On the equivariant \(2\)-type of a \(G\)-space (Q1295561)

A \(G\)-space \(X\) may be regarded as a diagram of its fixed point spaces \(X^H\) for \(H<G\) according to its equivariant homotopy type, since the homotopy category of \(G\)-spaces is equivalent to the homotopy category of \(O(G)^{op}\)-spaces where \(O(G)\) is the orbit category of \(G\). This paper uses the Grothendieck semidirect product construction of a small category by a functor, to define the concept of weak equivalence of small 2-categories as a functor. Finally, resting on an extension of the Grothendieck construction, the main theorem asserts that the equivariant homotopy type of a \(G\)-space \(X\) with trivial equivariant homotopy at dimensions \(g\geq 3\), is completely determined by a diagram of groupoids.