Mac Lane, Categories For The Working Mathematician Pdf
Categories for the Working MathematicianPreface to the Second EditionPreface to the First EditionContentsIntroductionI. Categories, Functors, and Natural TransformationsII. Constructions on CategoriesIII. Universals and LimitsIV. Monads and AlgebrasVII.
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Abelian CategoriesIX. Special LimitsX. Kan ExtensionsXI. Symmetry and Braidings in Monoidal CategoriesXII. Structure in CategoriesAppendix. FoundationsTable of Standard Categories: Objects and ArrowsTable of TerminologyBibliographyIndex.
Categories for the Working Mathematician provides 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.
The book 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 characterized 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 two new chapters on topics of active interest.
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Download Categories For The Working Mathematician Saunders Mac Lane Pdf Download Categories For The Working Mathematician Saunders Mac Lane free pdf. Buy, download and read Categories for the Working Mathematician ebook online in PDF format for iPhone, iPad, Android, Computer and Mobile readers. Author: Saunders Mac Lane.
One is on symmetric monoidal categories and braided monoidal categories and the coherence theorems for them. The second describes 2-categories and the higher dimensional categories which have recently come into prominence. The bibliography has also been expanded to cover some of the many other recent advances concerning categories.
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 offunctors with a natural isomorphism between corresponding sets of arrows.
All these forms, with their interrelations, are examined in Chapters III to V. The slogan is 'Adjoint functors arise everywhere'. Alternatively, the fundamental notion of category theory is that of a monoid -a set with a binary operation of multiplication which is associative and which has a unit; a category itself can be regarded as a sort of general ized monoid. Chapters VI and VII explore this notion and its generaliza tions. Its close connection to pairs of adjoint functors illuminates the ideas of universal algebra and culminates in Beck's theorem characterizing categories of algebras; on the other hand, categories with a monoidal structure (given by a tensor product) lead inter alia to the study of more convenient categories of topological spaces.