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Homology
In presenting this treatment of homological algebra, it is a pleasure to acknowledge the help and encouragement which I have had from all sides. Homological algebra arose from many sources in algebra and topology. Decisive examples came from the study of group extensions and their factor sets, a subject I learned in joint work with OTTO SCHIL LING. A further development of homological ideas, with a view to their topological applications, came in my long collaboration with SAMUEL ElLENBERG; to both collaborators, especial thanks. For many years the Air Force Office of Scientific Research supported my research projects on various subjects now summarized here; it is a pleasure to acknowledge th...
Algebra
Presents modern algebra from first principles. This title combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. It presents a conceptual approach to algebra that starts with a description of algebraic structures by means of axioms chosen to suit the examples.
Saunders Mac Lane
Saunders Mac Lane was an extraordinary mathematician, a dedicated teacher, and a good citizen who cared deeply about the values of science and education. In his autobiography, he gives us a glimpse of his "life and times," mixing the highly personal with professional observations. His recollections bring to life a century of extraordinary accomplis
Categories for the Working Mathematician
An array of general ideas useful in a wide variety of fields. Starting from the foundations, this book illuminates the concepts of category, functor, natural transformation, and duality. It then turns to adjoint functors, which provide a description of universal constructions, an analysis of the representations of functors by sets of morphisms, and a means of manipulating direct and inverse limits. These categorical concepts are extensively illustrated in the remaining chapters, which include many applications of the basic existence theorem for adjoint functors. The categories of algebraic systems are constructed from certain adjoint-like data and characterised by Beck's theorem. After considering a variety of applications, the book continues with the construction and exploitation of Kan extensions. This second edition includes a number of revisions and additions, including new chapters on topics of active interest: symmetric monoidal categories and braided monoidal categories, and the coherence theorems for them, as well as 2-categories and the higher dimensional categories which have recently come into prominence.
Categories for the Working Mathematician
Category Theory has developed rapidly. This book aims to present those ideas and methods which can now be effectively used by Mathe maticians working in a variety of other fields of Mathematical research. This occurs at several levels. On the first level, categories provide a convenient conceptual language, based on the notions of category, functor, natural transformation, contravariance, and functor category. These notions are presented, with appropriate examples, in Chapters I and II. Next comes the fundamental idea of an adjoint pair of functors. This appears in many substantially equivalent forms: That of universal construction, that of direct and inverse limit, and that of pairs offunct...
Selected Papers
Covering a period up to 1971, this selection of Saunders Mac Lane’s most distinguished papers takes the reader on a journey through the most important milestones of the mathematical world in the twentieth century. Mac Lane was an extraordinary mathematician and a dedicated teacher who cared earnestly about the values of science and education. His life spanned nearly a century of mathematical progress. In his earlier years, he participated in the exciting developments in Göttingen. He studied under David Hilbert, Hermann Weyl, and Paul Bernays. Later, he contributed to the more abstract and general mathematical viewpoints which emerged in the twentieth century. Perhaps the most outstanding accomplishment during his long and extraordinary career was the development of the concept and theory of categories, together with Samuel Eilenberg, which has broad applications in different areas, in particular in topology and the foundations of mathematics.
Algebra
This book presents modern algebra from first principles and is accessible to undergraduates or graduates. It combines standard materials and necessary algebraic manipulations with general concepts that clarify meaning and importance. This conceptual approach to algebra starts with a description of algebraic structures by means of axioms chosen to suit the examples, for instance, axioms for groups, rings, fields, lattices, and vector spaces. This axiomatic approach—emphasized by Hilbert and developed in Germany by Noether, Artin, Van der Waerden, et al., in the 1920s—was popularized for the graduate level in the 1940s and 1950s to some degree by the authors' publication of A Survey of Modern Algebra. The present book presents the developments from that time to the first printing of this book. This third edition includes corrections made by the authors.
Sheaves in Geometry and Logic
Sheaves arose in geometry as coefficients for cohomology and as descriptions of the functions appropriate to various kinds of manifolds. Sheaves also appear in logic as carriers for models of set theory. This text presents topos theory as it has developed from the study of sheaves. Beginning with several examples, it explains the underlying ideas of topology and sheaf theory as well as the general theory of elementary toposes and geometric morphisms and their relation to logic.
Selected Papers
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