Theory of categories

Although it is a relatively .l'oung branch of mathematics, category theorJ; has already achieved important results that are dispersed in a great number of papers and gathered in some monographs. For this reason, to write a new monograph on the theory of categories is easy, due to the abundance of material, but also difficult, due to the great quantity of ideas and results. In this work we try to give all exposition of some of the ideas and results of the theory of categories. We use current terminology and build as simple a framework as possible, but nevertheless sufficient to enable the reader of this book to understand most of the research papers deroted to this theory. In order to read this book e{fectil'ely, the reader is assumed to possess some knowledge of set theory, as well as some elementary facts from algebra and general topology. However, the reader should hare mathematical maturity. The first chapter deals with the general stud}' of categories. Here the basic notions are introduced and most of the fundamental results are proved. The second chapter discusses the problem of the completion of categories. Brief!:!-' said, this means the embedding of a gil'en category into a category that possesses some additional properties, e.lpeciallr those connected with the existence of limits and colimits. The third chapter covers the algebraic categories, i.e. categories of universal algebras and their general study. as well as the study of the functors between them.

Klappentext
Although it is a relatively .l'oung branch of mathematics, category theorJ; has already achieved important results that are dispersed in a great number of papers and gathered in some monographs. For this reason, to write a new monograph on the theory of categories is easy, due to the abundance of material, but also difficult, due to the great quantity of ideas and results. In this work we try to give all exposition of some of the ideas and results of the theory of categories. We use current terminology and build as simple a framework as possible, but nevertheless sufficient to enable the reader of this book to understand most of the research papers deroted to this theory. In order to read this book e{fectil'ely, the reader is assumed to possess some knowledge of set theory, as well as some elementary facts from algebra and general topology. However, the reader should hare mathematical maturity. The first chapter deals with the general stud}' of categories. Here the basic notions are introduced and most of the fundamental results are proved. The second chapter discusses the problem of the completion of categories. Brief!:!-' said, this means the embedding of a gil'en category into a category that possesses some additional properties, e.lpeciallr those connected with the existence of limits and colimits. The third chapter covers the algebraic categories, i.e. categories of universal algebras and their general study. as well as the study of the functors between them.

Inhalt
1 Categories and Functors.- 1.1. The notion of a category. Examples. Duality.- 1.2. Special morphisms in a category.- 1.3. Functors.- 1.4. Equivalence of categories.- 1.5. Equivalence relations on a category.- 1.6. Limits and colimits.- 1.7. Products and coproducts.- 1.8. Some special limits and colimits.- 1.9. Existence of limits and colimits.- 1.10. Limits and colimits in the category of functors.- 1.11. Adjoint functors.- 1.12. Commutation of functors with limits and colimits.- 1.13. Categories of fractions.- 1.14. Calculus of fractions.- 1.15. Existence of a coadjoint to the canonical functor P: ? ? ? (?-1).- 1.16. Subobjects and quotient objects.- 1.17. Intersections and unions of subobjects.- 1.18. Images and inverse images.- 1.19. Triangular decomposition of morphisms.- 1.20. Relative triangular decomposition of morphisms.- 2 Completion of Categories.- 2.1. Proper functors.- 2.2. The extension theorem.- 2.3. Dense functors.- 2.4. ?-sheaves.- 2.5. Topologies and sheaves.- 2.6. Some adjoint theorems.- 2.7. A generalization of the extension theorem.- 2.8. Completion of categories.- 2.9. Grothendieck topologies.- 3 Algebraic Categories.- 3.1. Algebraic theories.- 3.2. Algebraic categories.- 3.3. Algebraic functors.- 3.4. Coalgebras.- 3.5. Characterization of algebraic categories.- 4 Abelian Categories.- 4.1. Preadditive and additive categories.- 4.2. Abelian categories.- 4.3. The isomorphism theorems.- 4.4. Limits and colimits in abelian categories.- 4.5. The extension theorem in the additive case. A characterization of functor categories.- 4.6. Injective objects in abelian categories.- 4.7. Categories of additive fractions.- 4.8. Left exact functors. The embedding theorem.- References.

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