon categories we define the proper Killing form morphism and explore its role towards a structure theory of Lie algebras in this setting. Finally, the last paper is concerned with the computation of string amplitudes in four dimensional models with reduced supersymmetry. In particular, we develop general techniques to compute amplitudes involving gauge bosons and gravitons and explicitly compute the corresponding three- and four-point functions. On the one hand, these results can be used to extract important pieces of the effective actions that string theory dictates, on the other they can be used as a tool to compute the corresponding field theory amplitudes. / Over the last twenty years there have been spectacular observations and experimental achievements in fundamental physics. Nevertheless all the physical phenomena observed so far can still be explained in terms of two old models, namely the Standard Model of particle physics and the ΛCDM cosmological model. These models are based on profoundly different theories, quantum field theory and the general theory of relativity. There are many reasons to believe that the SM and the ΛCDM are effective models, that is they are valid at the energy scales probed so far but need to be extended and generalized to account of phenomena at higher energies. There are several proposals to extend these models and one promising theory that unifies all the fundamental interactions of nature: string theory. With the research documented in this thesis we contribute with four tiny drops to the filling of the fundamental physics research pot. When the pot will be saturated, the next fundamental discovery will take place.