Enrichment through variation

Abstract 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-category of closed W -representations and maps of such, which in turn is isomorphic to a full sub-2-category of Lax( W , Cat). We further show that, if ω is a locally dense subbicategory of W and W is biclosed, then the 2-category of W -categories having tensors with 1-cells of ω embeds fully into the 2-category of ω-representations. This allows us to generalize Gabriel-Ulmer duality to W -categories and to prove, for W -categories, that for locally finitely presentable A and for B admitting finite tensors and filtered colimits, the category of W -functors from A f to B is equivalent to that of finitary W -functors from A to B .