Re: [HoTT] Free higher groups

[Nicolai Kraus <nicolai.kraus@gmail.com>]

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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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