Permutation 2-groups I: structure and splitness

By a 2-group we mean a groupoid equipped with a weakened group structure. It is called split when it is equivalent to the semidirect product of a discrete 2-group and a one-object 2-group. By a permutation 2-group we mean the 2-group $\mathbb{S}ym(\mathcal{G})$ of self-equivalences of a groupoid $\mathcal{G}$ and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the symmetric groups $\mathsf{S}_n$, $n\geq 1$, obtained when $\mathcal{G}$ is a finite discrete groupoid. After introducing the wreath 2-product $\mathsf{S}_n\wr\wr\ \mathbb{G}$ of the symmetric group $\mathsf{S}_n$ with an arbitrary 2-group $\mathbb{G}$, it is shown that for any (finite type) groupoid $\mathcal{G}$ the permutation 2-group $\mathbb{S}ym(\mathcal{G})$ is equivalent to a product of wreath 2-products of the form $\mathsf{S}_n\wr\wr\ \mathbb{S}ym(\mathcal{B}\mathsf{G})$, where $\mathcal{B}\mathsf{G}$ is the delooping of $\mathsf{G}$. This is next used to compute the homotopy invariants of $\mathbb{S}ym(\mathcal{G})$ which classify it up to equivalence. In particular, we prove that $\mathbb{S}ym(\mathcal{G})$ can be non-split, and that the step from the trivial groupoid $\mathcal{B}\mathsf{1}$ to an arbitrary one-object groupoid $\mathcal{B}\mathsf{G}$ is in fact the only source of non-splitness. Various examples of permutation 2-groups are explicitly computed, in particular the permutation 2-group of the underlying groupoid of a (finite type) 2-group. It also follows from well known results about the symmetric groups that the permutation 2-group of the groupoid of all finite sets and bijections between them is equivalent to the direct product 2-group $\mathbb{Z}_2[1]\times\mathbb{Z}_2[0]$, where $\mathbb{Z}_2[0]$ and $\mathbb{Z}_2[1]$ stand for the group $\mathbb{Z}_2$ thought of as a discrete and a one-object 2-group, respectively.