Categories

Categorical methods of speaking and thinking are becoming more and more widespread in mathematics because they achieve a unifi­ cation of parts of different mathematical fields; frequently they bring simplifications and provide the impetus for new developments. The purpose of this book is to introduce the reader to the central part of category theory and to make the literature accessible to the reader who wishes to go farther. In preparing the English version, I have used the opportunity to revise and enlarge the text of the original German edition. Only the most elementary concepts from set theory and algebra are assumed as prerequisites. However, the reader is expected to be mathe­ to follow an abstract axiomatic approach. matically sophisticated enough The vastness of the material requires that the presentation be concise, and careful cooperation and some patience is necessary on the part of the reader. Definitions alway precede the examples that illuminate them, and it is assumed that the reader is familiar with some of the algebraic and topological examples (he should not let the other ones confuse him). It is also hoped that he will be able to explain the con­ cepts to himself and that he will recognize the motivation. Front Matter....Pages N1-XI Categories....Pages 1-5 Functors....Pages 5-16 Categories of Categories and Categories of Functors....Pages 16-24 Representable Functors....Pages 24-32 Some Special Objects and Morphisms....Pages 32-36 Diagrams....Pages 37-45 Limits....Pages 45-62 Colimits....Pages 62-69 Filtered Colimits....Pages 69-81 Setvalued Functors....Pages 81-96 Objects with an Algebraic Structure....Pages 96-110 Abelian Categories....Pages 110-123 Exact Sequences....Pages 123-139 Colimits of Monomorphisms....Pages 139-152 Injective Envelopes....Pages 152-166 Adjoint Functors....Pages 166-188 Pairs of Adjoint Functors between Functor Categories....Pages 188-220 Principles of Universal Algebra....Pages 220-256 Calculus of Fractions....Pages 256-291 Grothendieck Topologies....Pages 291-319 Triples....Pages 319-374 Back Matter....Pages 375-385