A central $U(1)$-extension of a double Lie groupoid

TL;DR

This paper introduces the concept of a central $U(1)$-extension for double Lie groupoids and demonstrates its relation to cocycles in a specific triple complex, advancing the understanding of their algebraic structure.

Contribution

It defines a new notion of central $U(1)$-extensions for double Lie groupoids and links it to cocycles in a triple complex, providing a novel mathematical framework.

Findings

Central $U(1)$-extensions are characterized as cocycles.

The notion connects geometric structures with algebraic cocycles.

Provides a foundation for further study of extensions in Lie groupoid theory.

Abstract

In this paper, we introduce a notion of a central $U(1)$-extension of a double Lie groupoid and show that it defines a cocycle in the certain triple complex.