Description

Book Synopsis
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.

Table of Contents
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.

Monoidal Topology A Categorical Approach to Order Metric and Topology 153 Encyclopedia of Mathematics and its Applications Series Number 153

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    A Hardback by Dirk Hofmann, Gavin J. Seal, Walter Tholen

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      Book details

      Publisher Cambridge University Press
      Published 7/31/2014 12:00:00 AM
      ISBN-13 9781107063945
      978-1107063945
      ISBN-10 1107063949
      Also in:
      Topology

      Description

      Book Synopsis
      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.

      Table of Contents
      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.

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