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The Funayama envelope as the $T_D$-hull of a frame

Guram Bezhanishvili, Ranjitha Raviprakash, Anna Laura Suarez, Joanne Walters-Wayland

TL;DR

This work extends pointfree topology by replacing strict frame morphisms with proximity morphisms between MT-algebras, yielding an equivalence $ extbf{Frm} o extbf{MT_P}$ via the Funayama envelope, which serves as a $T_D$-hull. It generalizes the $T_D$-duality from frames to MT-algebras, showing that spatial $T_D$-algebras form a reflective subcategory and that a duality with Top via sober maps captures both the pointfree and topological facets. The construction leverages MacNeille completions of Boolean envelopes and proximity-based morphisms to recover and extend classical dualities, including a pointfree account of $T_D$-coreflection. The framework unifies spectra, reflections, and dualities across MT-algebras, frames, and spatial topologies, offering a robust toolset for reasoning about lower separation axioms in a MT-algebraic setting. Overall, the paper demonstrates that MT-algebras and their proximity morphisms provide a natural, expressive arena for pointfree analogues of $T_D$-type separation phenomena and their spatial counterparts.

Abstract

We introduce proximity morphisms between MT-algebras and show that the resulting category is equivalent to the category of frames. This is done by utilizing the Funayama envelope of a frame, which is viewed as the $T_D$-hull. Our results have some spatial ramifications, including a generalization of the $T_D$-duality of Banaschewski and Pultr.

The Funayama envelope as the $T_D$-hull of a frame

TL;DR

This work extends pointfree topology by replacing strict frame morphisms with proximity morphisms between MT-algebras, yielding an equivalence via the Funayama envelope, which serves as a -hull. It generalizes the -duality from frames to MT-algebras, showing that spatial -algebras form a reflective subcategory and that a duality with Top via sober maps captures both the pointfree and topological facets. The construction leverages MacNeille completions of Boolean envelopes and proximity-based morphisms to recover and extend classical dualities, including a pointfree account of -coreflection. The framework unifies spectra, reflections, and dualities across MT-algebras, frames, and spatial topologies, offering a robust toolset for reasoning about lower separation axioms in a MT-algebraic setting. Overall, the paper demonstrates that MT-algebras and their proximity morphisms provide a natural, expressive arena for pointfree analogues of -type separation phenomena and their spatial counterparts.

Abstract

We introduce proximity morphisms between MT-algebras and show that the resulting category is equivalent to the category of frames. This is done by utilizing the Funayama envelope of a frame, which is viewed as the -hull. Our results have some spatial ramifications, including a generalization of the -duality of Banaschewski and Pultr.
Paper Structure (12 sections, 51 theorems, 77 equations, 1 figure, 4 tables)

This paper contains 12 sections, 51 theorems, 77 equations, 1 figure, 4 tables.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Frames and co-frames
  4. Interior algebras and MT-algebras
  5. Proximity morphisms between MT-algebras
  6. Funayama envelope
  7. Boolean envelope of a frame
  8. Funayama envelope of a frame
  9. --duality for MT-algebras
  10. --spectra of MT-algebras
  11. --reflection of MT-algebras and --coreflection of topological spaces
  12. Duality for spatial MT-algebras and proximity morphisms

Key Result

Theorem 2.3

$(\Omega,\mathop{\mathrm{\textit{pt}}}\nolimits_D)$ is an adjunction between $\mathbf{TDTop}\xspace$ and $\mathbf{Frm_D}\xspace^{\mathrm{op}}$, which restricts to an equivalence between $\mathbf{TDTop}\xspace$ and $\mathbf{TD- SFrm_D}\xspace^{\mathrm{op}}$. \begin{tikzcd} \TDTop && {\FrmD^{\mathrm{

Figures (1)

  • Figure 1: Relationship between categories

Theorems & Definitions (133)

  • Definition 2.1
  • Definition 2.2
  • Theorem 2.3: $T_D$-duality
  • Example 2.4
  • Remark 2.5
  • Definition 2.6
  • Definition 2.7
  • Definition 2.8
  • Definition 2.9
  • Theorem 2.10
  • ...and 123 more