On the construction of limits and colimits in ∞-categories

Emily Riehl, Dominic Verity

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    1 Citation (Scopus)

    Abstract

    In previous work, we introduce an axiomatic framework within which to prove theorems about many varieties of infinite-dimensional categories simultaneouslyIn this paper, we establish criteria implying that an ∞-category — for instance, a quasicategory, a complete Segal space, or a Segal category — is complete and cocompleteadmitting limits and colimits indexed by any small simplicial set. Our strategy is to build (co)limits of diagrams indexed by a simplicial set inductively from (co)limits of restricted diagrams indexed by the pieces of its skeletal filtration. We show directly that the modules that express the universal properties of (co)limits of diagrams of these shapes are reconstructible as limits of the modules that express the universal properties of (co)limits of the restricted diagrams. We also prove that the Yoneda embedding preserves and refiects limits in a suitable sense, and deduce our main theorems as a consequence.

    Original languageEnglish
    Pages (from-to)1101-1158
    Number of pages58
    JournalTheory and Applications of Categories
    Volume35
    Issue number30
    Publication statusPublished - 2020

    Keywords

    • infinity category
    • limit
    • colimits

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