Categorical structures enriched in a quantaloid: tensored and cotensored categories
Isar Stubbe · Theory and applications of categories · 2006
A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid Q.We show how cocomplete Q-categories are precisely those which are tensored and conically cocomplete, or alternatively, those which are tensored, cotensored and 'order-cocomplete'.In fact, tensors and cotensors in a Q-category determine, and are determined by, certain adjunctions in the category of Q-categories; some of these adjunctions can be reduced to adjuctions in the category of ordered sets.Bearing this in mind, we explain how tensored Q-categories are equivalent to order-valued closed pseudofunctors on Q op ; this result is then finetuned to obtain in particular that cocomplete Q-categories are equivalent to sup-lattice-valued homomorphisms on Q op (a.k.a.Q-modules).