ENRICHMENT THROUGH VARIATION

We show that, for a closed bicategory W, the 2-category of tensored W- categories and all W-functors between them is equivalent to the 2-cate gory of closed W-representations and maps of such, which in turn is is omorphic to a full sub-2-category of Lax(W, Cat). We further show that , if omega is a locally dense subbicategory of W and W is biclosed, th en the 2-category of W-categories having tensors with 1-cells of omega embeds fully into the 2-category of omega-representations. This allow s us to generalize Gabriel-Ulmer duality to W-categories and to prove, for W-categories, that for locally finitely presentable A and for B a dmitting finite tensors and filtered colimits, the category of W-funct ors from A(f) to B is equivalent to that of finitary W-functors from A to B. (C) 1997 Published by Elsevier Science B.V.