Thesis (Ph.D.)--Tufts University, 2004. / Adviser: Loring W. Tu. Submitted to the Dept. of Mathematics. Includes bibliographical references (leaves 43-45). Access restricted to members of the Tufts University community. Also available via the World Wide Web;

Algebraic theories and algebraic categories offer an innovative and revelatory description of the syntax and the semantics. An algebraic theory is a concrete mathematical object -- the concept -- namely a set of variables together with formal symbols and equalities between these terms; stated otherwise, an algebraic theory is a small category with finite products. An algebra or model of the theory is a set-theoretical interpretation -- a possible meaning -- or, more categorically, a finite product-preserving functor from the theory into the category of sets. We call the category of models of an algebraic theory an algebraic category. By generalising the theory we do generalise the models. This concept is the fascinating aspect of the subject and the reference point of our project. We are interested in the study of categories of models. We pursue our task by considering models of different theories and by investigating the corresponding categories of models they constitute. We analyse localizations (namely, fully faithful right adjoint functors whose left adjoint preserves finite limits) of algebraic categories and localizations of presheaf categories. These are still categories of models of the corresponding theory. We provide a classification of localizations and a classification of geometric morphisms (namely, functors together with a finite limit-preserving left adjoint), in both the presheaf and the algebraic context.

Coordenação de Aperfeiçoamento de Pessoal de Nível Superior - CAPES / The aim of this dissertation is to present a simpli cation of the proof of the following result obtained rst by Happel in [3]: If A is a nite-dimensional algebra over a eld algebraically closed K, then there is a triangulated, full and faithful functor of triangulated categories H : Db(modA) ! modA^, where A^ is the repetitive algebra obtained from A, which is also dense if A is of nite global dimension. We begin with a succinct presentation of the categorical language, approaching in general terms on the localization of categories, triangulated categories and their localizations, and nally derived categories, which are localized and triangulated categories. We also introduce the stable category of modules of a repetitive algebra A^. In the last chapter, we demonstrate the main result with the help of a result found in [8], in addition to the previously mentioned concepts. / O objetivo dessa disserta c~ao e trazer uma simpli ca c~ao da demonstra c~ao do seguinte resultado obtido primeiramente por Happel [3]: Se A e uma K- algebra de dimens~ao nita, ent~ao existe um funtor pleno, el e triangulado H : Db(modA) ! modA^, onde A^ e a a lgebra repetitiva obtida de A, que e tamb em denso se A e de dimensa~o global nita. Iniciamos com uma apresenta c~ao sucinta da linguagem categ orica, abordando de maneira geral sobre localiza c~ao de categorias, categorias trianguladas e suas localiza c~oes, e nalmente categorias derivadas, que s~ao categorias localizadas e trianguladas. Tamb em introduzimos a categoria est avel de m odulos da algebra repetitiva de A. No ultimo cap tulo, demonstramos o resultado principal com o aux lio de um resultado encontrado em [8], al em dos conceitos citados anteriormente. / São Cristóvão, SE