We prove that a model structure on a relative ∞-category (M,W) gives an efficient and computable way of accessing the hom-spaces hom_M[W^-1](x,y) in the localization. More precisely, we show that when the source x ∈ M is cofibrant and the target y ∈ M is fibrant, then this hom-space is a "quotient" of the hom-space hom_M(x,y) by either of a left homotopy relation or a right homotopy relation.

We wish to thank Omar Antolín-Camarena, Tobi Barthel, Clark Barwick, Rune Haugseng, Gijs Heuts, Zhen Lin Low, Mike Mandell, Justin Noel, and Aaron Royer for many very helpful conversations. Additionally, we gratefully acknowledge the financial support provided both by the NSF graduate research fellowship program (grant DGE-1106400) and by UC Berkeley's geometry and topology RTG (grant DMS-0838703) during the time that this work was carried out.

As this paper is the culmination of its series, we would also like to take this opportunity to extend our thanks to the people who have most influenced the entire project, without whom it certainly could never have come into existence: Bill Dwyer and Dans Kan and Quillen, for the model-categorical foundations; André Joyal and Jacob Lurie, for the ∞-categorical foundations; Zhen Lin Low, for countless exceedingly helpful conversations; Eric Peterson, for his tireless, dedicated, and impressive TeX support, and for listening patiently to far too many all-too-elaborate "here's where I'm stuck" monologues; David Ayala, for somehow making it through every last one of these papers and providing numerous insightful comments and suggestions, and for providing much-needed encouragement through the tail end of the writing process; Clark Barwick, for many fruitful conversations (including the one that led to the realization that there should exist a notion of "model ∞-categories" in the first place!), and for his consistent enthusiasm for this project; Peter Teichner, for his trust in allowing free rein to explore, for his generous support and provision for so many extended visits to so many institutions around the world, and for his constant advocacy on our behalf; and, lastly and absolutely essentially, Katherine de Kleer, for her boundless love, for her ample patience, and for bringing beauty and excitement into each and every day.