The 2-category theory of quasi-categories

Emily Riehl, Dominic Verity

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    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.

    LanguageEnglish
    Pages549-642
    Number of pages94
    JournalAdvances in Mathematics
    Volume280
    DOIs
    Publication statusPublished - 6 Aug 2015

    Fingerprint

    Category Theory
    Colimit
    Model Category
    Adjunction
    Categorical
    Alternate
    Immediately
    Diagram
    Generalise
    Theorem

    Keywords

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

    Cite this

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    The 2-category theory of quasi-categories. / Riehl, Emily; Verity, Dominic.

    In: Advances in Mathematics, Vol. 280, 06.08.2015, p. 549-642.

    Research output: Contribution to journalArticleResearchpeer-review

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    AU - Verity, Dominic

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