Elements of ∞-category theory

Emily Riehl, Dominic Verity

    Research output: Book/ReportBookpeer-review


    The language of ∞-categories provides an insightful new way of expressing many results in higher-dimensional mathematics but can be challenging for the uninitiated. To explain what exactly an ∞-category is requires various technical models, raising the question of how they might be compared. To overcome this, a model-independent approach is desired, so that theorems proven with any model would apply to them all. This text develops the theory of ∞-categories from first principles in a model-independent fashion using the axiomatic framework of an ∞-cosmos, the universe in which ∞-categories live as objects. An ∞-cosmos is a fertile setting for the formal category theory of ∞-categories, and in this way the foundational proofs in ∞-category theory closely resemble the classical foundations of ordinary category theory. Equipped with exercises and appendices with background material, this first introduction is meant for students and researchers who have a strong foundation in classical 1-category theory.
    Original languageEnglish
    Place of PublicationCambridge
    PublisherCambridge University Press (CUP)
    Number of pages760
    ISBN (Electronic)9781108936880
    ISBN (Print)9781108837989
    Publication statusPublished - 2022

    Publication series

    NameCambridge studies in advanced mathematics

    Bibliographical note

    This book has been reviewed and is under contract to Cambridge University Press, with final draft due for submission by the end of March 2021.


    • Mathematics (general)
    • Mathematics
    • Logic
    • Categories and Sets
    • Geometry and Topology


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