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

On the representations of 2-groups in {Baez-Crans} 2-vector spaces

We prove that the theory of representations of a finite 2-group $\mathbb{G}$ in Baez-Crans 2-vector spaces over a field $k$ of characteristic zero essentially reduces to the theory of $k$-linear representations of the group of isomorphism classes of objects of $\mathbb{G}$, the remaining homotopy invariants of $\mathbb{G}$ playing no role. It is also argued that a similar result is expected to hold for topological representations of compact topological 2-groups in suitable topological Baez-Crans 2-vector spaces.Comment: 9 page

The 2-group of symmetries of a split chain complex

We explicitly compute the 2-group of self-equivalences and (homotopy classes of) chain homotopies between them for any {\it split} chain complex $A_{\bullet}$ in an arbitrary \kb-linear abelian category (\kb any commutative ring with unit). In particular, it is shown that it is a {\it split} 2-group whose equivalence class depends only on the homology of $A_{\bullet}$, and that it is equivalent to the trivial 2-group when $A_\bullet$ is a split exact sequence. This provides a description of the {\it general linear 2-group} of a Baez and Crans 2-vector space over an arbitrary field $\mathbb{F}$ and of its generalization to chain complexes of vector spaces of arbitrary length.Preprin