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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.

http://www.tac.mta.ca/tac/volumes/16/14/16-14.dvi

http://www.tac.mta.ca/tac/volumes/16/14/16-14.ps

http://www.tac.mta.ca/tac/volumes/16/14/16-14.pdf

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/14/16-14.dvi

ftp://ftp.tac.mta.ca/pub/tac/html/volumes/16/14/16-14.ps

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