Differentiable Categories, gerbes and G-structures

TL;DR

This paper introduces the concept of differential categories to unify various geometric structures, providing new insights into characteristic classes and G-structures through gerbes, especially in non-smooth spaces like orbifolds.

Contribution

It develops the notion of differential categories, offering a unified framework to interpret characteristic classes and G-structures using gerbes in both smooth and singular spaces.

Findings

Unified geometric interpretation of 5-characteristic classes

Application of gerbes to classical G-structure problems

Extension of differential geometry tools to orbifolds and foliations

Abstract

The theories of strings and $D$-branes have motivated the development of non Abelian cohomology techniques in differential geometry, on the purpose to find a geometric interpretation of characteristic classes. The spaces studied here, like orbifolds are not often smooth. In classical differential geometry, non smooth spaces appear also naturally, for example in the theory of foliations, the space of leaves can be an orbifold with singularities. The scheme to study these structures is identical: classical tools used in differential geometry, like connections, curvature are adapted. The purpose of this paper is to present the notion of differential category which unifies all these points of view. This enables us to provide a geometric interpretation of 5-characteristic classes, and to interpret classical problems which appear in the theory of $G$-structures by using gerbes.