The complicial sets model of higher ∞-categories


Riehl, E. (2019). The complicial sets model of higher ∞-categories. Perimeter Institute. https://pirsa.org/19010001


Riehl, Emily. The complicial sets model of higher ∞-categories. Perimeter Institute, Jan. 09, 2019, https://pirsa.org/19010001


          @misc{ pirsa_PIRSA:19010001,
            doi = {10.48660/19010001},
            url = {https://pirsa.org/19010001},
            author = {Riehl, Emily},
            keywords = {Other},
            language = {en},
            title = {The complicial sets model of higher $\infty$-categories},
            publisher = {Perimeter Institute},
            year = {2019},
            month = {jan},
            note = {PIRSA:19010001 see, \url{https://pirsa.org}}

Emily Riehl Johns Hopkins University

Talk Type Scientific Series


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.