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A perspective on non-commutative frame theory

Ganna Kudryavtseva, Mark V. Lawson

Abstract

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We establish a duality between the category of complete restriction monoids and the category of étale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and étale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between étale localic categories and étale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral étale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.

A perspective on non-commutative frame theory

Abstract

This paper extends the fundamental results of frame theory to a non-commutative setting where the role of locales is taken over by étale localic categories. This involves ideas from quantale theory and from semigroup theory, specifically Ehresmann semigroups, restriction semigroups and inverse semigroups. We establish a duality between the category of complete restriction monoids and the category of étale localic categories. The relationship between monoids and categories is mediated by a class of quantales called restriction quantal frames. This result builds on the work of Pedro Resende on the connection between pseudogroups and étale localic groupoids but in the process we both generalize and simplify: for example, we do not require involutions and, in addition, we render his result functorial. We also project down to topological spaces and, as a result, extend the classical adjunction between locales and topological spaces to an adjunction between étale localic categories and étale topological categories. In fact, varying morphisms, we obtain several adjunctions. Just as in the commutative case, we restrict these adjunctions to spatial-sober and coherent-spectral equivalences. The classical equivalence between coherent frames and distributive lattices is extended to an equivalence between coherent complete restriction monoids and distributive restriction semigroups. Consequently, we deduce several dualities between distributive restriction semigroups and spectral étale topological categories. We also specialize these dualities for the setting where the topological categories are cancellative or are groupoids. Our approach thus links, unifies and extends the approaches taken in the work by Lawson and Lenz and by Resende.

Paper Structure

This paper contains 29 sections, 124 theorems, 199 equations.

Table of Contents

  1. Introduction and preliminaries
  2. Introduction
  3. Outline of the paper
  4. Preliminaries
  5. The commutative setting
  6. The Quantalization Theorem
  7. Ehresmann and restriction semigroups
  8. Ehresmann quantales and quantal frames
  9. Partial isometries
  10. The Quantalization Theorem: objects
  11. The Quantalization Theorem: morphisms
  12. The Correspondence Theorem
  13. Localic categories from Ehresmann quantal frames
  14. Ehresmann quantal frames from quantal localic categories
  15. Back and forth
  16. ...and 14 more sections

Key Result

Theorem 1.1

For appropriately defined classes of mor-phisms, the category of complete restriction monoids and the category of restriction quantal frames are equivalent.

Theorems & Definitions (235)

  • Theorem 1.1: Quantalization Theorem
  • Theorem 1.2: Correspondence Theorem
  • Theorem 1.3: Etale Correspondence Theorem
  • Theorem 1.4: Duality Theorem
  • Theorem 1.5: Adjunction Theorem
  • Theorem 1.6: Topological Duality Theorem
  • Theorem 1.7: Topological Duality Theorem II
  • Proposition 1.8
  • Remark 1.9
  • Theorem 1.10: Classical Adjunction Theorem
  • ...and 225 more