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Relating ample and biample topological categories with Boolean restriction and range semigroups

Ganna Kudryavtseva

TL;DR

The paper extends non-commutative Stone dualities by showing that Boolean range semigroups correspond to ample topological categories with an open range map, and that étale Boolean range semigroups correspond to biample categories. It develops adjunctions between preBoolean restriction semigroups with local units and ample categories, leading to several equivalences: Boolean restriction semigroups with local units ↔ ample categories, Boolean range semigroups ↔ strongly ample categories, and étale Boolean range semigroups ↔ biample categories; these are further connected to Boolean birestriction semigroups via the bideterministic elements. The construction relies on germs and slices to translate between algebraic structures and topological categories, and cofunctors to describe morphisms, enabling a coherent duality framework that encompasses inverse, restriction, and range semigroups. Applications include recovering and unifying known results on groupoids, inverse semigroups, and their associated operator-algebraic forms, with potential extensions to non-unital, non-commutative contexts relevant to Steinberg algebras and related C*-algebra constructions.

Abstract

We extend the equivalence by Cockett and Garner between restriction monoids and ample categories to the setting of Boolean range semigroups which are non-unital one-object versions of range categories. We show that Boolean range semigroups are equivalent to ample topological categories where the range map $r$ is open, and étale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between étale Boolean range semigroups and Boolean birestriction semigroups and a characterization of when a Boolean restriction semigroup admits a compatible cosupport operation. We also recover the equivalence between Boolean birestriction semigroups and biample topological categories by Kudryavtseva and Lawson. Our technique builds on the usual constructions relating inverse semigroups with ample topological groupoids via germs and slices.

Relating ample and biample topological categories with Boolean restriction and range semigroups

TL;DR

The paper extends non-commutative Stone dualities by showing that Boolean range semigroups correspond to ample topological categories with an open range map, and that étale Boolean range semigroups correspond to biample categories. It develops adjunctions between preBoolean restriction semigroups with local units and ample categories, leading to several equivalences: Boolean restriction semigroups with local units ↔ ample categories, Boolean range semigroups ↔ strongly ample categories, and étale Boolean range semigroups ↔ biample categories; these are further connected to Boolean birestriction semigroups via the bideterministic elements. The construction relies on germs and slices to translate between algebraic structures and topological categories, and cofunctors to describe morphisms, enabling a coherent duality framework that encompasses inverse, restriction, and range semigroups. Applications include recovering and unifying known results on groupoids, inverse semigroups, and their associated operator-algebraic forms, with potential extensions to non-unital, non-commutative contexts relevant to Steinberg algebras and related C*-algebra constructions.

Abstract

We extend the equivalence by Cockett and Garner between restriction monoids and ample categories to the setting of Boolean range semigroups which are non-unital one-object versions of range categories. We show that Boolean range semigroups are equivalent to ample topological categories where the range map is open, and étale Boolean range semigroups are equivalent to biample topological categories. These results yield the equivalence between étale Boolean range semigroups and Boolean birestriction semigroups and a characterization of when a Boolean restriction semigroup admits a compatible cosupport operation. We also recover the equivalence between Boolean birestriction semigroups and biample topological categories by Kudryavtseva and Lawson. Our technique builds on the usual constructions relating inverse semigroups with ample topological groupoids via germs and slices.

Paper Structure

This paper contains 35 sections, 61 theorems, 34 equations.

Table of Contents

  1. Introduction
  2. Motivation and purpose
  3. Structure and main results
  4. Stone duality for generalized Boolean algebras
  5. Restriction, birestriction and range semigroups
  6. Ehresmann, biEhresmann, restriction and birestriction semigroups
  7. Range semigroups
  8. Deterministic elements, the compatibility relations and joins
  9. Deterministic, codeterministic and bideterministic elements
  10. Partial isomorphisms
  11. Partial isomorphisms vs bideterministic elements in range semigroups
  12. The compatibility, cocompatibility and bicompatibility relations and joins
  13. Boolean restriction and birestriction semigroups
  14. Boolean and preBoolean restriction semigroups with local units
  15. Boolean birestriction semigroups
  16. ...and 20 more sections

Key Result

Theorem A

Theorems & Definitions (153)

  • Theorem A
  • Theorem B
  • Theorem 2.1
  • Definition 3.1: Ehresmann, coEhresmann and biEhresmann semigroups
  • Definition 3.2: Restriction, corestriction and birestriction semigroups
  • Remark 3.3
  • Definition 3.4: Natural partial orders
  • Remark 3.5
  • Lemma 3.6
  • proof
  • ...and 143 more