It is proved that the projective model structure of the category of topologically enriched diagrams of topological spaces over a locally contractible topologically enriched small category is Quillen equivalent to the standard Quillen model structure of topological spaces. We give a geometric interpretation of this fact in directed homotopy.

I thank Tim Campion for Theorem 3.2 and Tyler Lawson for Theorem 8.1. I also thank Asaf Karagila for [21] even if his contribution is not necessary anymore in this new version (it suffices in the proof of Theorem 3.2 to choose a regular cardinal μ > λ such that μ ⊳ λ instead of μ^λ = μ) and Tim Porter for valuable email discussions. I thank the anonymous referee for the detailed report.