The complicial sets model of higher ∞-categories

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While it’s undeniably sexy to work with infinite-dimensional categories “model-independently,” we contend there is a categorical imperative to familiarize oneself with at least one concrete model in order to check that proposed model-independent constructions interpret correctly. With this aim in mind, we recount the n-complicial sets model of (∞,n)-categories for 0 ≤ n ≤ ∞, the combinatorics of which are quite similar to its low-dimensional special cases: quasi-categories (n=1) and Kan complexes (n=0). We conclude by reporting on an encounter with 2-complicial sets in the wild, where a suitably-defined fibration of 2-complicial sets enables the comprehension construction introduced in joint work with Verity. Special cases of the comprehension construction can be used to “straighten” a co/cartesian fibration of (∞,1)-categories into a homotopy coherent functor, exhibit a quasi-categorical version of the “unstraightening” construction, and define an internal model of the Yoneda embedding for (∞,1)-categories.