Re: [HoTT] Re: Conjecture

[Favonia <fav...@gmail.com>]

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Yes, this has been known from Nicolai's work on truncation (e.g. "The
General Universal Property"). I just learned that this has been mechanized
in Agda by Nils as well.
http://www.cse.chalmers.se/~nad/listings/equality/H-level.Truncation.html#18216
By
the way, the condition (∏ x, c x x = idpath (f x)) is provable from
transitivity. -Favonia

On Sun, Apr 2, 2017 at 8:35 PM Daniel R. Grayson <
danielrich...@gmail.com> wrote:

Here's a fact related to the current discussion, which I have formalized
today.  I would appreciate knowing
whether it's already known.  It gives a criterion for factoring through the
propositional truncation
when the target is of h-level 3.

  Definition squash_to_HLevel_3 {X : UU} {Y : HLevel 3}
             (f : X -> Y) (c : ∏ x x', f x = f x') :
    (∏ x, c x x = idpath (f x)) ->
    (∏ x x' x'', c x x'' = c x x' @ c x' x'') ->
    ∥ X ∥ -> Y.

You may find it in WellOrderedSets.v
<https://github.com/DanGrayson/UniMath/blob/well-ordering-2/UniMath/Combinatorics/WellOrderedSets.v>
on
one of my branches.


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