Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions

Emily Riehl, Dominic Verity

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    Consider a diagram of quasi-categories that admit and functors that preserve limits or colimits of a fixed shape. We show that any weighted limit whose weight is a projective cofibrant simplicial functor is again a quasi-category admitting these (co)limits and that they are preserved by the functors in the limit cone. In particular, the Bousfield-Kan homotopy limit of a diagram of quasi-categories admits any (co)limits existing in and preserved by the functors in that diagram. In previous work, we demonstrated that the quasi-category of algebras for a homotopy coherent monad could be described as a weighted limit of this type, so these results specialise to (co)completeness results for quasi-categories of algebras. The second half of this paper establishes a further result in the quasi-categorical setting: proving, in analogy with the classical categorical case, that the monadic forgetful functor of the quasi-category of algebras for a homotopy coherent monad creates all limits that exist in the base quasi-category, regardless of whether its functor part preserves those limits. This proof relies upon a more delicate and explicit analysis of the particular weight used to define quasi-categories of algebras.

    LanguageEnglish
    Pages1-33
    Number of pages33
    JournalHomology, Homotopy and Applications
    Volume17
    Issue number1
    DOIs
    Publication statusPublished - 2015

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    Homotopy
    Completeness
    Functor
    Algebra
    Colimit
    Monads
    Diagram
    Categorical
    Analogy
    Cone

    Cite this

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    Completeness results for quasi-categories of algebras, homotopy limits, and related general constructions. / Riehl, Emily; Verity, Dominic.

    In: Homology, Homotopy and Applications, Vol. 17, No. 1, 2015, p. 1-33.

    Research output: Contribution to journalArticleResearchpeer-review

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