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

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

[Camell Kachour]

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

[Timothy Porter]

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

[Timothy Porter]

Ronnie,

The point about pointed spaces is perhaps not really that difficult to get
around at least partially. Using either the Bullejos-Cegarra-Duskin method
or my approach the formulae for simplicial groups extends more or less to
simplicially enriched groupoids and thus to all homotopy types.  This works
equally well for your point in the second paragraph, but the obscure part
is the following:

I will look back at the examples in arXiv:0903.2627v2, but I remember
discussing this with you and one of the differences is that in a
simplicially enriched groupoid, G, one has a simplicial groupoid with
constant object of objects. What this means for translating the simplicial
composition in G into structure in the corresponding cat^n objects is where
interesting things start happening so writing L for Loday's construction
one wants to use a composition
G(x,y)xG(y,z)-> G(x,z) for give something at the level of
LG(x,y)?LG(y,z)->LG(x,z), where ? is some construction not yet defined.

This should be  possible to analyse as the multiplication in a simplicial
group leads to structures in the corresponding cat^n group and back again,
so probably one could look at that as G(x,x)\times G(x,x)\to G(x,x) to see
what is happening???

This is getting a bit off-topic for the original messages so I will stop,
but would welcome any ideas either on a separate thread or via MathOverflow
perhaps.

Tim




On 15 July 2017 at 21:59, RONALD BROWN <ronnie.profbrown@btinternet.com>
wrote:

> Dear All,
>
> Loday's model is for pointed spaces, and Grothendieck was critical of this
> in a letter to me in 1983, of which I have quoted part in the Indag Paper
> on my preprint page.  I did not worry about this in the 1980s since the
> immediate consequences were quite novel. For example, Ellis and Steiner
> solved the old problem of the critical group for (n+1)-ads, and the
> nonabelian tensor product of groups has been well developed by group
> theorists (see www.groupoids.org.uk/nonabtens.html).
>
> What has not been looked at is an input of crossed modules over groupoids,
> instead of over groups, and considering first the work of Ellis-Steiner in
> that light. (crossed n-cubes of groupoids?)
>
> We know from examples that strict 2-fold groupoids are more complicated
> than homotopy 2-types, see my preprint  arXiv:0903.2627v2; and the van
> Kampen theorem with Loday has not so far been given a version with many
> base points, unlike the version in the book Nonabelian Algebraic Topology.
>
> The philosophy given in the Indag Paper has relatively  recently been put
> in this form, and so no part of it was discussed with Grothendieck, except
> the idea that n-fold groupoids model homotopy n-types, which, as said
> above, is not quite correct, though he thought it "absolutely beautiful".
> At that time, 1985,  he was starting to write "Recollte et Semaille", a
> task which seemed to lead him away from mathematics.
>
> The work with Loday shows in many explicit examples how low dimensional
> identifications in topology can give rise to high dimensional homotopy
> invariants, and there are explicit and precise calculations using the
> higher van Kampen theorems. Such calculation  is not the only aim, but it
> does give a useful test.
>
> Best
>
> Ronnie
>
>

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

[RONALD BROWN]

Dear All, 

Loday's model is for pointed spaces, and Grothendieck was critical of this in a letter to me in 1983, of which I have quoted part in the Indag Paper on my preprint page.  I did not worry about this in the 1980s since the  immediate consequences were quite novel. For example, Ellis and Steiner solved the old problem of the critical group for (n+1)-ads, and the nonabelian tensor product of groups has been well developed by group theorists (see www.groupoids.org.uk/nonabtens.html). 

What has not been looked at is an input of crossed modules over groupoids, instead of over groups, and considering first the work of Ellis-Steiner in that light. (crossed n-cubes of groupoids?)

We know from examples that strict 2-fold groupoids are more complicated than homotopy 2-types, see my preprint  arXiv:0903.2627v2; and the van Kampen theorem with Loday has not so far been given a version with many base points, unlike the version in the book Nonabelian Algebraic Topology. 

The philosophy given in the Indag Paper has relatively  recently been put in this form, and so no part of it was discussed with Grothendieck, except the idea that n-fold groupoids model homotopy n-types, which, as said above,  is not quite correct, though he thought it "absolutely beautiful". At that  time, 1985,  he was starting to write "Recollte et Semaille", a task which  seemed to lead him away from mathematics.  

The work with Loday shows in many explicit examples how low dimensional identifications in topology can give rise to high dimensional homotopy invariants, and there are explicit and precise calculations using the higher van Kampen theorems. Such calculation  is not the only aim, but it does give a useful test. 

Best 

Ronnie





----Original message----
From : t.porter.maths@gmail.com
Date : 15/07/2017 - 07:35 (GMTDT)
To : camell.kachour@gmail.com
Cc : categories@mta.ca
Subject : categories: Re: Homotopy hypothesis for contractible operad definitions of weak n-categories

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


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

[henry]

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
>
>
> [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
>



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Homotopy hypothesis for contractible operad definitions of weak n-categories

[Jamie Vicary]

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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