Enrichment through variation Original Research Article

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 Af to B is equivalent to that of finitary W-functors from A to B.