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nLab > Latest Changes: two-level type theory

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  1.  
    • CommentRowNumber1.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    A stub so far.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorUrs
    • CommentTimeApr 16th 2018
    • (edited Apr 16th 2018)

    Notice that we already have an entry Homotopy Type System, which is the same kind of idea.

    • CommentRowNumber3.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 16th 2018

    Ah ok. The second reference’s abstract says

    The idea of two-level type theory is heavily inspired by Voevodsky’s Homotopy Type System (HTS). Two-level type theory can be thought of as a version of HTS without equality reflection. We show that the lack of equality reflection does not hinder the development of the ideas that HTS was designed for.

    Does this merit a distinct entry then?

    • CommentRowNumber4.
    • CommentAuthorMike Shulman
    • CommentTimeApr 16th 2018

    Sorry, I forgot about that page.

    I would lean towards folding Homotopy Type System into two-level type theory. Unfortunately, “2LTT” seems to be being used both as a general term for type theories with both exact and path equality types, and also a name for the specific ACK system. The general term includes not just ACK and HTS but also the “computational cubical type theory” of Harper-Angiuli et. al. Nearly everything we currently have at Homotopy Type System applies more generally to all two-level type theories, and right now it seems that ACK and CCTT are receiving more active work than HTS. (In particular, HTS has the disadvantage of undecidable type-checking, which ACK restores by leaving out equality reflection in favor of UIP – for CCTT I’m not sure, that may not be a relevant question for the sort of type theory that it is.)

    • CommentRowNumber5.
    • CommentAuthorDavid_Corfield
    • CommentTimeApr 17th 2018

    I added a link to Homotopy Type System. I’ll leave it to experts to arrange what they take to be the proper relation between these pages.

    diff, v2, current

    • CommentRowNumber6.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018

    folded HTS into this page

    diff, v3, current

    • CommentRowNumber7.
    • CommentAuthorMike Shulman
    • CommentTimeApr 17th 2018

    I also added some discussion of the issue of fibrant vs non-fibrant nat.

  8. removed duplicate redirects (fibrant types used to redirect to here and to fibrant type)

    Anonymouse

    diff, v10, current

1 to 8 of 8

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