Local Theory for 2-Functors on Path 2-Groupoids

TL;DR

This paper develops a local theory for 2-functors on path 2-groupoids, establishing an equivalence between global 2-functors and local descent data, advancing the axiomatic understanding of connections on non-abelian gerbes.

Contribution

It provides the first part of an axiomatic framework for gerbes with connection, proving the equivalence between global 2-functors and local descent data.

Findings

Established an equivalence between global 2-functors and local descent data.

Contributed to the axiomatic formulation of connections on non-abelian gerbes.

Laid groundwork for relating descent data to various gerbe models.

Abstract

In this article we discuss local aspects of 2-functors defined on the path 2-groupoid of a smooth manifold; in particular, local trivializations and descent data. This is a contribution to a project that provides an axiomatic formulation of connections on (possibly non-abelian) gerbes in terms of 2-functors. The main result of this paper establishes the first part of this formulation: we prove an equivalence between the globally defined 2-functors and their locally defined descent data. The second part appears in a separate publication; there we prove equivalences between descent data, on one side, and various existing versions of gerbes with connection on the other side.