Categorical Foundations
Special Topics in Order, Topology, Algebra, and Sheaf Theory

The book offers categorical introductions to order, topology, algebra and sheaf theory, suitable for graduate students, teachers and researchers of pure mathematics.

Maria Cristina Pedicchio (Edited by), Walter Tholen (Edited by)

9780521834148, Cambridge University Press

Hardback, published 3 November 2003

442 pages, 300 b/w illus. 6 tables 200 exercises
23.4 x 15.6 x 2.9 cm, 0.86 kg

'The book will be very useful for graduate students and teachers, and inspiring for the researchers interested in the discussed topics as well as in category theory itself.' EMS Newsletter

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.

Subject Areas: Mathematical foundations [PBC]