Borel fibrations and G-spaces (Q1101738)

Let G be a topological group and X and Y G-spaces. It is well-known that a map \(f: X\to Y\) which is equivariant up to coherent homotopies induces an equivariant map \(f_{\infty}: X\times EG\to Y\) uniquely up to a contractible space of choices. Such maps \(f_{\infty}\) form a category \({\mathcal J}_{\infty}\) in an obvious way. Let \(X\to BX\to BG\) with \(BX=X\times_ GEG\) be the bundle with fibre X associated with the univeral G-bundle, and let \({\mathcal B}(G)\) be the category of such bundles and fibre preserving maps over BG. It is well known that the homotopy categories of \({\mathcal J}_{\infty}\) and \({\mathcal B}(G)\) are equivalent. The author gives a short proof asserting that \({\mathcal J}_{\infty}\) and \({\mathcal B}(G)\) themselves are equivalent if G is discrete.