Gabriel-Ulmer duality for categories enriched in bicategories Original Research Article

Gabriel-Ulmer duality is generalized to categories enriched in bicategories. We define weighted limits in a W-category for a locally finitely presentable bicategory W, and prove that a W-category has weighted limits if and only if it has cotensors and conical limits, which are defined to agree with the familiar definitions for one-object symmetric W. We define cocompleteness by duality, and define the notions of strong generator and finitely presentable object. These are used to generalize Gabriel-Ulmer duality. We also prove that these definitions and constructions agree with those given by regarding a W-category with finite tensors as a Wf-indexed category.