Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

The lattice of smooth sublocales as a Bruns-Lakser completion

Igor Arrieta, Anna Laura Suarez

TL;DR

The paper addresses lifting frame morphisms to the Bruns–Lakser completion underlying the lattice of smooth sublocales. It introduces the join-semilattice $\mathsf{LC}(L)$ of locally closed sublocales and proves that $\mathsf{S}_b(L)$ is isomorphic to the Bruns–Lakser completion $\mathcal{AU}(\mathsf{LC}(L))$, with $\mathsf{LC}(L)$ anti-isomorphic to the locally closed sublocales. A precise admissibility criterion is given: a frame map $f:L\to M$ lifts to $\mathsf{S}_b(-)$ if and only if the induced $\mathsf{LC}(f)$ is admissible (equivalently, satisfies the WD' condition). The results unify lifting phenomena across $\mathsf{S}_b(-)$, $\mathsf{S}_{\mathfrak{c}}(-)$, and $\mathsf{S}_o(-)$, and lay groundwork for a broader theory of morphism lifting in point-free topology. This provides a concrete, algebraic handle on when lattice-completion constructions admit functorial extensions along frame maps, with connections to the Funayama envelope and exactness concepts.

Abstract

We characterise the frame morphisms $f:L\to M$ that lift to frame maps $\overline{f}:\mathsf{S}_b(L)\to \mathsf{S}_b(M)$, where $\mathsf{S}_b(L)$ is the collection of joins of complemented sublocales of a frame $L$, or equivalently the Booleanization of the collection $\mathsf{S}(L)$ of all its sublocales. We do so by proving that $\mathsf{S}_b(L)$ is isomorphic to the Bruns--Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form $\mathfrak{c}(a)\cap \mathfrak{o}(b)$ for $a,b\in L$.

The lattice of smooth sublocales as a Bruns-Lakser completion

TL;DR

The paper addresses lifting frame morphisms to the Bruns–Lakser completion underlying the lattice of smooth sublocales. It introduces the join-semilattice of locally closed sublocales and proves that is isomorphic to the Bruns–Lakser completion , with anti-isomorphic to the locally closed sublocales. A precise admissibility criterion is given: a frame map lifts to if and only if the induced is admissible (equivalently, satisfies the WD' condition). The results unify lifting phenomena across , , and , and lay groundwork for a broader theory of morphism lifting in point-free topology. This provides a concrete, algebraic handle on when lattice-completion constructions admit functorial extensions along frame maps, with connections to the Funayama envelope and exactness concepts.

Abstract

We characterise the frame morphisms that lift to frame maps , where is the collection of joins of complemented sublocales of a frame , or equivalently the Booleanization of the collection of all its sublocales. We do so by proving that is isomorphic to the Bruns--Lakser completion of the meet-semilattice formed by the locally closed sublocales, i.e. the sublocales of the form for .
Paper Structure (11 sections, 18 theorems, 51 equations)

This paper contains 11 sections, 18 theorems, 51 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Some Heyting rules
  4. Sublocales
  5. The Boolean algebra of smooth sublocales.
  6. The Bruns--Lakser completion of a join-semilattice
  7. Joins of closed sublocales and the Bruns--Lakser completion
  8. The join-semilattice of locally closed sublocales
  9. Admissible meets of locally closed sublocales
  10. Smooth sublocales as admissible upper sets
  11. Lifting of morphisms

Key Result

Lemma 2.1

(notesball). If $L$ is a frame, a family $\{a_i\}_{i\in I}\subseteq L$ is exact if and only if the sublocale $\mathop{ \bigvee }_{i\in I}\mathfrak{c}(a_i)$ is closed.

Theorems & Definitions (30)

  • Lemma 2.1
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • Lemma 3.3
  • proof
  • Lemma 3.4
  • Example 4.1
  • Proposition 4.2
  • proof
  • ...and 20 more