QQ-closure spaces

For a small quantaloid QQ, a QQ-closure space is a small category enriched in QQ equipped with a closure operator on its presheaf category. We investigate QQ-closure spaces systematically with specific attention paid to their morphisms and, as preordered fuzzy sets are a special kind of quantaloid-enriched categories, in particular fuzzy closure spaces on fuzzy sets are introduced as an example. By constructing continuous relations that naturally generalize continuous maps, it is shown (in the generality of the QQ-version) that the category of closure spaces and closed continuous relations is equivalent to the category of complete lattices and sup-preserving maps.