Completely distributive enriched categories are not always continuous

TL;DR

The paper investigates the relationship between complete distributivity and continuity in enriched categories, revealing that complete distributivity does not always imply continuity, unlike in lattices, and provides conditions for when it does.

Contribution

It demonstrates that complete distributivity in enriched categories over certain t-norms does not guarantee continuity and establishes necessary and sufficient conditions for this implication.

Findings

Complete distributivity does not imply continuity in enriched categories.

Conditions are identified under which complete distributivity implies continuity.

The result contrasts with the lattice case where these properties coincide.

Abstract

In contrast to the fact that every completely distributive lattice is necessarily continuous in the sense of Scott, it is shown that complete distributivity of a category enriched over the closed category obtained by endowing the unit interval with a continuous t-norm does not imply its continuity in general. Necessary and sufficient conditions for the implication are presented.