Frames and Locales


Category theory

Topos Theory

This entry provides a hyperlinked index for the book

on basics of locales and pointfree topology.


The following lists chapterwise linked lists of keywords to relevant and related existing entries, as far as they already exist.

I. Spaces and Lattices of Open Sets

  1. Sober spaces?
  2. The axiom T DT_D: another case of spaces easy to reconstruct
  3. Summing up
  4. Aside: several technical properties of T DT_D-spaces

II. Frames? and Locales?. Spectra

  1. Frames?
  2. Locales? and localic maps?
  3. Points
  4. Spectra
  5. The unit σ and spatiality?
  6. The unit λ and sobriety?

III. Sublocales?

  1. Extremal monomorphisms? in Loc
  2. Sublocales?
  3. The co-frame of sublocales
  4. Images and preimages
  5. Alternative representations of sublocales
  6. Open and closed sublocales
  7. Open and closed localic maps?
  8. Closure?
  9. Preimage as a homomorphism
  10. Other special sublocales: one-point sublocales, and Boolean ones
  11. Sublocales? as quotients. Factorizing frames is surprisingly easy

IV. Structure of Localic Morphisms?. The Categories Loc and Frm

  1. Special morphisms. Factorizing in Loc and Frm
  2. The down-set functor and free constructions
  3. Limits? and a colimit in Frm
  4. Coproducts? of frames
  5. More on the structure of coproduct
  6. Epimorphisms in Frm

V. Separation Axioms?

  1. Instead of T 1T_1: subfit? and fit?
  2. Mimicking the Hausdorff axiom?
  3. I-Hausdorff frames and regular monomorphisms
  4. Aside: Raney identity
  5. Quite like the classical case: Regular?, completely regular and normal
  6. The categories RegLoc?, CRegLoc?, HausLoc? and FitLoc?

VI. More on Sublocales?

  1. Subspaces and sublocales of spaces
  2. Spatial and induced sublocales
  3. Complemented sublocales? of spaces are spatial
  4. The zero-dimensionality? of Sl(L) opSl(L)^op and a few consequences
  5. Difference and pseudodifference, residua
  6. Isbell’s Development Theorem?
  7. Locales with no non-spatial sublocales
  8. Spaces with no non-induced sublocales

VII. Compactness? and Local Compactness?

  1. Basics, and a technical lemma
  2. Compactness? and separation?
  3. Kuratowski-Mrówka characterization
  4. Compactification
  5. Well below? and rather below?. Continuous completely regular frames
  6. Continuous is the same as locally compact. Hofmann-Lawson duality?
  7. One more spatiality theorem
  8. Supercompactness?. Algebraic, superalgebraic and supercontinuous frames?

VIII. (Symmetric) Uniformity and Nearness

  1. Background
  2. Uniformity? and nearness? in the point-free context
  3. Uniform homomorphisms. Modelling embeddings. Products
  4. Aside: admitting nearness in a weaker sense
  5. Compact uniform and nearness frames. Finite covers
  6. Completeness? and completion
  7. Functoriality. CUniFrm? is coreflective in UniFrm?
  8. An easy completeness criterion

IX. Paracompactness?

  1. Full normality?
  2. Paracompactness?, and its various guises
  3. An elegant, specifically point-free, characterization of paracompactness
  4. A pleasant surprise: paracompact (co)reflection

X. More about Completion

  1. A variant of the completion of uniform frames?
  2. Two applications
  3. Cauchy points and the resulting space
  4. Cauchy spectrum?
  5. Cauchy completion. The case of countably generated uniformities
  6. Generalized Cauchy points

XI. Metric Frames?

  1. Diameters and metric diameters
  2. Metric spectrum
  3. Uniform Metrization Theorem?
  4. Metrization theorems? for plain frames
  5. Categories of metric frames?

XII. Entourages?. Asymmetric Uniformity?

  1. Entourages?
  2. Uniformities? via entourages
  3. Entourages? versus covers
  4. Asymmetric uniformity?: the classical case
  5. Biframes?
  6. Quasi-uniformity? in the point-free context via paircovers?
  7. The adjunction QUnifQUniFrmQUnif \leftrightarrows QUniFrm
  8. Quasi-uniformity? in the point-free context via entourages

XIII. Connectedness?

  1. A few observations about sublocales
  2. Connected? and disconnected locales
  3. Locally connected? locales
  4. A weird example
  5. A few notes

XIV. Frame of Reals and Real Functions

  1. The frame L(R)L(R) of reals?
  2. Properties of L(R)L(R)
  3. L(R)L(R) versus the usual space of reals
  4. The metric uniformity of L(R)L(R)
  5. Continuous real functions
  6. Cozero? elements
  7. More general real functions
  8. Notes

XV. Localic Groups?

  1. Basics
  2. The category of localic groups
  3. Closed Subgroup Theorem?
  4. The multiplication μ is open. The semigroup of open parts
  5. Uniformities
  6. Notes

Appendix I. Posets

  1. Basics
  2. Zorn’s Lemma?
  3. Suprema? and infima
  4. Semilattices?, lattices and complete lattices. Completion
  5. Galois connections (adjunctions)
  6. (Semi)lattices as algebras. Distributive lattices?
  7. Pseudocomplements? and complements. Heyting and Boolean algebras

Appendix II. Categories

  1. Categories
  2. Functors? and natural transformations
  3. Some basic constructions
  4. More special morphisms. Factorization
  5. Limits? and colimits
  6. Adjunction?
  7. Adjointness? and (co)limits
  8. Reflective and coreflective subcategories
  9. Monads?
  10. Algebras in a category
category: reference