Monoidal Topology
A Categorical Approach to Order, Metric, and Topology

Based on lax-algebraic and categorical methods, Monoidal Topology provides a unified theory for metric and topological structures with far-reaching applications.

Dirk Hofmann (Edited by), Gavin J. Seal (Edited by), Walter Tholen (Edited by)

9781107063945, Cambridge University Press

Hardback, published 31 July 2014

518 pages, 225 exercises
24 x 16 x 3.3 cm, 0.93 kg

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.

Subject Areas: Topology [PBP], Set theory [PBCH], Mathematical logic [PBCD]