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I would like to know some open problems in HoTT for Homotopy theorist (without knowledge of logic). With such a Homotopy theorist I mean someone who takes a serious reading ("at the level") of Cisinski's Univalent universes for elegant models of Homotopy types.

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    $\begingroup$ Asking merely for a list of open problems seems the wrong approach if one wants to actually understand the mathematics... $\endgroup$
    – Yemon Choi
    Jul 28, 2017 at 3:30
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    $\begingroup$ If someone just drops a bunch of open problems on your head that won't be very useful. I'd recommend joining the HoTT list and having an informed discussion. I am sure the crowd will be very welcoming to a homotopy theorist who wants to get into HoTT. $\endgroup$ Jul 28, 2017 at 9:03
  • $\begingroup$ Given @AndrejBauer's comment, would you like to clarify a bit more what actual research background you have in homotopy theory? $\endgroup$
    – Yemon Choi
    Jul 30, 2017 at 3:31

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The introduction of the HoTT book ends with an "Open problems" section that certainly answers your question.

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