## October 2, 2008

### More Model ω-Questions

#### Posted by Urs Schreiber

Here are some further questions related to model category theory in general and the folk model structure on $\omega$-categories in particular, concerned with

- fibrant objects in $\omega$-categories;

- model structure on $\omega$-groupoids and crossed complexes;

- general question about weak universal property of homotopy (co)limits.

Fibrant objects and $\infty$-groupoids

It seems that generically in model categories of “higher structures” the fibrant objects are close to being $\infty$-groupoids.

This is true for at least one model structure on simplicial sets: fibrant objects here are precisely the Kan complexes. It seems to be also true in $1Cat$ and $2Cat$, though I need to check that reference again.

So I am wondering whether the fibrant objects in $\omega$-categories are related to the $\omega$-groupoids (well, either those with strict or with weak inverses, I am not sure yet). Is anything known in general?

Model structure on crossed complexes

I have only just started looking at

R. Brown, M. Golasinski, A model structure for the homotopy theory of crossed complexes which describes parts of a model category structure on the category of crossed complexes.

Now, of course by R. Brown and P. Higgins On the algebra of cubes it is well known that crossed complexes are precisely equivalent to $\omega$-groupoids, i.e. to those $\omega$-categories for which each cell has a strict inverse cell.

So the obvious question is: How is the Brown-Higgins model structure on crossed complexes related to the folk model structure on $\omega$-categories? One would hope that the former is precisely the restriction of the latter to $\omega$-groupoids. Is anything known about this?

Homotopy colimits

I was reading about homotopy (co)limits in Chachólski-Scherer’s very nice Homtopy theory of diagrams especially around page 15 which reviews the special simple case of ordinary push-outs.

I am wondering: suppose the case mentioned in the middle of this page, that the homotopy colimit happens to be weakly equivalent to the ordinary colimit. Then intuitively the following ought to be right:

given any pushout cone which commutes only up to left or right homotopy in the model category sense, does the expected universal arrow exist anyway, up to left/right homotopy, making the expected diagrams commute, up to left/right homotopy?

This must be an elementary thing, I’d be grateful for a pointer to the literature.

Posted at October 2, 2008 1:52 PM UTC

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### Re: More Model ω-Questions

Concerning the thirdd question, on the relation between homotopy colimits and weak/lax colimits I am being pointed to the classical result by Thomason

R. Thomason, Homotopy colimits in the category of all small categories

reviewed for instance in

Murray Heggie, Homotopy colimits in presheaf categories appearing on p. 5 there,

which says that the Grothendieck construction on a weak/lax functor $K \to Cat$, which is the weak/lax colimit of that functor, is equivalent to the homotopy colimit of that functor

using a model structure which in Thomason’s original article was induced from one on simplicial sets (after passing to nerves), and which has meanwhile be turned into a model structure on Cat called the Thomason model structure.

This has recently been generalized to $n$-fold categories (strict cubical $n$-categories) in

As far as I understand this “topological” Thomason-style model structure on Cat, 2Cat and cubical $n$-cat is different from the “folklore” Lack-style model structure on Cat, 2Cat and $\omega Cat$ that I think I am interested in.

Posted by: Urs Schreiber on October 2, 2008 5:25 PM | Permalink | Reply to this

### Re: More Model ω-Questions

The Thomason-style model structures on 1Cat and 2Cat are Quillen equivalent to the usual model structure on simplicial sets, so in at least one sense they model $\infty$-groupoids. However I think it’s a bit of a stretch to say that even the fibrant objects (or even the fibrant-cofibrant objects, since not all their objects are cofibrant) in these model categories “are” $\infty$-groupoids.

On the other hand, in the folk/Lack model structures on 1Cat and 2Cat, all objects are fibrant. And 1-categories and 2-categories are certainly different beasts from $\infty$-groupoids.

From a higher-categorical point of view, a model category can be regarded as a presentation of an $(\infty,1)$-category. Since the most basic, and most studied, $(\infty,1)$-category is the $(\infty,1)$-category of $\infty$-groupoids, it makes sense that it admits many different presentations (topological spaces, simplicial sets, Thomason model structures, cubical and symmetric simplicial thingummies, etc. etc.), which is probably what you are seeing. However, there are also many other $(\infty,1)$-categories having little or no relation to $\infty$Gpd.

I don’t know anything about crossed complexes. I do know something about homotopy colimits, but I don’t think I understand your third question enough to be able to answer it.

Posted by: Mike Shulman on October 2, 2008 6:00 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Hi Mike,

thanks once again for the wealth of useful information!

Let me just quickly try to reply to this here:

I don’t think I understand your third question enough to be able to answer it.

Let there be a diagram and its homotopy colomit.

Now suppose we have a cone over the diagram which does not necessarily commute, but such that all triangles commute up to left and/or right homotopy in the model category theoretic sense (i.e. going around any triangle in two ways is different, but there is a map into a path object or out of a cylinder object which relates the two ways).

Then, given such a “weak” or “lax” cone, do we still get a morphism into the homotopy colimit out of the tip of the cone, satisfying the usual properties in an analogously weakened way?

Posted by: Urs Schreiber on October 2, 2008 6:30 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Merely having each triangle commute up to a specified homotopy is not very much… I can’t imagine getting your hoped-for result from that condition. One usually wants much more. For example, when four triangles forming the faces of a tetrahedron commute up to specified homotopies, these homotopies, thought of as 2-morphisms, should make that tetrahedron commute…

… at least up to a specified homotopy!

And so on, ad infinitum. Then we have a ‘homotopy coherent’ diagram, and these are nicely related to homotopy limits and colimits.

I don’t know the best references, but it can’t hurt to peek at these:

• J.-M. Cordier and T. Porter, Vogt’s theorems on categories of homotopy coherent diagrams, Math. Proc. Camb. Phil. Soc. 100 (1986), 65–90.
• J.-M. Cordier, Sur les limites homotopiques de diagrammes homotopiquement cohérents, Comp. Math. 62 (1987), 367–388.
• J.-M. Cordier and T. Porter, Fibrant diagrams, rectifications and a construction of Loday, J. Pure Appl. Alg 67 (1990), 111–124.

Tim Porter is clearly a good guy to ask about homotopy coherent diagrams! He has some review articles about such things.

Posted by: John Baez on October 2, 2008 9:42 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Merely having each triangle commute up to a specified homotopy is not very much… I can’t imagine getting your hoped-for result from that condition.

Sure, I was just thinking of a simple pushout diagram. For that case of “very small colimit diagrams” this page 15 in Homotopy theory of diagrams which I mentioned gives an easily applied prescription, whereas the general case, according to this article, is way more involved.

I have just a morphism $p : C \to D$ of $\omega$-groups and can compute in various low dimensional special cases that its ordinary kernel $ker(p) \to C$, the pullback along the map from the point, has not just the usual property you would expect from an ordinary kernel, but the property that

given a morphism $A \to C$ such that $A \to C \stackrel{p}{\to} D$ is just homtopic to the map that factors through the point, then already it lifts, up to homotopy, to a map that factors through $\mathrm{ker}(p)$.

I can do this computation explicitly using transformations, modifications, etc, of $n$-functors, which quickly gets a bit out of control as $n$ grows.

Now, I suspect that the model structure comes to the rescue: namely I can easily show using that prescription on that p. 15 that in the special case that I am looking at the ordinary kernel $ker(p)$ is already weakly equivalent to the homotopy hernel $hoker(p)$.

So, if on top of all this which I already know the statement holds which I expect but don’t know, namely that “homotopy coherent cones”, as I take it they are to be called, have up to homotopy unique maps into their tip out of the homotopy colimit, then that would not only sumarize the partial results I have but also give the expected statement for all $n$.

Then we have a ‘homotopy coherent’ diagram, and these are nicely related to homotopy limits and colimits.

Good. That’s what i am expecting. i just didn’t know where to find the precise statement. Thanks for providing some! I’ll have a look.

Posted by: Urs Schreiber on October 2, 2008 11:28 PM | Permalink | Reply to this

### Re: More Model ω-Questions

John,
Thanks for the plug!

I would suggest that for homotopy limits and colimits it is also worth looking at the paper by Bourn and Cordier on indexed limits.

D. Bourn and J.-M. Cordier, A general formulation of homotopy limits , J. Pure Appl. Algebra, 29, (1983), 129–141.

Posted by: Tim Porter on October 3, 2008 2:41 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Other people have already given lots of references, but let me add that in the case of a coherently-homotopy-commutative cone under a strictly commuting diagram, you may find section 10 of this paper helpful.

(And, as proven by various other people elsewhere, diagrams that commute up to coherent homotopy are the same as ones that commute strictly of a different (enriched) diagram shape.)

Posted by: Mike Shulman on October 3, 2008 6:44 PM | Permalink | Reply to this

### Re: More Model ω-Questions

After having taken care of a couple of duties, I am now coming back to this discussion here.

Mike Shulman kindly wrote:

you may find section 10 of this paper [Shulman: Homotopy limits and colimits and enriched homotopy theory] helpful.

Yes, indeed, I am finding this quite helpful. In particular, I find in this the general statement that I was looking for, on p. 29:

We […] make precise the notion of “homotopy coherent cones” […]. A homotopy coherent cone under a diagram $F$ should consist of a vertex $C$ and a coherent transformation $F \sim\to G$, where by $C$ we mean the diagram constant at the vertex $C$. […] Therefore, a representing object for homotopy coherent cones under $F$ is precisely […] the homotopy colimit of $F$. Thus the latter really does have a “homotopical universal property”.

Now I need to go back to the previous sections to unwrap this.

While I am busy learning this stuff, maybe somebody can help me by just saying whether or not the answer to my concrete problem on a simple special case is Yes or No:

Consider an ordinary pushout cone $\array{ X &\to& B \\ \downarrow && \downarrow \\ A &\to& col }$ in some category. $col$ indicating the ordinary colimit. Now assume there is some model category structure around and that it so happens that with respect to that $col$ happens to be weakly equivalent to the homotopy colimit

$hocolim \left( \array{ X &\to& B \\ \downarrow && \\ A } \right) \stackrel{\simeq}{\to} col \,.$

Assume in this case furthermore that we have morphisms $\array{ && B \\ &&\downarrow \\ A &\to& V }$ which do not quite make a cone, but make a “homotopy coherent cone” in that the outermost boundaries of the following diagram

are homotopic - as indicated by the long double arrow.

Then: is it true or not that this alone is sufficient and necessary for the indicated dashed arrow to exist, such that it makes the indicated triangles commute up to homotopy, as indicated by the two short double arrows?

Posted by: Urs Schreiber on October 6, 2008 8:52 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Hi, Urs.

The answer to your question is “Yes”, if X,A,B are cofibrant and V is fibrant. If they are not, your question is not completely correct because the homotopies between non-fibrant, non-cofibrant objects in a model category are not defined (there are left and right homotopies, they do not compose etc..)

Michael.

Posted by: Michael on October 7, 2008 4:57 AM | Permalink | Reply to this

### Re: More Model ω-Questions

The answer to your question is “Yes”, if $X,A,B$ are cofibrant and $V$ is fibrant.

If they are not, your question is not completely correct because the homotopies between non-fibrant, non-cofibrant objects in a model category are not defined (there are left and right homotopies, they do not compose etc..)

Ah, I see. Thanks again, I wasn’t paying attention to this.

So for maps out of cofibrant into fibrant objects the notions of left and right homotopies coincide?

Posted by: Urs Schreiber on October 7, 2008 10:52 AM | Permalink | Reply to this

### Re: More Model ω-Questions

I have another question on Mike Shulman’s Homotopy limits and colimits and enriched homotopy theory:

When it comes to enriched homotopy theory from section 11 on, there is, stated on page 34, the assumption that the category $V$ that one is enriching over is related by a strong monoidal adjunction $V \leftrightarrow SimpSet$ to simplicial sets.

I suppose for $V := \omega Cat$, the $\omega$-nerve $N : \omega Cat \to SimpSet$ does satisfy the required conditions?

(In the sense of the discussion on page 33 this comes from the canonical functor $O(\Delta^{(-)}) : \Delta \to \omega Cat$ provided by the orientals.)

If this is correct, that $V = \omega Cat$ satisfies the assumptions in section 13, would this generalize to $\omega$-categories internal to any category of sheaves?

Posted by: Urs Schreiber on October 7, 2008 11:47 AM | Permalink | Reply to this

### Re: More Model ω-Questions

Dear Urs,

you can find more sophisticated version of homotopy limit theory in
Cisinski’s paper

“Propriétés universelles
et extensions de Kan dérivées”
in his webpage
http://www.math.univ-paris13.fr/~cisinski/publications.html

And also in his book.

Michael Batanin

Posted by: Michael on October 3, 2008 3:38 AM | Permalink | Reply to this

### Re: More Model ω-Questions

I am on my way back from Zagreb. Will try to look at this as soon as possible and maybe get back with some more questions. Thanks again.

Posted by: Urs Schreiber on October 3, 2008 2:18 PM | Permalink | Reply to this

### Re: More Model ω-Questions

Dear Urs,

Nicola Gambino has two papers which carefully explain what is going on with homotopy limits and colimits for 2-categories, and for simplicial model categories, using tools of enriched category theory. I found them quite enlightening.

They are available from his webpage.

Posted by: Richard Garner on October 20, 2008 12:12 PM | Permalink | Reply to this

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