The comprehension construction

Emily Riehl*, Dominic Verity

*Corresponding author for this work

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    In this paper we construct an analogue of Lurie’s “unstraightening” construction that we refer to as the comprehension construction. Its input is a cocartesian fibration p: E ↠ B between ∞-categories together with a third ∞-category A. The comprehension construction then defines a map from the quasi-category of functors from A to B to the large quasi-category of cocartesian fibrations over A that acts on f : A → B by forming the pullback of p along f. To illustrate the versatility of this construction, we define the covariant and contravariant Yoneda embeddings as special cases of the comprehension functor. We then prove that the hom-wise action of the comprehension functor coincides with an “external action” of the hom-spaces of B on the fibres of p and use this to prove that the Yoneda embedding is fully faithful, providing an explicit equivalence between a quasi-category and the homotopy coherent nerve of a Kan-complex enriched category.
    Original languageEnglish
    Pages (from-to)116-190
    Number of pages75
    JournalHigher Structures
    Issue number1
    Publication statusPublished - 2018

    Bibliographical note

    Copyright the Author(s) 2018. Version archived for private and non-commercial use with the permission of the author/s and according to publisher conditions. For further rights please contact the publisher.


    • ∞-categories
    • straightening
    • unstraightening
    • comprehension
    • Yoneda embedding


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