Monoidal Topology

A Categorical Approach to Order, Metric, and Topology

Dirk Hofmann, Gavin J. Seal, Walter Tholen

• Publisher: Cambridge University Press
• Year: 2014
• Print ISBN:
  9781107063945
• eBook ISBN:
  9781139990998
• Topic: Mathematics
• Subtopic: Logic in Mathematics
• Index: Mathematics

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

• Cover
• Half-title page
• Series page
• Title page
• Copyright page
• Dedication
• Summary of contents
• Contents
• Preface
• I Introduction
• II Monoidal structures
• III Lax algebras
• IV Kleisli monoids
• V Lax algebras as spaces
• Bibliography
• Selected categories
• Selected functors
• Selected symbols
• Index