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The free objects and the product of subcategories


Author: Ţurcanu Alina
Degree:doctor of physics and mathematics
Speciality: 01.01.06 - Mathematical logics, algebra and theory of numbers
Year:2012
Scientific adviser: Dumitru Botnaru
doctor habilitat, associate professor (docent), Tiraspol State University
Institution: Technical University of Moldova
Scientific council: DH 01-01.01.06
Institute of Mathematics and Computer Science of the ASM

Status

The thesis was presented on the 25 August, 2012
Approved by NCAA on the 9 October, 2012

Abstract

Adobe PDF document0.25 Mb / in romanian

Keywords

the free functor; reflective functor; coreflective functor; factorization structures; left (right) product of two subcategories; injective objects; category of Hausdorff vectorial topological locally convex spaces; catagory of Tihonovspaces

Summary

The structure of the thesis: the thesis is written in romanian and consists of an introduction, three chapters, general conclusions and recommendations, a bibliography including 118 bibliographical titles, 120 pages of basic text, 105 figures. The results obtained have been published in 11 scientific works. The field of study of the thesis: the theory of categories. The purpose and the objectives of the work: defining and studying the properties of the left and right products of two subcategories; examining the relations between these products and: the factorization of the reflective functor, the pairs of the conjugate subcategories, the theories of relative torsion, the semireflexive subcategories; applying the factorized right product to describe the Hewitt spaces.

The novelty and the scientific originality of the obtained results is determined by the fact of solving the following concrete problems: general methods were elaborated to apply the factorization structures while studying the reflective and coreflective categories; there were determined the conditions necessary for the class of the P-reflective and I-reflective subcategories to be determined by its smallest element; a method was elaborated to decompose a reflective functor of the R(P, I) class as a composition of a reflective functor of the R(P) class and a reflective functor of the R(I) class; it was pointed to the necessary conditions sufficient to make the right product of two subcategories be a reflective subcategory; the domain of values of the right product of two subcategories was established; the relations between these products and: the factorization of the reflective functor, the pairs of the conjugate subcategories, the theories of relative torsion, as well as the semireflexive subcategories, were examined; the factorized right product was applied to describe the Hewitt spaces.

The solved scientific problem consists in the study of the properties of the free objects via the adjunct functors, the elaboration of a general concept of the left and the right product of two subcategories and the applications of these products in the theory of the categories.

The theoretical importance and the applicative value of the work: the work is a theoretical one. The obtained results can be applied in the theory of categories, the general topology and the category of the locally convex spaces.

The implementation of the scientific results: the results contained in the thesis can constitute the contents of some special courses for mathematics students and be a support to Master Degree theses.