Fixed points of adjoint functors enriched in a quantaloid

TL;DR

This paper develops a general framework for representation theorems of fixed points of adjoint functors in categories enriched over a quantaloid, unifying and extending concepts in formal concept analysis and rough set theory.

Contribution

It introduces a broad, unified approach to representation theorems for concept lattices in FCA and RST, including new results and generalizations to multi-typed and multi-valued contexts.

Findings

Concept lattices in RST can be represented as those in FCA via relative pseudo-complements.

The framework extends to multi-typed and multi-valued contexts.

Representation theorems are unified under a common categorical setting.

Abstract

Representation theorems are established for fixed points of adjoint functors between categories enriched in a small quantaloid. In a very general setting these results set up a common framework for representation theorems of concept lattices in formal concept analysis (FCA) and rough set theory (RST), which not only extend the realm of formal contexts to multi-typed and multi-valued ones, but also provide a general approach to construct various kinds of representation theorems. Besides incorporating several well-known representation theorems in FCA and RST as well as formulating new ones, it is shown that concept lattices in RST can always be represented as those in FCA through relative pseudo-complements of the given contexts, especially if the contexts are valued in a non-Girard quantaloid.