The contribution of this thesis is a novel framework for rewriting in higher categories. Its theoretical foundation is the theory of quasistrict higher categories and the practical realisation is a proof assistant Globular. The framework introduces the notions of diagrams and signatures as new mutually-recursive structures that give the algebraic basis for the approach. These structures are related the notion of an n-polygraph, but allow reasoning about quasistrict higher categorical structures in a way amenable to computer implementation. Building on this, we propose a new definition of a quasistrict 4-category, and prove a result that in a quasistrict 4-category, an adjunction of 1-morphisms gives rise to a coherent adjunction satisfying the butterfly equations.

abstract: Zwischen in the German language means `between,' and over the past century, as operatic voices have evolved in both range and size, the voice classification of Zwischenfach has become much more relevant - particularly to the female voice. Identifying whether nineteenth century composers recognized the growing opportunities for vocal drama, size, and range in singers and therefore wrote roles for `between' singers; or conversely whether, singers began to challenge and develop their voices to sing the new influx of romantic, verismo and grand repertoire is difficult to determine. Whichever the case, teachers and students should not be surprised about the existence of this nebulous Fach. A clear and concise definition of the word Fach for the purpose of this paper is as follows: a specific voice classification. Zwischenfach is an important topic because young singers are often confused and over-eager to self-label due to the discipline's excessive labeling of Fachs. Rushing to categorize a young voice ultimately leads to misperceptions. To address some of the confusion, this paper briefly explores surveys of the pedagogy and history of the Fach system. To gain insights into the relevance of Zwischenfach in today's marketplace, I developed with my advisors, colleagues and students a set of subjects willing to fill out questionnaires. This paper incorporates current interviews from two casting directors of national and international opera houses, an emerging American mezzo-soprano, a mid-career working European mezzo-soprano, an operatic stage director, an education director for opera houses and a composer. These interviews, along with modern examples of zwischenfach voices are analyzed and discussed. / Dissertation/Thesis / D.M.A. Music 2012

We construct a semi-orthogonal decomposition for the equivariant derived category of a hypersurface associated to the sum of two potentials. More specifically, if $f,g$ are two homogeneous poynomials of degree $d$ defining smooth Calabi-Yau or general type hypersurfaces in $\mathbb{P}^n$, we construct a semi-orthogonal decomposition of $D[V(f\oplus g)/\mu_d]$. Moreover, every component of the semi-orthogonal decomposition is explicitly given by Fourier-Mukai functors.

For each abelian category A, there is a category D(A), called the derived category of A, whose objects are complexes of objects of A, and whose morphisms are formal fractions of homotopy classes of complex morphisms having as denominators homotopy classes inducing isomorphisms in cohomology. If F : A →B is an additive functor between abelian categories, then under suitable conditions on A, there is a functor RF : D(A) → D(B) with the property that if objects X of A are considered as complexes concentrated at degree 0, then there are isomorphisms [formula omitted] for all n, where [formula omitted] is the ordinary [formula omitted] right derived functor of F. RF is called the derived functor of F, and one may look upon it as a kind of extension of F. / Science, Faculty of / Mathematics, Department of / Graduate

This thesis is concerned With showing the relations among Boolean lattices, Boolean rings and Boolean spaces, It establishes that the categories of Boolean lattices and proper Boolean lattice homomorphisms, Boolean spaces and proper continuous maps, Boolean rings and proper ring homomorphisms are equivalent to each other, In the final chapter the notion of a Boolean semi-group is used to obtain an alternate characterization of a Boolean lattice. / Thesis / Master of Science (MS)

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The German word Zwischenfach often refers to opera roles and singers whose voices lie between the categories of mezzo-soprano and soprano. While the term is not universally accepted as a voice category, Zwischenfach voices and roles are being discussed more openly and with more specificity in collegiate and professional circles. This document includes a discussion on the challenges of categorizing dramatic voices, mezzo-soprano voices, and those who could be considered Zwischenfach, taking into consideration the inherent ambiguity and flexibility within these voice categories. The elements that have led to developmental changes in opera voices and their categories over the centuries provide insight and context on how Zwischenfach has become a term that describes the ambiguity and challenge of classifying opera voices in the twenty-first century. A main focus of this document is a discussion of eleven pieces from twenty-first century American operas which a Zwischenfach singer could consider for auditions and performances. Operas included are: Dead Man Walking by Jake Heggie, The Grapes of Wrath by Ricky Ian Gordon, After Life and Glory Denied by Tom Cipullo, Lysistrata by Mark Adamo, Dinner at Eight by William Bolcom, and Fantastic Mr. Fox by Tobias Picker. My hope is that this document will give Zwischenfach voices a resource when choosing twenty-first century repertoire and promote further discussion and acceptance of the Zwischenfach voice category.

The purpose of this paper is to examine some basic topics in category theory. A category consists of a class of mathematical objects along with a morphism class having an associative composition. The paper is divided into two chapters. Chapter I deals with intrinsic properties of categories. Various "sub-objects" and properties of morphisms are defined and examples are given. Chapter II deals with morphisms between categories called functors and the natural transformations between functors. Special types of functors are defined and examples are given.