Categorical Structures Enriched in a Quantaloid: Tensored and Cotensored Categories

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). Introduction The concept of “category enriched in a bicategory W” is as old as the definition of bicategory itself [Bénabou, 1967]; however, J. Bénabou called them “polyads”. Taking a W with only one object gives a monoidal category, and for symmetric monoidal closed V the theory of V-categories is well developed [Kelly, 1982]. But also categories enriched in a W with more than one object are interesting. R. Walters [1981] observed that sheaves on a locale give rise to bicategory-enriched categories: “variation” (sheaves on a locale Ω) is related to “enrichment” (categories enriched in Rel(Ω)). This insight was further developed in [Walters, 1982], [Street, 1983] and [Betti et al., 1983]. Later [Gordon and Power, 1997, 1999] complemented this work, stressing the important rôle of tensors in bicategory-enriched categories. Here we wish to discuss “variation and enrichment” in the case of a base quantaloid Q (a small sup-lattice-enriched category). This is, of course, a particular case of the above, but we believe that it is also of particular interest; many examples of bicategoryenriched categories (like Walters’) are really quantaloid-enriched. Since in a quantaloid Q every diagram of 2-cells commutes, many coherence issues disappear, so the theory of Qenriched categorical structures is very transparent. Moreover, by definition a quantaloid Q has stable local colimits, hence (by local smallness) it is closed; this is of great help when working with Q-categories. The theory of quantaloids is documented in [Rosenthal, 1996]; examples and applications of quantaloids abound in the literature; and [Stubbe, Received by the editors 2005-03-14 and, in revised form, 2006-06-14. Transmitted by Ross Street. Published on 2006-06-19. 2000 Mathematics Subject Classification: 06F07, 18D05, 18D20.