So, here is the key insight that migrates from the modern approach to the post-modern: do not look at a single cat alone, but see it as as only the ground floor of a higher and higher groupoid (paths, paths of paths, etc.) Rather than being the bedrock of Homotopy, model structures become models, or presentations, of the REAL OBJECT of Homotopy, the invariant infinity groupoid in all its splendor. That basic insight is already in Grothendieck, around 1983, maybe earlier, but has blossomed into an entire field thanks to Voevodsky, Lurie, ReskRezk, etc.