Categorical Foundations

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.