Homotopy hypothesis for contractible operad definitions of weak n-categories
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-... [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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-...
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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. [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
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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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 [For admin and other information see: http://www.mta.ca/~cat-dist/ ]
participants (5)
-
Camell Kachour -
henry＠phare.normalesup.org -
Jamie Vicary -
RONALD BROWN -
Timothy Porter