Re: Homotopy hypothesis for contractible operad definitions of weak n-categories

[henry@phare.normalesup.org]

Hi Jamie,

For a long time, the only reasons to believe it where the fact that
Grothendieck original definition was a rather natural one (as well as
Batanin's and other subsequent definitions), and that the result had been
checked I believe up to dimension 3, with the only thing preventing to go
higher being the combinatorial explosion in the definition of weak
infinity groupoid.

There also have been some (related) work by Clemens Berger and
Denis-Charles Cisinski toward the homotopy hypothesis:
http://math1.unice.fr/~cberger/nerve.pdf
https://arxiv.org/abs/math/0604442


Finally, I have a very recent work (only on the arxiv at the present time)
which is Higly relevant for your question:

"Algebraic models of homotopy types and the homotopy hypothesis"
https://arxiv.org/abs/1609.04622

on which I can say a little bit more:

As the title suggest, the paper is generally interested in producing
algebraic model for representing homotopy type, and in particular two
results are obtained which brings some light on this question:


- The first is that the following very natural conjecture implies the
homotopy hypothesis:

Conjecture:
Let $X$ be a free finitely generated Grothendieck infinity groupoid (i.e.
$X$ is constructed from the empty groupoid by iteratively freely adding
cells). Let $a$ be a $n$-cell on $X$, and consider the Grothendieck
infinity groupoid $X+$ obtained from $X$ by freely adding one cell $a'$
parallel de $a$ and one cell $b$ between $a$ and $a'$.
Then the natural map from $X$ to $X+$ is a homotopy equivalence (in the
sense that it induces a bijection on all the homotopy groups)


Note that as the cell $b$ is automatically an isomorphism because we are
working with groupoids, a failure of this conjecture would indicate that
Grothendieck infinity groupoids are very poorly behaved with respect to
free construction.


If this conjecture is true, then one can construct a semi-model structure
on the the category of Grothendieck infinity groupoid and it is shown in
my paper that this semi model structure is Quillen equivalent to the usual
model structure on topological space or on simplicial sets.


Note that thank's to Dimitri Ara's Phd Thesis, the relation between
Grothendieck-Maltsiniotis's defintion and Batanin-Leinster defintion is
rather well understood, so even if it is not detailled in the paper one
also have a similar result for Batanin-Leinster type definition.


- The second is that the homotopy hypothesis hold for similar structure:
One can also define a new a notion of infinity groupoid which are globular
sets endowed with all the operation that can be defined on a type in
homotopy type theory using only identity types (more prececely, a weak
version of identity type). See my paper for the precise definition. I have
proved in the paper that for this notion of infinity groupoids on has the
homotopy hypothesis.

This also say something (informal) about the fact that infinity groupoid
do have a somehow weaker structure than what type have in homotopy type
theory...


Best regards,
Simon


> Hi,
>
> Batanin, Leinster and other have presented related definitions of weak
> n-groupoid in terms of contractible globular operads. I personally find
> these definitions of "contractible n-groupoids" extremely beautiful. I am
> interested to learn what evidence we have that the homotopy hypothesis
> might be true for (at least one of) these definitions.
>
> Some good evidence is provided by Peter LeFanu Lumsdaine's [1] proof that
> a
> homotopy type gives rise to an infinity-groupoid in the sense of Leinster.
> There is other work along similar lines.  But, as far as I am aware, it
> remains possible that contractible n-groupoids might in general be weaker
> structures than homotopy n-types.
>
> A fun way to investigate this would be to verify small instances of
> phenomena associated to the periodic table in contractible n-groupoids.
> For
> example, Christoph Dorn has shown me a proof that the Eckmann-Hilton
> argument holds in a Leinster 2-category; that is, for an object X, and for
> 2-morphisms f,g:id[X]-->id[X], we have f.g=g.f, thereby establishing one
> of
> the first phenomena predicted by the periodic table.
>
> Have any higher phenomena from the periodic table been verified? Or, is
> there other evidence that contractible n-groupoids behave "homotopically"
> in general?
>
> Best wishes,
> Jamie
>
> [1]
> http://peterlefanulumsdaine.com/research/Lumsdaine-Weak-omega-cats-from-ITT-LMCS.pdf
>
>
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