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nLab > Latest Changes: bracket type

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  1.  
    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2011
    • (edited Sep 26th 2012)

    started bracket type, just for completeness, but don’t really have time for it

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2011

    had the chance to add a bit more to bracket type: now there is an Idea-section a Semantics-section.

    I have also added links to bracket type to relevant entries, in particular to types and logic - table.

    Experts please check. I am pretty sure I know what I am talking about here, but I may not be using language in the standard way.

    • CommentRowNumber3.
    • CommentAuthorUrs
    • CommentTimeDec 1st 2011

    I am somewhat surprised that the idea of “bracket types” is not already in the original articles on propositions as types. I had thought that taking bracket types is implicitly understood all along.

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeDec 3rd 2011
    Thanks! I edited bracket type a bit. Bracket types were actually already referred to at propositions as types, only not by that name -- in the final paragraph of the Idea section under the phrase "propositions as some types". I added extra clarification there as well.
    • CommentRowNumber5.
    • CommentAuthorUrs
    • CommentTimeSep 26th 2012
    • (edited Sep 26th 2012)

    I have added to bracket type a brief section Definition in homotopy type theory with the definition of supp(A)supp(A).

    Also added four references on this. I see in Mike’s lectures the construction is attributed to Lumsdaine. However in the recent post by Brunerie, it seems to be attributed to Voevodsky. I don’t know. The entry currently does not cite Lumsdaine yet, but probably it should.

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeSep 26th 2012
    • (edited Sep 26th 2012)

    Voevodsky defined it, as Guillaume said, “using impredicative quantification and resizing rules” (and also univalence). I believe his definition simply mimics the classical proof that a (1-)topos is a regular category — univalence plus a resizing rule is just the way that you get a subobject classifier in HoTT.

    The definition as an HIT is what is due to Peter Lumsdaine.

    • CommentRowNumber7.
    • CommentAuthorDavid_Corfield
    • CommentTimeSep 26th 2017

    I added the recursion principle for supp(A)supp(A). I also added to the notations for the bracket type. We’re up to 6 now.

  8. fixed HTML entities to UTF-8 characters in Agda snippet

    anqurvanillapy

    diff, v17, current

  9. added section relating propositional truncations to localizations.

    Anonymous

    diff, v23, current

    • CommentRowNumber10.
    • CommentAuthorUrs
    • CommentTimeFeb 20th 2023

    added pointer to

    and doi-link to:

    diff, v24, current

    • CommentRowNumber11.
    • CommentAuthorUrs
    • CommentTimeFeb 21st 2023

    added pointer to:

    Have only scanned the first few pages, but: Is this not secretly rediscovering the notion of anafunctor?

    diff, v25, current

    • CommentRowNumber12.
    • CommentAuthorGuest
    • CommentTimeMar 10th 2023

    added definition of bracket type using a type of all propositions.

    diff, v27, current

  13. Added definition of propositional truncation as a sequential colimit and added the reference where the definition came from

    Anonymouse

    diff, v31, current

    • CommentRowNumber14.
    • CommentAuthorncfavier
    • CommentTimeJul 11th 2025

    Removed the incorrect rules for the path constructor (∀ (x y : A) → [x] = [y] instead of ∀ (x y : [A]) → x = y) as well as the now mostly incorrect Properties section and the redundant “Using functions from the type of booleans” section.

    Replaced the missing Coq HoTT reference with a 1Lab one.

    diff, v44, current

    • CommentRowNumber15.
    • CommentAuthorlinuxmetel
    • CommentTimeFeb 27th 2026
    • (edited Feb 27th 2026)

    What are B([x]) and B([a]) in Inference rules? Is the [] necessary or just mistake?

    • CommentRowNumber16.
    • CommentAuthorlinuxmetel
    • CommentTimeFeb 27th 2026
    • (edited Feb 27th 2026)

    I understand it.

    • CommentRowNumber17.
    • CommentAuthorlinuxmetel
    • CommentTimeFeb 27th 2026
    • (edited Feb 27th 2026)

    I still don’t understand how B([x]) is defined when only B(x) for x : A is assumed.

    • CommentRowNumber18.
    • CommentAuthorUrs
    • CommentTimeFeb 27th 2026

    True, this does look like a typing error.

    Beware that these inference rules (as much else in the entry) was written (revision 40) by “Anonymouse”, notorious for low level edits and formally blocked from editing the nLab.

    Maybe all this material deserves to be deleted.

    • CommentRowNumber19.
    • CommentAuthorlinuxmetel
    • CommentTimeFeb 27th 2026

    I see. Thank you for your reply. I don’t know how to fix it so I would appreciate it if someone could fix it.

  20. In definition 14.2.2 of Egbert Rijke’s textbook, the induction principle states that “for any family QQ of types over |A|\vert A \vert”, so the rules should contain Γ,x:[A]⊢B(x)\Gamma, x:[A] \vdash B(x) instead of Γ,x:A⊢B(x)\Gamma, x:A \vdash B(x).

    Peter Anderson

    diff, v45, current

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