Categorical foundations

Walter Tholen

"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 most 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 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 via their interaction with other objects, or via their topoi of set-valued sheaves. Likewise, "algebras" are treated as both models for Lawvere's algebraic theories and 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...

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