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Monoidal Topology

Book Description

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

  1. Cover
  2. Half-title page
  3. Series page
  4. Title page
  5. Copyright page
  6. Dedication
  7. Summary of contents
  8. Contents
  9. Preface
  10. I. Introduction
    1. I.1 The ubiquity of monoids and their actions
      1. I.1.1 Monoids and their actions in algebra
      2. I.1.2 Orders and metrics as monoids and lax algebras
      3. I.1.3 Topological and approach spaces as monoids and lax algebras
      4. I.1.4 The case for convergence
      5. I.1.5 Filter convergence and Kleisli monoids
    2. I.2 Spaces as categories, and categories of spaces
      1. I.2.1 Ordinary small categories
      2. I.2.2 Considering a space as a category
      3. I.2.3 Moving to the large category of all spaces
    3. I.3 Chapter highlights and dependencies
  11. II. Monoidal structures
    1. II.1 Ordered sets
      1. II.1.1 The Cartesian structure of sets and its monoids
      2. II.1.2 The compositional structure of relations
      3. II.1.3 Orders
      4. II.1.4 Modules
      5. II.1.5 Adjunctions
      6. II.1.6 Closure operations and closure spaces
      7. II.1.7 Completeness
      8. II.1.8 Adjointness criteria
      9. II.1.9 Semilattices, lattices, frames, and topological spaces
      10. II.1.10 Quantales
      11. II.1.11 Complete distributivity
      12. II.1.12 Directed sets, filters, and ideals
      13. II.1.13 Ultrafilters
      14. II.1.14 Natural and ordinal numbers
      15. Exercises
    2. II.2 Categories and adjunctions
      1. II.2.1 Categories
      2. II.2.2 Functors
      3. II.2.3 Natural transformations
      4. II.2.4 The Yoneda embedding
      5. II.2.5 Adjunctions
      6. II.2.6 Reflective subcategories, equivalence of categories
      7. II.2.7 Initial and terminal objects, comma categories
      8. II.2.8 Limits
      9. II.2.9 Colimits
      10. II.2.10 Construction of limits and colimits
      11. II.2.11 Preservation and reflection of limits and colimits
      12. II.2.12 Adjoint Functor Theorem
      13. II.2.13 Kan extensions
      14. II.2.14 Dense functors
      15. Exercises
    3. II.3 Monads
      1. II.3.1 Monads and adjunctions
      2. II.3.2 The Eilenberg–Moore category
      3. II.3.3 Limits in the Eilenberg–Moore category
      4. II.3.4 Beck’s monadicity criterion
      5. II.3.5 Duskin’s monadicity criterion
      6. II.3.6 The Kleisli category
      7. II.3.7 Kleisli triples
      8. II.3.8 Distributive laws, liftings, and composite monads
      9. II.3.9 Distributive laws and extensions
      10. Exercises
    4. II.4 Monoidal and ordered categories
      1. II.4.1 Monoidal categories
      2. II.4.2 Monoids
      3. II.4.3 Actions
      4. II.4.4 Monoidal closed categories
      5. II.4.5 Ordered categories
      6. II.4.6 Lax functors, pseudo-functors, 2-functors, and their transformations
      7. II.4.7 Maps
      8. II.4.8 Quantaloids
      9. II.4.9 Kock–Zöberlein monads
      10. II.4.10 Enriched categories
      11. Exercises
    5. II.5 Factorizations, fibrations, and topological functors
      1. II.5.1 Factorization systems for morphisms
      2. II.5.2 Subobjects, images, and inverse images
      3. II.5.3 Factorization systems for sinks and sources
      4. II.5.4 Closure operators
      5. II.5.5 Generators and cogenerators
      6. II.5.6 U-initial morphisms and sources
      7. II.5.7 Fibrations and cofibrations
      8. II.5.8 Topological functors
      9. II.5.9 Self-dual characterization of topological functors
      10. II.5.10 Epireflective subcategories
      11. II.5.11 Taut Lift Theorem
      12. Exercises
    6. Notes on Chapter II
  12. III. Lax algebras
    1. III.1 Basic concepts
      1. III.1.1 V-relations
      2. III.1.2 Maps in V-Rel
      3. III.1.3 V-categories, V-functors, and V-modules
      4. III.1.4 Lax extensions of functors
      5. III.1.5 Lax extensions of monads
      6. III.1.6 (T, V)-categories and (T, V)-functors
      7. III.1.7 Kleisli convolution
      8. III.1.8 Unitary (T, V)-relations
      9. III.1.9 Associativity of unitary (T, V)-relations
      10. III.1.10 The Barr extension
      11. III.1.11 The Beck–Chevalley condition
      12. III.1.12 The Barr extension of a monad
      13. III.1.13 A double-categorical presentation of lax extensions
      14. Exercises
    2. III.2 Fundamental examples
      1. III.2.1 Ordered sets, metric spaces, and probabilistic metric spaces
      2. III.2.2 Topological spaces
      3. III.2.3 Compact Hausdorff spaces
      4. III.2.4 Approach spaces
      5. III.2.5 Closure spaces
      6. Exercises
    3. III.3 Categories of lax algebras
      1. III.3.1 Initial structures
      2. III.3.2 Discrete and indiscrete lax algebras
      3. III.3.3 Induced orders
      4. III.3.4 Algebraic functors
      5. III.3.5 Change-of-base functors
      6. III.3.6 Fundamental adjunctions
      7. Exercises
    4. III.4 Embedding lax algebras into a quasitopos
      1. III.4.1 (T, V)-graphs
      2. III.4.2 Reflecting (T, V)-RGph into (T, V)-Cat
      3. III.4.3 Coproducts of (T, V)-categories
      4. III.4.4 Interlude on partial products and local Cartesian closedness
      5. III.4.5 Local Cartesian closedness of (T, V)-Gph
      6. III.4.6 Local Cartesian closedness of subcategories of (T, V)-Gph
      7. III.4.7 Interlude on subobject classifiers and partial-map classifiers
      8. III.4.8 The quasitopos (T, V)-Gph
      9. III.4.9 Final density of (T, V)-Cat in (T, V)-Gph
      10. Exercises
    5. III.5 Representable lax algebras
      1. III.5.1 The monad T on V-Cat
      2. III.5.2 T-algebras in V-Cat
      3. III.5.3 Comparison with lax algebras
      4. III.5.4 The monad T on (T, V)-Cat
      5. III.5.5 Dualizing (T, V)-categories
      6. III.5.6 The ultrafilter monad on Top
      7. III.5.7 Representable topological spaces
      8. III.5.8 Exponentiable topological spaces
      9. III.5.9 Representable approach spaces
      10. Exercises
    6. Notes on Chapter III
  13. IV. Kleisli monoids
    1. IV.1 Kleisli monoids and lax algebras
      1. IV.1.1 Topological spaces via neighborhood filters
      2. IV.1.2 Power-enriched monads
      3. IV.1.3 T-monoids
      4. IV.1.4 The Kleisli extension
      5. IV.1.5 Topological spaces via filter convergence
      6. Exercises
    2. IV.2 Lax extensions of monads
      1. IV.2.1 Initial extensions
      2. IV.2.2 Sup-dense and interpolating monad morphisms
      3. IV.2.3 (S, 2)-categories as Kleisli monoids
      4. IV.2.4 Strata extensions
      5. IV.2.5 (S, V)-categories as Kleisli towers
      6. Exercises
    3. IV.3 Lax algebras as Kleisli monoids
      1. IV.3.1 The ordered category (T, V)-URel
      2. IV.3.2 The discrete presheaf monad
      3. IV.3.3 Approach spaces
      4. IV.3.4 Revisiting change of base
      5. Exercises
    4. IV.4 Injective lax algebras as Eilenberg–Moore algebras
      1. IV.4.1 Eilenberg–Moore algebras as Kleisli monoids
      2. IV.4.2 Monads on categories of Kleisli monoids
      3. IV.4.3 Eilenberg–Moore algebras over S-Mon
      4. IV.4.4 Continuous lattices
      5. IV.4.5 Kock–Zöberlein monads on T-Mon
      6. IV.4.6 Eilenberg–Moore algebras and injective Kleisli monoids
      7. Exercises
    5. IV.5 Domains as lax algebras and Kleisli monoids
      1. IV.5.1 Modules and adjunctions
      2. IV.5.2 Cocontinuous ordered sets
      3. IV.5.3 Observable realization spaces
      4. IV.5.4 Observable realization spaces as lax algebras
      5. IV.5.5 Observable specialization systems
      6. IV.5.6 Ordered abstract bases and round filters
      7. IV.5.7 Domains as Kleisli monoids of the ordered-filter monad
      8. IV.5.8 Continuous dcpos as sober domains
      9. IV.5.9 Cocontinuous lattices among lax algebras
      10. Exercises
    6. Notes on Chapter IV
  14. V. Lax algebras as spaces
    1. V.1 Hausdorff separation and compactness
      1. V.1.1 Basic definitions and properties
      2. V.1.2 Tychonoff Theorem, Čech–Stone compactification
      3. V.1.3 Compactness for Kleisli-extended monads
      4. V.1.4 Examples involving monoids
      5. Exercises
    2. V.2 Low separation, regularity, and normality
      1. V.2.1 Order separation
      2. V.2.2 Between order separation and Hausdorff separation
      3. V.2.3 Regular spaces
      4. V.2.4 Normal and extremally disconnected spaces
      5. V.2.5 Normal approach spaces
      6. Exercises
    3. V.3 Proper and open maps
      1. V.3.1 Finitary stability properties
      2. V.3.2 First characterization theorems
      3. V.3.3 Notions of closure
      4. V.3.4 Kuratowski–Mrówka Theorem
      5. V.3.5 Products of proper maps
      6. V.3.6 Coproducts of open maps
      7. V.3.7 Preservation of space properties
      8. Exercises
    4. V.4 Topologies on a category
      1. V.4.1 Topology, fiberwise topology, derived topology
      2. V.4.2 P-compactness, P-Hausdorffness
      3. V.4.3 A categorical characterization theorem
      4. V.4.4 P-dense maps, P-open maps
      5. V.4.5 P-Tychonoff and locally P-compact Hausdorff objects
      6. Exercises
    5. V.5 Connectedness
      1. V.5.1 Extensive categories
      2. V.5.2 Connected objects
      3. V.5.3 Topological connectedness governs
      4. V.5.4 Products of connected spaces
      5. Exercises
    6. Notes on Chapter V
  15. Bibliography
  16. Selected categories
  17. Selected functors
  18. Selected symbols
  19. Index