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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→ℬ, representing type families (living in the fibers of ℰ) indexed by contexts (living in 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→B has a fibered terminal object if (e.g. p. 6 here) E and B have terminal objects and p 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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