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• 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 $supp(A)$.

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

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

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