Sheaves on a bicategory

TL;DR

This paper develops a comprehensive theory of sheaves on bicategories by generalizing enrichment, defining complete B-categories, and establishing an adjunction with 2-presheaves, unifying and extending existing sheaf and enriched category theories.

Contribution

It introduces a new framework for sheaf theory on bicategories through enrichment, generalizes known results, and connects sheaves with 2-presheaves via an adjunction.

Findings

Established an adjunction between complete B-categories and 2-presheaves.

Unified sheaf theory with enriched category theory in the bicategorical setting.

Reproduced known results for quantaloids and analyzed fixed points in the monoidal case.

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.