Categorical structures enriched in a quantaloid: Tensored and cotensored categories

Abstract

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 Qop; this result is then finetuned to obtain in particular that cocomplete Q-categories are equivalent to sup-lattice-valued homomorphisms on Qop (a.k.a. Q-modules).