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.Comment: 45 pages; v2, expository and language improvement

Einstein's accelerated reference systems and Fermi-Walker coordinates

We show that the uniformly accelerated reference systems proposed by Einstein when introducing acceleration in the theory of relativity are Fermi-Walker coordinate systems. We then consider more general accelerated motions and, on the one hand we obtain Thomas precession and, on the other, we prove that the only accelerated reference systems that at any time admit an instantaneously comoving inertial system belong necessarily to the Fermi-Walker class.Comment: 15 pages, 1 figur