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Algebraic classification of equivariant homotopy 2-types. I. (English) Zbl 0787.55008

The authors construct an algebraic model for the homotopy category of \(G\)-spaces having homotopy \(G\)-dimension 2.
MacLane and Whitehead showed that the category of pointed connected spaces of homotopy dimension 2 is equivalent to the category of crossed modules. This result is first extended here to an equivalence between the homotopy category of spaces having homotopy dimension 2 for every choice of basepoint, and the homotopy category of 2 groupoids; this generalizes to a corresponding equivalence between diagrams of spaces and diagrams of 2-groupoids.
A \(G\)-space of homotopy \(G\)-dimension 2 is a \(G\)-space \(X\) which satisfies \(\pi_ i(X^ H,x) = 0\) for all subgroups \(H\subseteq G\), all \(x\in X^ H\) and all \(i\geq 3\). Here \(X^ H\) is the fixed-point set. The previous results are next extended to an equivalence between the homotopy category of equivariant \(G\)-spaces of homotopy \(G\)-dimension 2, and the homotopy category of diagrams of 2-groupoids over the orbit category of the group \(G\).

MSC:

55P15 Classification of homotopy type
55P91 Equivariant homotopy theory in algebraic topology
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