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nLab > Latest Changes: clan

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 24th 2023

    Started a stub.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023
    • (edited Aug 24th 2023)

    I have expanded and cross-linked a little more, to make clear that this definition didn’t happen in an intellectual vacuum (did this definition really not have another name before, in the literature?)

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023
    • (edited Aug 24th 2023)

    added this remark on terminology:

    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 Σ\Sigma-types and ∏\prod-types coincides with his notion of π\pi-clan.

    diff, v2, current

    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023

    added the remark that the definition is due to:

    • Paul Taylor, §4.3.2 in: Recursive Domains, Indexed Category Theory and Polymorphism, Cambridge (1983-7) [pdf]

    I think we should merge the entry clan into display map.

    diff, v3, current

    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeAug 24th 2023

    I am all for finding better – namely more evocative – words for existing definitions (eg. I have sympathy for logos) but I admit with “clan” I am at a loss.

    • CommentRowNumber6.
    • CommentAuthorvarkor
    • CommentTimeAug 24th 2023

    I think we should merge the entry clan into display map.

    I agree. The terminology “clan” is less descriptive than existing terminology, and doesn’t appear evocative of its nature.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeAug 24th 2023

    Is there a more descriptive name for “tribe” as well? You see at least how Joyal was looking to link the two – clan/tribe.

    • CommentRowNumber8.
    • CommentAuthorvarkor
    • CommentTimeAug 24th 2023
    • (edited Aug 24th 2023)

    Is there a more descriptive name for “tribe” as well? You see at least how Joyal was looking to link the two – clan/tribe.

    Yes, as mentioned on the tribe page, this is the same as a display map category with ∑\sum-types and identity types.

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

    BTW, I had added this remark to tribe already.

    I can see that in many situations the term “display category with XYZ-types” may be too clunky

    (though it nicely indicates that this is a generic notion where one is not supposed to take credit of originality just for making choices of XYZ…)

    on the other hand, we have terms like ΠW-pretopos, so we might as well speak of ΠI-display categories (instead of “tribes, in this case).

    Or why not just shorten it to “display”

    (which can sensibly refer to the entire category as much as to its class of fibrations):

    this would make “tribes” be ΠI-displays. Maybe not too bad.

  10. fixed the definition of clan — they need not have all finite limits, but only a terminal object and base change of display maps

    Jonas Frey

    diff, v4, current

1 to 10 of 10

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