A structure theorem for 2-hypergroupoids with topological applications (Q1077008)

\textit{H. Brandt} gave, in 1940, a structure theorem for connected groupoids G, which in modern terms says that any vertex group G(x) of G is a strong deformation rectract of G [Vjschr. Naturforsch. Ges. Zürich 85, Beibl. No. 32, 95-104 (1940; Zbl 0023.21404)]. In recent years, various higher dimensional versions of groupoids have found uses, for example the n-hypergroupoids of Duskin and Schanuel [cf. \textit{P. G. Glenn}, J. Pure Appl. Alg. 25, 33-105 (1982; Zbl 0487.18015)] and various kinds of multiple groupoids [cf. the reviewer's survey in Categorical Topology, Proc. int. Conf., Toledo/Ohio 1983, Sigma Ser. Pure Math. 5, 108-146 (1984; Zbl 0558.55001)]. The aim of this paper is to prove a 2- dimensional version of Brandt's theorem, namely that if G is a Kan 1-connected 2-hypergroupoid, then for every \(*\in G_ 0\), \(K(\pi_ 2(G,*),2)\) is a deformation retract of G. This implies the Hurewicz theorem in dimension 2.