For categories enriched over the unit interval with a continuous t-norm, complete distributivity implies continuity exactly when the t-norm's implication is continuous away from the diagonal, equivalently when no non-trivial Lukasiewicz component appears above zero.
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Completely distributive enriched categories are not always continuous
For categories enriched over the unit interval with a continuous t-norm, complete distributivity implies continuity exactly when the t-norm's implication is continuous away from the diagonal, equivalently when no non-trivial Lukasiewicz component appears above zero.