It has been almost ten years since Lurie published Higher Topos Theory, where (following Joyal and probably others) he set up foundations for higher category theory via quasicategories. My impression is that this field is not yet completely settled, and so one might expect there to have been significant simplifications of this material since.

I know of one possible example (though I am not sure if it is best described as a simplification, rather than an alternative route), namely the proof of the equivalence between quasicategories and simplicial categories given in https://arxiv.org/abs/0911.0469. What other foundational advances have happened after the publication of HTT?