# The congruence biframe as a quasi-uniform bicompletion

## TL;DR

This paper establishes a pointfree analogue of a known sobriety characterization for $T_0$ spaces, showing that strictly zero-dimensional biframes are bicomplete in a specific quasi-uniformity, leading to new insights on congruence frames.

## Contribution

It introduces a pointfree version of the bicompletion characterization, linking congruence biframes with bicompleteness in quasi-uniformity, and provides a new proof of ultraparacompactness of congruence frames.

## Key findings

- A pointfree analogue of the sobriety characterization is proven.
- Bicompletion of a quasi-uniform biframe is constructed as a quotient of the Samuel compactification.
- A new proof of ultraparacompactness of congruence frames is obtained.

## Abstract

K\"unzi and Ferrario have shown that a $T_0$ space is sober if and only if it is bicomplete in the well-monotone quasi-uniformity. We prove a pointfree version of this result: a strictly zero-dimensional biframe is a congruence biframe if and only if it is bicomplete in the same quasi-uniformity. As a corollary we obtain a new proof of a result of Plewe that a congruence frame is ultraparacompact. The main result makes use of a new construction of the bicompletion of a quasi-uniform biframe as a quotient of the Samuel compactification.