Categorical structures enriched in a quantaloid: tensored and cotensored categories

Isar Stubbe

Theory and applications of categories, Tome 16 (2006), pp. 283-306

Résumé

A quantaloid is a sup-lattice-enriched category; our subject is that of categories, functors and distributors enriched in a base quantaloid $\mathcal{Q}$. We show how cocomplete $\mathcal{Q}$-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 $\mathcal{Q}$-category determine, and are determined by, certain adjunctions in the category of $\mathcal{Q}$-categories; some of these adjunctions can be reduced to adjuctions in the category of ordered sets. Bearing this in mind, we explain how tensored $\mathcal{Q}$-categories are equivalent to order-valued closed pseudofunctors on $\mathcal{Q}^{op}$; this result is then finetuned to obtain in particular that cocomplete $\mathcal{Q}$-categories are equivalent to sup-lattice-valued homomorphisms on $\mathcal{Q}^{op}$ (a.k.a.\ $\mathcal{Q}$-modules).

Détail

Classification : 06F07, 18D05, 18D20
Keywords: quantaloid, enriched category, weighted (co)limit, module