On the Gottlieb group, Drinfeld centre and the centre of a crossed module

TL;DR

This paper introduces the concept of the centre of a crossed module, linking it to the Gottlieb group, Drinfeld centre, and group cohomology, providing new insights into the structure of 2-types.

Contribution

It defines the centre of a crossed module and connects it to the Gottlieb group, Drinfeld centre, and group cohomology, offering a novel perspective on 2-types.

Findings

The centre of a crossed module is closely related to the Gottlieb group.

The centre is connected to the Drinfeld centre of a monoidal category.

A relationship between the Whitehead centre and the Gottlieb group of a 2-type is established.

Abstract

This new version includes a connection of the main construction to the Gottlieb group, which was absent in the previous versions. However, the first version included material about Lie algebras which will become available soon as a separate paper. The aim of this paper is to introduce the concept of the centre of a crossed module $\G_* = (\G_1\to \G_0)$. This centre is closely related to the Gottlieb group of the classifying space of a crossed module and also to the Drinfeld centre of a monoidal category introduced independently by Drinfeld and Joyal and Street. Our definition of the centre is based on certain crossed homomorphisms $\G_0\to \G_1$, which makes it easy to relate it to group cohomology. This connection is used to relate the Gottlieb group of a 2-type to its Whitehead centre.