Kan extensions and the calculus of modules for ∞–categories

Emily Riehl, Dominic Verity

    Research output: Contribution to journalArticleResearchpeer-review

    Abstract

    Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.
    LanguageEnglish
    Pages189-271
    Number of pages83
    JournalAlgebraic and geometric topology
    Volume17
    Issue number1
    DOIs
    Publication statusPublished - 2017

    Keywords

    • ∞–categories
    • modules
    • profunctors
    • virtual equipment
    • pointwise Kan extension
    • Virtual equipment
    • Modules
    • Profunctors
    • Pointwise Kan extension

    Cite this

    @article{36f573c9a0f74d54b0d22a38a7e73fd6,
    title = "Kan extensions and the calculus of modules for ∞–categories",
    abstract = "Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.",
    keywords = "∞–categories, modules, profunctors, virtual equipment, pointwise Kan extension, Virtual equipment, Modules, Profunctors, Pointwise Kan extension",
    author = "Emily Riehl and Dominic Verity",
    year = "2017",
    doi = "10.2140/agt.2017.17.189",
    language = "English",
    volume = "17",
    pages = "189--271",
    journal = "Algebraic and geometric topology",
    issn = "1472-2739",
    publisher = "Mathematical Sciences Publishers",
    number = "1",

    }

    Kan extensions and the calculus of modules for ∞–categories. / Riehl, Emily; Verity, Dominic.

    In: Algebraic and geometric topology, Vol. 17, No. 1, 2017, p. 189-271.

    Research output: Contribution to journalArticleResearchpeer-review

    TY - JOUR

    T1 - Kan extensions and the calculus of modules for ∞–categories

    AU - Riehl,Emily

    AU - Verity,Dominic

    PY - 2017

    Y1 - 2017

    N2 - Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.

    AB - Various models of (∞,1)-categories, including quasi-categories, complete Segal spaces, Segal categories, and naturally marked simplicial sets can be considered as the objects of an ∞-cosmos. In a generic ∞-cosmos, whose objects we call ∞-categories, we introduce modules (also called profunctors or correspondences) between ∞-categories, incarnated as as spans of suitably-defined fibrations with groupoidal fibers. As the name suggests, a module from A to B is an ∞-category equipped with a left action of A and a right action of B, in a suitable sense. Applying the fibrational form of the Yoneda lemma, we develop a general calculus of modules, proving that they naturally assemble into a multicategory-like structure called a virtual equipment, which is known to be a robust setting in which to develop formal category theory. Using the calculus of modules, it is straightforward to define and study pointwise Kan extensions, which we relate, in the case of cartesian closed ∞-cosmoi, to limits and colimits of diagrams valued in an ∞-category, as introduced in previous work.

    KW - ∞–categories

    KW - modules

    KW - profunctors

    KW - virtual equipment

    KW - pointwise Kan extension

    KW - Virtual equipment

    KW - Modules

    KW - Profunctors

    KW - Pointwise Kan extension

    UR - http://purl.org/au-research/grants/arc/DP130101969

    UR - http://www.scopus.com/inward/record.url?scp=85011875134&partnerID=8YFLogxK

    U2 - 10.2140/agt.2017.17.189

    DO - 10.2140/agt.2017.17.189

    M3 - Article

    VL - 17

    SP - 189

    EP - 271

    JO - Algebraic and geometric topology

    T2 - Algebraic and geometric topology

    JF - Algebraic and geometric topology

    SN - 1472-2739

    IS - 1

    ER -