Infinity category theory from scratch

Emily Riehl, Dominic Verity

    Research output: Other contributionResearch

    Abstract

    We use the terms "∞-categories" and "∞-functors" to mean the objects and morphisms in an "∞-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, θn-spaces, and fibered versions of each of these are all ∞-categories in this sense. We show that the basic category theory of ∞-categories and ∞-functors can be developed from the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts.
    In the first lecture, we define an ∞-cosmos and introduce its "homotopy 2-category," using formal category theory to define and study equivalences and adjunctions between ∞-categories. In the second lecture, we study (co)limits of diagrams taking values in an ∞-category and the relationship between (co)limits and adjunctions. In the third lecture, we introduce comma ∞-categories, which are used to encode the universal properties of (co)limits and adjointness and prove "model independence" results. In the fourth lecture, we introduce (co)cartesian fibrations, describe the calculus of "modules" between ∞-categories, and use this framework to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of ∞-functors.
    LanguageEnglish
    TypeArXiv Preprint
    Number of pages53
    Publication statusUnpublished - 19 Aug 2016

    Fingerprint

    Category Theory
    Infinity
    Colimit
    Functor
    Adjunction
    Independence Results
    Simplicial Set
    Fibration
    Morphisms
    Cartesian
    Categorical
    Axioms
    Homotopy
    Lemma
    Calculus

    Bibliographical note

    These lecture notes were written to accompany a mini course given at the 2015 Young Topologists' Meeting at EPFL, videos of which can be found at https://hessbellwald-lab.epfl.ch/ytm2015, which summarizes the content of arXiv:1306.5144, arXiv:1310.8279, arXiv:1401.6247, arXiv:1506.05500, and arXiv:1507.01460.

    Cite this

    Riehl, E., & Verity, D. (2016, Aug 19). Infinity category theory from scratch. Unpublished.
    @misc{8dee5848abfd4fe6acc44cfbe91edd9a,
    title = "Infinity category theory from scratch",
    abstract = "We use the terms {"}∞-categories{"} and {"}∞-functors{"} to mean the objects and morphisms in an {"}∞-cosmos.{"} Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, θn-spaces, and fibered versions of each of these are all ∞-categories in this sense. We show that the basic category theory of ∞-categories and ∞-functors can be developed from the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts. In the first lecture, we define an ∞-cosmos and introduce its {"}homotopy 2-category,{"} using formal category theory to define and study equivalences and adjunctions between ∞-categories. In the second lecture, we study (co)limits of diagrams taking values in an ∞-category and the relationship between (co)limits and adjunctions. In the third lecture, we introduce comma ∞-categories, which are used to encode the universal properties of (co)limits and adjointness and prove {"}model independence{"} results. In the fourth lecture, we introduce (co)cartesian fibrations, describe the calculus of {"}modules{"} between ∞-categories, and use this framework to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of ∞-functors.",
    author = "Emily Riehl and Dominic Verity",
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    Infinity category theory from scratch. / Riehl, Emily; Verity, Dominic.

    53 p. 2016, ArXiv Preprint.

    Research output: Other contributionResearch

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    AU - Riehl, Emily

    AU - Verity, Dominic

    N1 - These lecture notes were written to accompany a mini course given at the 2015 Young Topologists' Meeting at EPFL, videos of which can be found at https://hessbellwald-lab.epfl.ch/ytm2015, which summarizes the content of arXiv:1306.5144, arXiv:1310.8279, arXiv:1401.6247, arXiv:1506.05500, and arXiv:1507.01460.

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    N2 - We use the terms "∞-categories" and "∞-functors" to mean the objects and morphisms in an "∞-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, θn-spaces, and fibered versions of each of these are all ∞-categories in this sense. We show that the basic category theory of ∞-categories and ∞-functors can be developed from the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts. In the first lecture, we define an ∞-cosmos and introduce its "homotopy 2-category," using formal category theory to define and study equivalences and adjunctions between ∞-categories. In the second lecture, we study (co)limits of diagrams taking values in an ∞-category and the relationship between (co)limits and adjunctions. In the third lecture, we introduce comma ∞-categories, which are used to encode the universal properties of (co)limits and adjointness and prove "model independence" results. In the fourth lecture, we introduce (co)cartesian fibrations, describe the calculus of "modules" between ∞-categories, and use this framework to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of ∞-functors.

    AB - We use the terms "∞-categories" and "∞-functors" to mean the objects and morphisms in an "∞-cosmos." Quasi-categories, Segal categories, complete Segal spaces, naturally marked simplicial sets, iterated complete Segal spaces, θn-spaces, and fibered versions of each of these are all ∞-categories in this sense. We show that the basic category theory of ∞-categories and ∞-functors can be developed from the axioms of an ∞-cosmos; indeed, most of the work is internal to a strict 2-category of ∞-categories, ∞-functors, and natural transformations. In the ∞-cosmos of quasi-categories, we recapture precisely the same theory developed by Joyal and Lurie, although in most cases our definitions, which are 2-categorical rather than combinatorial in nature, present a new incarnation of the standard concepts. In the first lecture, we define an ∞-cosmos and introduce its "homotopy 2-category," using formal category theory to define and study equivalences and adjunctions between ∞-categories. In the second lecture, we study (co)limits of diagrams taking values in an ∞-category and the relationship between (co)limits and adjunctions. In the third lecture, we introduce comma ∞-categories, which are used to encode the universal properties of (co)limits and adjointness and prove "model independence" results. In the fourth lecture, we introduce (co)cartesian fibrations, describe the calculus of "modules" between ∞-categories, and use this framework to prove the Yoneda lemma and develop the theory of pointwise Kan extensions of ∞-functors.

    M3 - Other contribution

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