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    • CommentRowNumber1.
    • CommentAuthorMike Shulman
    • CommentTimeJan 14th 2019

    A stub with a few references.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorsamwinnick
    • CommentTimeJan 17th 2022

    I was just fed this great presentation:

    https://www.youtube.com/watch?v=HRBShaxIblI

    I just did a ctrl+f in the HoTT book and didn’t find “clan” or “tribe”. I also don’t see any pages on it here. Have these concepts proven to be useful in understanding the trinity? Any chance on getting an update on these pages?

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2022

    As far as I am aware, the terms “tribe” et al. are essentially synonyms for what most people had already called by other names, such as “display map categories” and others. Probably best to start with the nnLab entry categorical model of dependent types.

    • CommentRowNumber4.
    • CommentAuthorDavid_Corfield
    • CommentTimeJan 17th 2022

    Maybe the references at tribe will help.

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeJan 17th 2022

    I’d also think that the statement in #3 follows by immediate inspection of the definition. For a more explicit quote from a random reference on the matter:

    A tribe in the sense of Joyal is a display map category which models Σ-types and Id-types

    (arXiv:1901.03567, p. 6)

    • CommentRowNumber6.
    • CommentAuthorsamwinnick
    • CommentTimeJan 17th 2022

    Thanks for these. I hadn’t yet come across categorical model of dependent types.

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeJan 18th 2022

    However, you won’t find these things under any other names in the HoTT Book either. The reason is that the HoTT Book is not about semantics at all, only about doing mathematics internal to HoTT.

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023

    I have adjusted wording and referencing to make it clearer that the terminology may be new, but the basic idea isn’t (as far as I am aware).

    diff, v3, current

    • CommentRowNumber9.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023
    • (edited Aug 24th 2023)

    added this remark:


    The type-theory-literature traditionally refers to such categories generically as display map/fibration categories with such-and-such types, eg.

    North 2019, Rem. 2.4:

    The definitions in this section are relatively standard in the literature. A display map category which models Σ\Sigma-types coincides with Joyal’s notion of clan, and a display map category which models \sum-types and \prod-types coincides with his notion of π\pi-clan. […]

    A tribe in the sense of Joyal is a display map category which models Σ\Sigma-types and Paulin-Mohring Id-types.


    diff, v3, current