Chu connections and back diagonals between $\mathcal{Q}$-distributors

TL;DR

This paper introduces Chu connections and back diagonals as morphisms for distributors in categories enriched over a small quantaloid, establishing their categorical properties and applications to formal concept analysis.

Contribution

It defines new morphisms for enriched distributors, explores their categorical relationships, and applies these concepts to reduce formal contexts in formal concept analysis.

Findings

Category of complete $ ext{Q}$-categories is a retract of the dual of $ ext{Q}$-distributors with Chu connections.

Duality established between $ ext{Q}$-distributors with back diagonals and the category of complete $ ext{Q}$-categories.

Characterization of reducts of formal contexts using Chu connections.

Abstract

Chu connections and back diagonals are introduced as morphisms for distributors between categories enriched in a small quantaloid $\mathcal{Q}$. These notions, meaningful for closed bicategories, dualize the constructions of arrow categories and the Freyd completion of categories. It is shown that, for a small quantaloid $\mathcal{Q}$, the category of complete $\mathcal{Q}$-categories and left adjoints is a retract of the dual of the category of $\mathcal{Q}$-distributors and Chu connections, and it is dually equivalent to the category of $\mathcal{Q}$-distributors and back diagonals. As an application of Chu connections, a postulation of the intuitive idea of reduction of formal contexts in the theory of formal concept analysis is presented, and a characterization of reducts of formal contexts is obtained.