The 2-category theory of quasi-categories

Emily Riehl*, Dominic Verity

*Corresponding author for this work

    Research output: Contribution to journalArticlepeer-review

    22 Citations (Scopus)


    In this paper we re-develop the foundations of the category theory of quasi-categories (also called ∞-categories) using 2-category theory. We show that Joyal's strict 2-category of quasi-categories admits certain weak 2-limits, among them weak comma objects. We use these comma quasi-categories to encode universal properties relevant to limits, colimits, and adjunctions and prove the expected theorems relating these notions. These universal properties have an alternate form as absolute lifting diagrams in the 2-category, which we show are determined pointwise by the existence of certain initial or terminal vertices, allowing for the easy production of examples.All the quasi-categorical notions introduced here are equivalent to the established ones but our proofs are independent and more "formal". In particular, these results generalise immediately to model categories enriched over quasi-categories.

    Original languageEnglish
    Pages (from-to)549-642
    Number of pages94
    JournalAdvances in Mathematics
    Publication statusPublished - 6 Aug 2015


    • Quasi-categories
    • 2-Category theory
    • Formal category theory


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