| Description |
1 online resource (xvii, 443 pages) : illustrations |
| Series |
Encyclopedia of mathematics and its applications ; 51 |
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Encyclopedia of mathematics and its applications ; 51.
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| Contents |
Cover; Half-title; Title; Copyright; Dedication; Contents; Preface to volume 2; Introduction to this handbook; 1 Abelian categories; 1.1 Zero objects and kernels; 1.2 Additive categories and biproducts; 1.3 Additive functors; 1.4 Abelian categories; 1.5 Exactness properties of abelian categories; 1.6 Additivity of abelian categories; 1.7 Union of subobjects; 1.8 Exact sequences; 1.9 Diagram chasing; 1.10 Some diagram lemmas; 1.11 Exact functors; 1.12 Torsion theories; 1.13 Localizations of abelian categories; 1.14 The embedding theorem; 1.15 Exercises; 2 Regular categories |
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2.1 Exactness properties of regular categories2.2 Definition in terms of strong epimorphisms; 2.3 Exact sequences; 2.4 Examples; 2.5 Equivalence relations; 2.6 Exact categories; 2.7 An embedding theorem; 2.8 The calculus of relations; 2.9 Exercises; 3 Algebraic theories; 3.1 The theory of groups revisited; 3.2 A glance at universal algebra; 3.3 A categorical approach to universal algebra; 3.4 Limits and colimits in algebraic categories; 3.5 The exactness properties of algebraic categories; 3.6 The algebraic lattices of subobjects; 3.7 Algebraic functors; 3.8 Freely generated models |
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3.9 Characterization of algebraic categories3.10 Commutative theories; 3.11 Tensor product of theories; 3.12 A glance at Morita theory; 3.13 Exercises; 4 Monads; 4.1 Monads and their algebras; 4.2 Monads and adjunctions; 4.3 Limits and colimits in categories of algebras; 4.4 Characterization of monadic categories; 4.5 The adjoint lifting theorem; 4.6 Monads with rank; 4.7 A glance at descent theory; 4.8 Exercises; 5 Accessible categories; 5.1 Presentable objects in a category; 5.2 Locally presentable categories; 5.3 Accessible categories; 5.4 Raising the degree of accessibility |
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5.5 Functors with rank5.6 Sketches; 5.7 Exercises; 6 Enriched category theory; 6.1 Symmetric monoidal closed categories; 6.2 Enriched categories; 6.3 The enriched Yoneda lemma; 6.4 Change of base; 6.5 Tensors and cotensors; 6.6 Weighted limits; 6.7 Enriched adjunctions; 6.8 Exercises; 7 Topological categories; 7.1 Exponentiable spaces; 7.2 Compactly generated spaces; 7.3 Topological functors; 7.4 Exercises; 8 Fibred categories; 8.1 Fibrations; 8.2 Cartesian functors; 8.3 Fibrations via pseudo-functors; 8.4 Fibred adjunctions; 8.5 Completeness of a fibration; 8.6 Locally small fibrations |
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8.7 Definability8.8 Exercises; Bibliography; Index |
| Summary |
The Handbook of Categorical Algebra is designed to give, in three volumes, a detailed account of what should be known by everybody working in, or using, category theory. As such it will be a unique reference. The volumes are written in sequence. The second, which assumes familiarity with the material in the first, introduces important classes of categories that have played a fundamental role in the subject's development and applications. In addition, after several chapters discussing specific categories, the book develops all the major concepts concerning Benabou's ideas of fibred categories. There is ample material here for a graduate course in category theory, and the book should also serve as a reference for users |
| Bibliography |
Includes bibliographical references (pages 436-438) and index |
| Notes |
Print version record |
| Subject |
Categories (Mathematics)
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Algebra, Homological.
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Abelian categories.
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MATHEMATICS -- Essays.
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MATHEMATICS -- Pre-Calculus.
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MATHEMATICS -- Reference.
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Abelian categories
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Algebra, Homological
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Categories (Mathematics)
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Catégories (mathématiques)
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Algèbre homologique.
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Catégories abéliennes.
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| Form |
Electronic book
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| ISBN |
9781461941453 |
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1461941458 |
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9780511525865 |
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0511525869 |
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9781139881975 |
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1139881973 |
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