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    • CommentRowNumber1.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2019

    now that Mike announced a proof, and hearing Steve’s comment, I felt it would be nice to have a name for conjecture (partially) proven thereby, for ease of communiucating it to the rest of the world. Just a start, please edit the entry as need be.

    v1, current

    • CommentRowNumber2.
    • CommentAuthorDavid_Corfield
    • CommentTimeMar 13th 2019

    What of the elementary/Grothendieck distinction? Who first articulated claims about what this distinction means for the internal logic?

    And when do we get to see slides, or better a recording, Mike?

    • CommentRowNumber3.
    • CommentAuthorAlizter
    • CommentTimeMar 13th 2019

    Here are the slides of Mike’s second talk. I don’t think there exists a better recording apart from the one Felix took.

    • CommentRowNumber4.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 13th 2019

    So the strict univalent universes correspond to which type universes out of: Russell, Tarski, weak Tarski ? I guess not the last… Or is this an orthogonal notion? This should be made explicit in the page where Mike’s results are discussed.

    • CommentRowNumber5.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 13th 2019

    And what was Steve’s comment?

    • CommentRowNumber6.
    • CommentAuthorUrs
    • CommentTimeMar 13th 2019

    Steve just highlighted that it’s a breakthrough (on the off-chance that anyone didn’t get it) and amplified that it finally goes to confirm his 10 year old conjecture.

  1. this was not a precise conjecture, but more of a research proposal.

    steveawodey

    diff, v3, current

    • CommentRowNumber8.
    • CommentAuthorUrs
    • CommentTimeMar 14th 2019

    upon request, I added pointer to p. 9 of

    diff, v5, current

    • CommentRowNumber9.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2019

    Re #4, I would say these universes correspond most directly to strictly Tarski ones, although the distinction between Russell and strict-Tarski universes is mostly lost when passing from syntax to categorical models (even strict ones such as CwFs). I think the usual way to interpret Russell universes is to first “desugar” them to Tarski universes and then interpret those.

    • CommentRowNumber10.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2019

    I don’t have time to add it right now, but we should also be clear about the difference between the conjectures “type theory can be interpreted in all Grothendieck (,1)(\infty,1)-toposes” and “the homotopy theory of type theories is equivalent to the homotopy theory of elementary (,1)(\infty,1)-toposes”. The former is what I announced, but the latter is still quite open, although Kapulkin-Sumilo have proven the analogue for (,1)(\infty,1)-categories with finite limits, corresponding to type theories with Σ\Sigma and identity types only, no Π\Pis or universes, and it seems likely that the tools I used in the Grothendieck case may be useful in proving versions of the stronger conjecture.

    • CommentRowNumber11.
    • CommentAuthorMike Shulman
    • CommentTimeMar 14th 2019

    Also here are my slides from the Midwest HoTT last weekend, which have some more detail. I’m hoping to put up the preprint within a week or two.

    • CommentRowNumber12.
    • CommentAuthorDavidRoberts
    • CommentTimeMar 14th 2019

    @Mike thanks!

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