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

[Timothy Porter <t.porter.maths@gmail.com>]

Dear All,

Can I ask why Loday's cat^n groups are not mentioned? (They have been
now.)  I know they are not globular, but by spreading out the `weakness' of
the higher groupoid structures the axioms end up being strict (and very
simple as they are really just abstractions of classical commutator
identities).  Surely they deserve to be used as a reference point to
compare some of the other candidates. Loday's models work for *all *n-types
for finite n. (I do not know how to handle general homotopy types using any
similar methodology.)

Tim

On 13 July 2017 at 23:19, Camell Kachour <camell.kachour@gmail.com> wrote:

>
> Hi Jamie,
>
> You said : "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.
>
> To be more precise they gave an operadic approach of weak higher
> categories with which we can extract a definition of weak n-groupoids and
> can say :
> a weak n-groupoid is a specific algebra for the operad K of weak higher
> categories (build first by Batanin). However it is important to know that
> neither Batanin or Leinster have defined a monad,
> specific to higher groupoids,
> which algebras are models of globular weak higher groupoids. However this
> was done
> in my work here :
>
> http://www.tac.mta.ca/tac/volumes/30/22/30-22.pdf
>
> where in particular I proved that my models of weak higher groupoids are
> also
> algebras for the operad K of Batanin (which algebras are his definition of
> weak
> higher categories).
>
> Remark : And with similar methods we can go beyond, and build
> cubical and multiple weak higher groupoids, but this is an other story ...
> (see
> my arxived work ...)
>
> The homotopy hypothesis for these globular weak higher groupoids (those
> defined
> by Batanin in 1998, or the definition of Grothendieck-Maltsiniotis, or my
> approach), seems to be a difficult problem (for that it is good to see the
> work of Ara (thesis), Tuy=C3=A9ras (thesis) and Simon Henry), and it is not
> evident at all that the homotopy hypothesis is in fact true. However we
> suspect it to be true
> only based on the fact that Kan-complexes models homotopy of spaces, and
> we suspect that there is a Quillen model structure on the category of weak
> globular higher groupoids which is Quillen equivalent to the category of
> Kan-complexes equipped with the induced model structure on the category of
> simplicial sets.
>
> In fact, in=C2=A0http://www.tac.mta.ca/tac/volumes/30/22/30-22.pdf, I
> said =
> that
> we have a generalized
> version of the homotopy hypothesis of Grothendieck, which is the statement
> that the category of globular weak (infinity,N)-categories (which is the
> category of algebras for a fixed monad, for each integer N; and these
> algebras are still algebras for the operad K of Batanin !), should be
> Quillen equivalent to the category of other simplicial models of
> (infinity,N)-categories.
>
> Best,
> Camell.
>

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