[Paper Review] Stone duality for topological theories

Dirk Hofmann, Isar Stubbe|arXiv (Cornell University)|Apr 1, 2010

Homotopy and Cohomology in Algebraic Topology15 references3 citations

TL;DR

This paper establishes a $Ø$-categorical generalization of Stone duality by introducing $Ø$-colimits in $Ø$-categories, showing that Cauchy completeness of a $Ø$-category corresponds precisely to sobriety. It extends classical Stone duality between topological spaces and frames to enriched categorical settings, particularly for quantales like $Ø = \mathbf{2}$ and $[0,\infty]$, and proves that frame homomorphisms preserving finite suprema and infima correspond to continuous maps in the dual category.

ABSTRACT

In the context of categorical topology, more precisely that of T-categories [Hofmann, 2007], we define the notion of T-colimit as a particular colimit in a V-category. A complete and cocomplete V-category in which limits distribute over T-colimits, is to be thought of as the generalisation of a (co-)frame to this categorical level. We explain some ideas on a T-categorical version of "Stone duality", and show that Cauchy completeness of a T-category is precisely its sobriety.

Motivation & Objective

• To generalize Stone duality from topological spaces and frames to the setting of $Ø$-categories and $Ø$-frames.
• To define $Ø$-colimits as a categorical generalization of finite suprema in enriched categories.
• To establish that Cauchy completeness in a $Ø$-category is equivalent to sobriety, generalizing a key property in classical Stone duality.
• To show that for quantales $Ø = \mathbf{2}$ and $[0,\infty]$, the dual adjunction between $Ø$-categories and $Ø$-frames recovers standard dualities for co-frames and approach frames.

Proposed method

• Introduces $Ø$-colimits as a special class of colimits in $Ø$-categories, generalizing finite suprema in lattices.
• Defines $Ø$-frames as complete and cocomplete $Ø$-categories where limits distribute over $Ø$-colimits, generalizing frames and co-frames.
• Uses the $Ø$-functor $Ø = [Ø, -]$ to represent $Ø$-functors preserving infima, tensors, and cotensors.
• Applies a filter-ideal separation argument via Lemma 3.1 to construct an ultrafilter $Ø$ with $a(Ø, -) = \varphi$ and $k = \xi \cdot U\varphi(\u00d8)$ under conditions on $Ø$.
• Demonstrates that under suitable conditions on $Ø$, preservation of finite suprema implies preservation of $Ø$-suprema, enabling simplification of $Ø$-frame structure.
• Establishes a dual adjunction between $Ø$-categories and $Ø$-frames, recovering classical dualities for $Ø = \mathbf{2}$ and $[0,\infty]$.

Experimental results

Research questions

• RQ1 How can Stone duality be generalized from topological spaces and frames to the setting of $Ø$-categories and $Ø$-frames?
• RQ2 What is the categorical role of $Ø$-colimits in generalizing finite suprema in enriched category theory?
• RQ3 Under what conditions does a $Ø$-functor preserving finite suprema also preserve $Ø$-suprema?
• RQ4 Is Cauchy completeness in a $Ø$-category equivalent to sobriety, generalizing the classical duality for topological spaces?
• RQ5 How do the dualities for $Ø = \mathbf{2}$ and $[0,\infty]$ compare to existing notions like approach frames in [BLVO06]?

Key findings

• Cauchy completeness of a $Ø$-category is precisely equivalent to its sobriety, generalizing a key result in classical Stone duality.
• For $Ø = \mathbf{2}$, the category $Ø_{\mathbf{2}}$-Frm is equivalent to the category of co-frames and co-frame homomorphisms, recovering the standard dual adjunction with $Ø_{\mathbf{2}}$-Cat = Top.
• For $Ø = [0,\infty]$, $Ø_{[0,\infty]}$-frames are complete metric spaces where finite colimits commute with arbitrary limits, and morphisms are contraction maps preserving all limits and finite colimits.
• Under the conditions that $Ø$ satisfies $⊤ = k$, the set $\{u \in \u00d8 \mid u \ll k\}$ is directed, and $k \leq u \vee v$ implies $k \leq u$ or $k \leq v$, a $Ø$-functor preserving finite suprema also preserves $Ø$-suprema.
• The construction of an ultrafilter $Ø$ with $a(Ø, x) = \varphi(x)$ and $k = \xi \cdot U\varphi(\u00d8)$ relies on separating a filter base $\{A_u \mid u \ll k\}$ from an ideal $\{B \mid \varphi \not\leq \varphi_B\}$ using Lemma 3.1.
• The dual adjunction between $Ø$-categories and $Ø$-frames for $Ø = \mathbf{2}$ and $[0,\infty]$ generalizes classical Stone duality and provides a framework for extending it to enriched settings.