Sheaves on a bicategory

Abstract: We give a detailed account of the theory of enrichment over a bicategory and show that it establishes a two-fold generalization of enrichment over both quantaloids and monoidal categories. We define complete B-categories, a generalization of Cauchy-complete enriched categories serving as a basis for the development of sheaf theory in the enriched setting. We prove an adjunction between complete B-categories and 2-presheaves on the category Map(B) of left adjoints in B. We express conditions under which this adjunction becomes a left-exact reflection, yielding back the usual results linking sheaves on sites and enriched categories. We prove that our adjunction recovers the already existing results about quantaloids, and discuss the fixed points of the adjunction in the monoidal case.