How to enhance categories, and why

It has become accepted wisdom by now that when you localize a category with respect to a class of morphisms, what you get is not just a category but a category "with a homotopical enhancement".  Typically, the latter is made precise through the machinery of "infinity-categories", or "quasicategories", but this is quite heavy technically and not really optimal from the conceptual point of view.  I am going to sketch an alternative technique based on Grothendieck's idea of a "derivator".

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