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 oneobject 2-group. By a permutation 2-group we mean the 2-group Sym(G) of self-equivalences of a groupoid G and natural isomorphisms between them, with the product given by composition of self-equivalences. These generalize the symmetric groups Sn, n ≥ 1, obtained when G is a finite discrete groupoid. After introducing the wreath 2-product Sn ≀ ≀ G of the symmetric group Sn with an arbitrary 2-group G, it is shown that for any (finite type) groupoid G the permutation 2-group Sym(G) is equivalent to a product of wreath 2-products of the form Sn≀≀ Sym(G) for a group G thought of as a one-object groupoid. This is next used to compute the homotopy invariants of Sym(G) which classify it up to equivalence. Using a previously shown splitness criterion for strict 2-groups, it is then proved that Sym(G) can be non-split, and that the step from the trivial groupoid to an arbitrary one-object groupoid is 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 Z2[1]×Z2[0], where Z2[0] and Z2[1] stand for the group Z2 thought of as a discrete and a one-object 2-group, respectively.