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nLab > Latest Changes: definability (fibred category theory)

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  1.  
    • CommentRowNumber1.
    • CommentAuthormattecapu
    • CommentTimeMar 28th 2023

    Started page on definability à là Benabou

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeMar 28th 2023

    In the first sentence

    Suppose we have a fibration p:ℰ→ℬp \colon \mathcal{E} \to \mathcal{B}, representing type families (living in the fibers of ℰ\mathcal{E}) indexed by contexts (living in ℬ\mathcal{B}).

    probably “fibration” means “Grothendieck fibration”? (I have now hyperlinked it as such). But then referring to this as a “type family” is unusual – unless maybe you want to speak of directed type theory?

    I think this should be clarified.

    diff, v2, current

    • CommentRowNumber3.
    • CommentAuthormattecapu
    • CommentTimeMar 29th 2023
    • (edited Mar 29th 2023)
    Yeah it means Grothendieck fibration. Maybe the phrasing is slightly odd, I mean to say objects in E are type families.
    • CommentRowNumber4.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2023

    Oh, I see. So I misread what you mean. (Haven’t take much time with the entry, admittedly).

    But then lets’ say this in the entry: “whose objects are type families”.

    And add a link to category with families! (or whatever it may be)

    • CommentRowNumber5.
    • CommentAuthormattecapu
    • CommentTimeMar 29th 2023
    Ok great, I'll rephrase it.
    • CommentRowNumber6.
    • CommentAuthormattecapu
    • CommentTimeMar 29th 2023

    Changed opening sentence as suggested by Urs

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorUrs
    • CommentTimeMar 29th 2023

    Thanks. So I have appended pointer to categorical semantics of dependent types.

    diff, v4, current

    • CommentRowNumber8.
    • CommentAuthormattecapu
    • CommentTimeMar 29th 2023

    Changed context dropdown to ’fibred category theory’

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthormattecapu
    • CommentTimeApr 3rd 2023

    Added definition of definable collections of functors, vertical morphisms, subfibrations

    diff, v6, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeApr 3rd 2023

    I have made the link to “fibered terminal object” that you request here point to fibered limit.

    But now I realize it may not be the pointer you intend(?):

    In the usual language a fibered category p:E→Bp \colon E \to B has a fibered terminal object if (e.g. p. 6 here) EE and BB have terminal objects and pp preserves the terminal object. To get from there to the condition you state one needs an extra assumption, I suppose, such as the base change preserving limits, hence such as the fibration being a bifibration. No? Maybe you could expand on this point in the entry.

1 to 10 of 10

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