We apply the Dwyer–Kan theory of homotopy function complexes in model categories to the study of mapping spaces in quasi-categories. Using this, together with our work on rigidification from [Alg. Geom. Topol. 11 (2011) 225–261], we give a streamlined proof of the Quillen equivalence between quasi-categories and simplicial categories. Some useful material about relative mapping spaces in quasi-categories is developed along the way.
"Mapping spaces in quasi-categories." Algebr. Geom. Topol. 11 (1) 263 - 325, 2011. https://doi.org/10.2140/agt.2011.11.263