Categorical Foundations
Special Topics in Order, Topology, Algebra, and Sheaf Theory
Seiten
2003
Cambridge University Press (Verlag)
978-0-521-83414-8 (ISBN)
Cambridge University Press (Verlag)
978-0-521-83414-8 (ISBN)
The book gives a categorical introduction to some of the key areas of modern mathematics. Researchers, teachers and graduate students in algebra and topology familiar with the very basic notions of category theory will find all the advanced tools needed for their subjects, without being forced to study category theory for its own sake.
The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. Readers familiar with the very basic notions of category theory will learn about the main tools that are used in modern categorical mathematics but are not readily available in the literature. Hence, in eight rather independent chapters the reader will encounter various ways of how to study 'spaces': order-theoretically via their open-set lattices, as objects of a fairly abstract category merely via their interaction with other objects, or via their topoi of set-valued sheaves. Likewise, 'algebras' are treated both as models for Lawvere's algebraic theories and as Eilenberg-Moore algebras for monads, but they appear also as the objects of an abstract category with various levels of 'exactness' conditions. The abstract methods are illustrated by applications which, in many cases, lead to results not yet found in more traditional presentations of the various subjects, for instance on the exponentiability of spaces and embeddability of algebras.
The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics. Readers familiar with the very basic notions of category theory will learn about the main tools that are used in modern categorical mathematics but are not readily available in the literature. Hence, in eight rather independent chapters the reader will encounter various ways of how to study 'spaces': order-theoretically via their open-set lattices, as objects of a fairly abstract category merely via their interaction with other objects, or via their topoi of set-valued sheaves. Likewise, 'algebras' are treated both as models for Lawvere's algebraic theories and as Eilenberg-Moore algebras for monads, but they appear also as the objects of an abstract category with various levels of 'exactness' conditions. The abstract methods are illustrated by applications which, in many cases, lead to results not yet found in more traditional presentations of the various subjects, for instance on the exponentiability of spaces and embeddability of algebras.
Introduction Walter Tholen; 1. Ordered sets via adjunction R. J. Wood; 2. Locales Jorge Picado, Ales Pultr and Anna Tozzi; 3. A functional approach to general topology Maria Manuel Clementino, Eraldo Giuli and Walter Tholen; 4. Regular, protomodular and abelian categories Dominique Bourn and Marino Gran; 5. Aspects of monads John MacDonald and Manuela Sobral; 6. Algebraic categories Maria Cristina Pedicchio and Fabrizio Rovatti; 7. Sheaf theory Claudia Centazzo and Enrico M. Vitale; 8. Beyond barr exactness: effective descent morphisms George Janelidze, Manuela Sobral and Walter Tholen.
Erscheint lt. Verlag | 3.11.2003 |
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Reihe/Serie | Encyclopedia of Mathematics and its Applications |
Zusatzinfo | Worked examples or Exercises; 6 Tables, unspecified; 300 Line drawings, unspecified |
Verlagsort | Cambridge |
Sprache | englisch |
Maße | 156 x 234 mm |
Gewicht | 860 g |
Themenwelt | Mathematik / Informatik ► Mathematik ► Allgemeines / Lexika |
ISBN-10 | 0-521-83414-7 / 0521834147 |
ISBN-13 | 978-0-521-83414-8 / 9780521834148 |
Zustand | Neuware |
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