A Bicategory Approach to Differential Cohomology

TL;DR

This paper introduces a bicategory framework for differential cohomology, proving the uniqueness of differential refinements and providing a method to construct them for any generalized cohomology theory.

Contribution

It develops a bicategory-based approach to differential cohomology, establishing uniqueness results and a general construction method for refinements.

Findings

Differential refinements are unique up to equivalence of symmetric monoidal groupoids.

The bicategory formalism simplifies the construction of differential refinements.

A refinement of the Chern character is used to build differential refinements for all generalized cohomology theories.

Abstract

A bicategory approach to differential cohomology is presented. Based on the axioms of Bunke-Schick, a symmetric monoidal groupoid is associated to differential refinements of cohomology theories. It is proven that such differential refinements are unique up to equivalence of the corresponding symmetric monoidal groupoids and the existing uniqueness results for rationally-even theories are interpreted in this framework. Moreover we show how the bicategory formalism may be used to give a simple construction of a differential refinement for any generalized cohomology theory, based on a refinement of the Chern character to a strict transformation of bicategories.