Categorical Foundations

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

Maria Cristina Pedicchio, Walter Tholen

• Publisher: Cambridge University Press
• Year: 2003
• Print ISBN: 9780521834148
• eBook ISBN: 9781107299535
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

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.

Table of contents

• Cover
• Series Page
• Title
• Copyright
• Dedication
• Summary of Contents
• Preface
• Table of Contents
• Introduction
• I Ordered Sets via Adjunctions
• II Locales
• III A Functional Approach to General Topology
• IV Regular, Protomodular, and Abelian Categories
• V Aspects of Monads
• VI Algebraic Categories
• VII Sheaf Theory
• VIII Beyond Barr Exactness: Effective Descent Morphisms
• Index