Handbook of Categorical Algebra: Volume 2, Categories and Structures (Encyclopedia of Mathematics and its Applications, Series Number 51)

The second volume, 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 fibered categories. Cover......Page A001.djvu Title page......Page all_11227_to_0066.cpc.djvu Contents......Page all_11227_to_0066.cpc0005.djvu Preface to volume 2......Page all_11227_to_0066.cpc0008.djvu Introduction to this handbook......Page all_11227_to_0066.cpc0011.djvu Contents of the three volumes......Page all_11227_to_0066.cpc0013.djvu 1.1 Zero objects and kernels......Page all_11227_to_0066.cpc0014.djvu 1.2 Additive categories and biproducts......Page all_11227_to_0066.cpc0016.djvu 1.3 Additive functors......Page all_11227_to_0066.cpc0021.djvu 1.4 Abelian categories......Page all_11227_to_0066.cpc0026.djvu 1.5 Exactness properties of abelian categories......Page all_11227_to_0066.cpc0029.djvu 1.6 Additivity of abelian categories......Page all_11227_to_0066.cpc0034.djvu 1.7 Union of subobjects......Page all_11227_to_0066.cpc0039.djvu 1.8 Exact sequences......Page all_11227_to_0066.cpc0045.djvu 1.9 Diagram chasing......Page all_11227_to_0066.cpc0047.djvu 1.10 Some diagram lemmas......Page all_11227_to_0066.cpc0053.djvu 1.11 Exact functors......Page all_11227_to_0066.cpc0062.djvu 1.12 Torsion theories......Page all_11227_to_0066.cpc0064.djvu 1.13 Localizations of abelian categories......Page all_11227_to_0132.cpc0009.djvu 1.14 The embedding theorem......Page all_11227_to_0132.cpc0018.djvu 1.15 Exercises......Page all_11227_to_0132.cpc0033.djvu 2.1 Exactness properties of regular categories......Page all_11227_to_0132.cpc0036.djvu 2.2 Definition in terms of strong epimorphisms......Page all_11227_to_0132.cpc0039.djvu 2.3 Exact sequences......Page all_11227_to_0132.cpc0042.djvu 2.4 Examples......Page all_11227_to_0132.cpc0045.djvu 2.5 Equivalence relations......Page all_11227_to_0132.cpc0048.djvu 2.6 Exact categories......Page all_11227_to_0132.cpc0052.djvu 2.7 An embedding theorem......Page all_11227_to_0132.cpc0057.djvu 2.8 The calculus of relations......Page all_11227_to_0132.cpc0060.djvu 2.9 Exercises......Page all_11227_to_0198.cpc.djvu 3.1 The theory of groups revisited......Page all_11227_to_0198.cpc0003.djvu 3.2 A glance at universal algebra......Page all_11227_to_0198.cpc0006.djvu 3.3 A categorical approach to universal algebra......Page all_11227_to_0198.cpc0011.djvu 3.4 Limits and colimits in algebraic categories......Page all_11227_to_0198.cpc0018.djvu 3.5 The exactness properties of algebraic categories......Page all_11227_to_0198.cpc0020.djvu 3.6 The algebraic lattices of subobjects......Page all_11227_to_0198.cpc0022.djvu 3.7 Algebraic functors......Page all_11227_to_0198.cpc0024.djvu 3.8 Freely generated models......Page all_11227_to_0198.cpc0027.djvu 3.9 Characterization of algebraic categories......Page all_11227_to_0198.cpc0039.djvu 3.10 Commutative theories......Page all_11227_to_0198.cpc0047.djvu 3.11 Tensor product of theories......Page all_11227_to_0198.cpc0054.djvu 3.12 A glance at Morita theory......Page all_11227_to_0198.cpc0060.djvu 3.13 Exercises......Page all_11227_to_0198.cpc0063.djvu 4 Monads......Page all_11227_to_0264.cpc.djvu 4.1 Monads and their algebras......Page all_11227_to_0264.cpc0003.djvu 4.2 Monads and adjunctions......Page all_11227_to_0264.cpc0008.djvu 4.3 Limits and colimits in categories of algebras......Page all_11227_to_0264.cpc0012.djvu 4.4 Characterization of monadic categories......Page all_11227_to_0264.cpc0027.djvu 4.5 The adjoint lifting theorem......Page all_11227_to_0264.cpc0036.djvu 4.6 Monads with rank......Page all_11227_to_0264.cpc0046.djvu 4.7 A glance at descent theory......Page all_11227_to_0264.cpc0052.djvu 4.8 Exercises......Page all_11227_to_0330.cpc.djvu 5 Accessible categories......Page all_11227_to_0330.cpc0003.djvu 5.1 Presentable objects in a category......Page all_11227_to_0330.cpc0004.djvu 5.2 Locally presentable categories......Page all_11227_to_0330.cpc0005.djvu 5.3 Accessible categories......Page all_11227_to_0330.cpc0012.djvu 5.4 Raising the degree of accessibility......Page all_11227_to_0330.cpc0016.djvu 5.5 Functors with rank......Page all_11227_to_0330.cpc0021.djvu 5.6 Sketches......Page all_11227_to_0330.cpc0026.djvu 5.7 Exercises......Page all_11227_to_0330.cpc0038.djvu 6 Enriched category theory......Page all_11227_to_0330.cpc0040.djvu 6.1 Symmetric monoidal closed categories......Page all_11227_to_0330.cpc0041.djvu 6.2 Enriched categories......Page all_11227_to_0330.cpc0049.djvu 6.3 The enriched Yoneda lemma......Page all_11227_to_0330.cpc0058.djvu 6.4 Change of base......Page all_11227_to_0330.cpc0062.djvu 6.5 Tensors and cotensors......Page all_11227_to_0396.cpc0003.djvu 6.6 Weighted limits......Page all_11227_to_0396.cpc0008.djvu 6.7 Enriched adjunctions......Page all_11227_to_0396.cpc0023.djvu 6.8 Exercises......Page all_11227_to_0396.cpc0030.djvu 7 Topological categories......Page all_11227_to_0396.cpc0032.djvu 7.1 Exponentiable spaces......Page all_11227_to_0396.cpc0033.djvu 7.2 Compactly generated spaces......Page all_11227_to_0396.cpc0042.djvu 7.3 Topological functors......Page all_11227_to_0396.cpc0049.djvu 7.4 Exercises......Page all_11227_to_0396.cpc0054.djvu 8 Fibred categories......Page all_11227_to_0396.cpc0056.djvu 8.1 Fibrations......Page all_11227_to_0396.cpc0057.djvu 8.2 Cartesian functors......Page all_11227_to_0396.cpc0065.djvu 8.3 Fibrations via pseudo-functors......Page all_11227_to_0456.cpc0004.djvu 8.4 Fibred adjunctions......Page all_11227_to_0456.cpc0011.djvu 8.5 Completeness of a fibration......Page all_11227_to_0456.cpc0016.djvu 8.6 Locally small fibrations......Page all_11227_to_0456.cpc0029.djvu 8.7 Definability......Page all_11227_to_0456.cpc0041.djvu 8.8 Exercises......Page all_11227_to_0456.cpc0049.djvu Bibliography......Page all_11227_to_0456.cpc0053.djvu Index......Page all_11227_to_0456.cpc0056.djvu 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.