It is often said that model categories are but a shadow of an $\infty$-category. It is also often said that model categories should give rise to an $\infty$-category via their homotopies. In fact, this ought to be the quintessential example of an $\infty$-category. Unfortunately I have yet to find a place where the author bothered to give the definition. I am also unable to give the definition myself, as things start to get fuzzy beyond 2-simplices. What is the definition?
Edit: The only place that gets close is Hirschhorn's book on model categories, which defines a simplicial structure on the Hom-sets of a model category. The definition spans multiple chapters, and involves Reedy model structures, cosimplicial approximations, fibrant replacements, and function complexes, none of which I understand. I am certain that this is just another case of making things a thousand times harder than it needs to be.