The aim of this work is to establish precisely what are the functorial connections between the abstract theories of quadratic forms, as well as, to create a short and introductory path from the classic theory to the abstract ones. There is a large amount of literature developed about classic and abstract theories but does note relate them ``geographically\'\'. In this perspective, we discuss the fundamental aspects of the classic and reduced theory of quadratic forms, and sum up the theories of Quaternionic Structures, Cordes Schemes, Abstract Witt Rings, Abstract Ordering Spaces, Special Groups, Abstract Real Spectra and Real Semigroups in a functorial picture, inserting the new aspects involve the recent theory of Multirings and Multifields. / O principal objetivo deste trabalho é estabelecer precisamente quais são as conexões funtoriais entre as teorias abstratas de formas quadráticas, criando uma via introdutória entre a teoria clássica e as abstratas durante este processo. Há uma gama de literatura desenvolvida tanto na teoria clássica quanto nas abstratas, mas nenhuma intercalando-as ``geograficamente\'\'. Nesta perspectiva, discutiremos os aspectos fundamentais da teoria clássica e reduzida de formas quadráticas, encapsulando as teorias das Estruturas Quaterniônicas, Esquemas de Cordes, Anéis de Witt Abstratos, Espaços de Ordens Abstratos, Grupos Especiais, Espectro Real Abstratos e Semigrupos Reais em um quadro funtorial, inserindo os novos elementos envolvendo a teoria recente dos Multi-anéis e Multi-corpos.

We study three category-theoretic types in the context of functional query languages (typed lambda-calculi extended with additional operations for bulk data processing). The types we study are: / Engineering and Applied Sciences

Title: Categorial methods in structure theory Author: Jakub Opršal Department / Institute: Mathematical Institute, Charles University Supervisor of the master thesis: prof. RNDr. Věra Trnková, DrSc. Abstract: In the first part of the thesis we investigate functor algebras. Initial algebras have distin- guished role in the study of these structures, and it can be constructed by certain transfinite construction, which is called initial algebra construction. Sooner this year Adámek and Trnková have prooved, that the construction stops in either at most three, or in κ steps where κ is a regular cardinal. We continue with their work, and we study the relation between the size of the algebra and the length of the convergence. We prove that the length of the convergence never exceeds the cardinality of the initial algebra. Another transfinite construction has been studied in 1980 by Kelly. He has described the construction of free algebras for a pointed functor and defined a class of well-pointed functors for which the construction is especially simple (and is in fact special case of the construction of relatively terminal coalgebra which has been recently defined by Adámek and Trnková). In the last chapter we describe all well-pointed functors in categories of sets and the dual category, and we provide list of...