Re: [HoTT] Free higher groups

[Dan Christensen <jdc@uwo.ca>]

A not-so-interesting answer to Mike's question is the type of deloopings
of S^3.  The reason this isn't so interesting is that it's in the image
of the natural functor from Spaces to any oo-topos, so it's true just
because it is true for Spaces.  Similarly, a statement asserting that
pi_42(S^17) = (insert what it is) is true in any oo-topos.  Another
reason these aren't interesting is that I expect that they are provable
in HoTT with enough work.

So, I'll second Mike's question, with the extra condition that it would
be good to have a type for which there is some reason to doubt that it
is provably inhabited in HoTT.

Oh, what about whether the hypercomplete objects are the modal objects
for a modality?  I'm throwing this out there without much thought...

Dan

On Apr 24, 2023, Michael Shulman <shulman@sandiego.edu> wrote:

> This is fantastic, especially the simplicity of the construction.  As
> Peter said, a wonderful way to commemorate the 10th anniversary of the
> special year and the release of the HoTT Book.
>
> Relatedly to Nicolai's question, this question also has an easy proof
> in any Grothendieck infinity-topos.  Now that we know it also has a
> proof in HoTT, do we know of any type in HoTT whose interpretation in
> any Grothendieck infinity-topos is known to be inhabited, but which
> isn't known to be inhabited in HoTT?
>
> On Fri, Apr 21, 2023 at 5:25 PM Nicolai Kraus
> <nicolai.kraus@gmail.com> wrote:
>
>     Hi David,
>
>     Congratulations (again)! I find it very interesting that this
>     question has a positive answer. I had suspected that it might
>     separate HoTT from Voevodsky's HTS (aka 2LTT with a fibrancy
>     assumption on strict Nat). Since this isn't the case, do we know
>     of another type in HoTT that is inhabited in HTS, while we don't
>     know whether we can construct an inhabitant in HoTT?
>
>     Best,
>     Nicolai
>
>     On Fri, Apr 21, 2023 at 8:30 PM Jon Sterling
>     <jon@jonmsterling.com> wrote:
>
>         Dear David,
>
>         Congratulations on your beautiful result; I'm looking forward
>         to understanding the details. Recently I had been wondering if
>         anyone had proved this, and I am delighted to see that it is
>         now done.
>
>         Best wishes,
>         Jon
>
>         On 21 Apr 2023, at 12:04, David Wärn wrote:
>
>         > Dear all,
>         >
>         > I'm happy to announce a solution to one of the oldest open
>         problems in synthetic homotopy theory: the free higher group
>         on a set is a set.
>         >
>         > The proof proceeds by describing path types of pushouts as
>         sequential colimits of pushouts, much like the James
>         construction. This description should be useful also in many
>         other applications. For example it gives a straightforward
>         proof of Blakers-Massey.
>         >
>         > Best wishes,
>         > David
>         >
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