# Categorical structures enriched in a quantaloid: tensored and cotensored categories

## Isar Stubbe

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).

Keywords: quantaloid, enriched category, weighted (co)limit, module

2000 MSC: 06F07, 18D05, 18D20

Theory and Applications of Categories, Vol. 16, 2006, No. 14, pp 283-306.

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