On the Gottlieb group, Drinfeld centre and the centre of a crossed module
Mariam Pirashvili
TL;DR
The paper defines the centre ${\mathcal{Z}}_*({\mathtt G}_*)$ of a crossed module and establishes its deep connections to the Gottlieb group and the Drinfeld-Joyal-Street centre of the associated monoidal category. It develops a concrete algebraic model ${\bf Z}_1({\mathtt G}_*)$ and shows that ${\mathcal Z}_*({\mathtt G}_*)$ is a braided crossed module, with low-dimensional homotopy groups tied to group cohomology via exact sequences ${\bf H}^1$ and ${\bf H}^2$. The work also compares with Norrie’s centre, computes explicit examples such as ${\sf AUT}(D_4)$, and proves a topological application: for crossed modules with free ${\mathtt G}_1$, the classifying space $B{\mathtt G}_*$ satisfies ${\mathcal Z}(B{\mathtt G}_*) \simeq B({\mathcal Z}_*({\mathtt G}_*))$, while the Gottlieb group $G(X,x_0)$ is the kernel of a cohomological map $g$, equating to a concrete subgroup of the Whitehead centre. The framework uses crossed complexes over groupoids to connect algebraic structures with mapping spaces, and suggests potential extensions to higher ${\sf cat}^n$-groups and further 2-mategorical generalisations.
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.