Abstract: Firstly, by introducing the concepts of a weakly idempotent quantale and a weakly idempotent nucleus on a quantale, we gave an equivalent characterization for the largest weakly idempotent quantale quotient. Secondly, we proved that the category of weakly idempotent quantales was a reflective subcategory of the category of quantales, the category of idempotent quantales was a reflective subcategory of the category of weakly idempotent quantales. Finally, we obtained that the category of idempotent quantales was a reflective subcategory of the category of quantales.

Key words: quantale quotient, weakly idempotent quantale, weakly idempotent nucleus, category