Monoidal Topology A Categorical Approach to Order, Metric, and Topology Encyclopedia of Mathematics and its Applications Series

Monoidal Topology describes an active research area that, after various past proposals on how to axiomatize 'spaces' in terms of convergence, began to emerge at the beginning of the millennium. It combines Barr's relational presentation of topological spaces in terms of ultrafilter convergence with Lawvere's interpretation of metric spaces as small categories enriched over the extended real half-line. Hence, equipped with a quantale V (replacing the reals) and a monad T (replacing the ultrafilter monad) laxly extended from set maps to V-valued relations, the book develops a categorical theory of (T,V)-algebras that is inspired simultaneously by its metric and topological roots. The book highlights in particular the distinguished role of equationally defined structures within the given lax-algebraic context and presents numerous new results ranging from topology and approach theory to domain theory. All the necessary pre-requisites in order and category theory are presented in the book.

Preface; 1. Introduction Robert Lowen and Walter Tholen; 2. Monoidal structures Gavin J. Seal and Walter Tholen; 3. Lax algebras Dirk Hofmann, Gavin J. Seal and Walter Tholen; 4. Kleisli monoids Dirk Hofmann, Robert Lowen, Rory Lucyshyn-Wright and Gavin J. Seal; 5. Lax algebras as spaces Maria Manuel Clementino, Eva Colebunders and Walter Tholen; Bibliography; Tables; Index.

Dirk Hofmann is an Assistant Professor at the University of Aveiro, Portugal. He received his PhD from the University of Bremen (Germany) in 1999. His research interests focus on the development and application of categorical methods in mathematics, primarily in algebra and topology, but also in logic and computer science. Over the past ten years he has contributed significantly to the development of the theory presented in this book, and beyond. He is also well known for his contributions to duality theory.
Gavin J. Seal is a lecturer at the Swiss Federal Institute in Lausanne (EPFL), where he takes part in the activities of the Homotopy Theory Group and the Euler course, a program for talented youth in mathematics. He established a Fundamental Theorem of Polar Geometry for his PhD Thesis in 2000 at the Université Libre de Bruxelles, before pursuing his research in category theory at York and McGill Universities in Canada, as well as at Georgia Southern University in the USA. He has contributed to fundamental aspects of the theory of lax algebras, especially through the development of its Kleisli monoid facet.
Walter Tholen is a Professor of Mathematics at York University, Toronto and an internationally recognized specialist of category theory and its applications to algebra, topology and computer science. His work encompasses some 120 published papers on various subjects, ranging from the fundamental study of categories (in particular, of monads, factorization systems and closure operators) to their applications (in particular, in general topology, homotopy theory, duality theory). Twelve students wrote their PhD theses under his supervision. He has co-authored several books and serves on various editorial boards. He is also an engaged academic administrator, currently serving as Associate Vice-President Research of the university.