Homotopy classification of braided graded categorical groups
Let \(G\) be a group, \(M\) and \(N\) be \(G\)-modules, \(H^n_{G,ab}(M,-)\) be the Quillen cohomology groups of the pointed \(G\)-space \(K(M, 2)\) in the homotopy category of \(G\)-spaces, \(Ho\mathbb{S}^G_*\), with respect to the closed model structure where weak equivalences are those \(G\)-equivariant pointed maps that are weak equivalences on the underlying spaces, that is, \[ H^n_{G,ab}(M, N)= \text{Hom}_{Ho\mathbb{S}^G_*}(K(M, 2), K(N, n+1)). \] This paper shows that equivariant weak homotopy types of pointed \(G\)-spaces \(X\) with only two non-trivial homotopy groups at dimensions 2 and \(n\) are classified by the elements of the cohomology groups \(H^n_{G,ab}(\pi_2 X,\pi_n X)\) and then, the authors get their main result as follows: The homotopy category of \(G\)-equivariant pointed simply-connected 3-types is equivalent to the homotopy category of braided \(G\)-graded categorical groups.
scientific article; zbMATH DE number 5129950
scientific article