## September 27, 2008

### Some Model ω-Questions

#### Posted by Urs Schreiber

Here are some questions related to generalizing the model structure on $\omega Cat$, the closed 1-category of strict $\infty$-categories, as described in A folk model structure on $\omega Cat$, to the case of $\omega$-categories internal to some presheaf topos.

a) First of all, the authors of the A folk model structure on $\omega Cat$ prove the model structure properties by checking that the hypothesis of “Smith’s theorem” are satisfied.

What is a reference for “Smith’s theorem”? They don’t give any.

b) Let $Spaces$ denote some presheaf category. I am interested in $\omega$-categories internal to presheaves, $\omega Cat(Spaces)$. Or equivalently: $\omega$-category valued presheaves.

Is there a general way to lift the model structure on a category to one on presheaves with values in that category?

I know that for cases such as presheaves with values in the model category of simplicial sets, the model structure on the presheaf category is obtained by applying the model structure of simplicial sets stalkwise. I.e. a weak equivalence of presheaves with values in $C$ is a morphism of presheaves such that for each stalk it restricts to a weak equivalence in $C$.

How general is that? Does the stalkwise definition of w-equivalences/fibrations/cofibrations for presheaves in a model category always yield an induced model structure?

Something very similar happens for the model structure on categories enriched over a monoidal model category: the model structure is essentially inherited Hom-wise in this case. Again, at least for the case of simplicially enriched categories, as reviewed for instance on p. 5 of Bergner’s Survey of $(\infty,1)$-categories.

How general is that phenomenon? Is there a natural model structure on every category of categories enriched over a given model category?

If the answer to all this is: “No, stalkwise and homwise induction of model structures does not work generally.” then my question is: what about the special case of $\omega$-category valued presheaves and $\omega$-category enriched categories?

Switching back from the perspective of $\omega Cat$-valued presheaves to $\omega$-categories internal to presheaves: can one use alternatively the fact that we are then internal to a topos to see if and how the model structure lifts? I suppose if we had a proof for the model structure on $\omega Cat$ which is intuitionistic (do we?), then we’d be guaranteed that it induces a model structure on $\omega Cat(Spaces)$? Zoran Škoda points me to work by Durov who discusses a related problem not for internal $\omega$-categories but for internal rings. In doing so, Durov seems to encounter some obstacles. But I have to have a closer look at that.

3) Thanks to David Roberts, it seems we can show the following:

Let a weak equivalence $C \to D$ in $\omega Cat(Spaces)$ be a stalkwise weak equivalence with respect to the folk model structure on $\omega Cat$. Then there exists an $\omega$-anafunctor $D \to C$ being the weak inverse of this.

Here an $\omega$-anafunctor $D \to C$ is defined to be an $\omega$-functor out of an acyclic fibration over $Y \to D$ over $D$ (again, using the fiberwise definition). There is an $\omega$-category (internal to $Spaces$) of $\omega$-anafunctors from $D$ to $C$, defined by the colimit $colim_{Y \in Hypercovers(D)} \; \; \omega Cat(Spaces)(Y,C) \,,$ where $Hypercovers(D)$ is the category whose objects are acyclic fibrations $Y \stackrel{\simeq}{ \to \gt} D$ over $D$ and whose morphisms are commuting triangles between these.

I suppose this induces an $\omega Cat(Spaces)$-enriched category whose objects are the objects of $\omega Cat(Spaces)$ and whose Hom-objects are the above colimit $\omega$-categories of $\omega$-anafunctors. I am inclined to call this $\mathbf{Ho}(\omega Cat(Spaces)) \,,$ where the boldface is supposed to be read as “weak homotopy category”, because there seems to be a canonical functor of $\omega Cat(Spaces)$-enriched categories $\omega Cat(Spaces) \to \mathbf{Ho}(\omega Cat(Spaces))$ which is the identity on objects and which, by the above statement, is such that every weak equivalence is sent – not to an isomorphism but – to an object with a homotopy inverse (namely that weakly inverse $\omega$-anafunctor), and such that $\mathbf{Ho}(\omega Cat(Spaces))$ is universal for this property.

Anything known about this? This concept of a “weak homotopy category” must have appeared before. Can anyone give me references? Igor Baković tells me about the work by Moerdijk and Pronk on homotopy categories in a 2-categorical context. But here I need more and less: the inverse $\omega$-anafunctor may be inverse only up to equivalence involving arbitrarily high cells and this is detected by the model structure on the enriching category.

I’d be very grateful for whatever comment you may have.

Posted at September 27, 2008 5:03 PM UTC

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### Re: Some Model ω-Questions

Have a look at Tibor Beke’s paper “Sheafifiable homotopy model categories I” for a reference to Smith’s theorem: given a set of generating cofibrations and class of weak equivalences in a locally presentable category, it guarantees the existence of a cofibrantly generated model category structure when certain conditions are satisfied.

Posted by: Sam Isaacson on September 27, 2008 7:03 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Is there a general way to lift the model structure on a category to one on presheaves with values in that category?

If $M$ is cofibrantly generated, then the category of functors $M^I$ is a model category with weak equivalences and fibrations defined stalkwise. The cofibrations are determined by the left lifting property, and can also be described in terms of a specific set of generating cofibrations.

For a general model category $M$, under some assumptions on $I$, again $M^I$ is a model category.

For general $M$ and $I$, it is an open question whether $M^I$ is always a model category, but the consensus is that it is not. Nevertheless, the “homotopy theory” of $M^I$ is well-defined. For example, there is a “model approximation” for $M^I$. (See, for example, http://arxiv.org/abs/math/0110316, by Chacholski and Scherer.)

If the answer to all this is: “No, stalkwise and homwise induction of model structures does not work generally.” then my question is: what about the special case of ω-category valued presheaves and ω-category enriched categories?

I haven’t looked at the paper you refer to, but it appears that $\omega$Cat is cofibrantly generated, so the above result should apply.

Posted by: Dan Christensen on September 28, 2008 2:10 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Sam Isaacson wrote:

Have a look at Tibor Beke’s paper “Sheafifiable homotopy model categories I

Dan Christensen wrote:

See, for example, http://arxiv.org/abs/math/0110316, by Chacholski and Scherer.

This is most helpful indeed! Thanks a lot.

Yesterday I read Tibor Beke, will have to read Chacholski and Scherer now.

Taken everything together what I have learned from Beke’s article and now from Dan’s comment

If $M$ is cofibrantly generated, then the category of functors $M^I$ is a model category with weak equivalences and fibrations defined stalkwise.

it seems to me (but this is tentative, please correct me) that the following is true:

There is a model structure on $\omega$-categories internal to sheaves on some site $S$ $\omega Cat(Spaces) := \omega Cat(Sheaves(S)) \simeq Sh(S,\omega Cat)$ given as follows:

1) $I$-fibrations / surjective equivalences

An $\omega$-functor $f : C \to D$ is called $(k+1)$-surjective for $k \in \mathbb{N}$ if the universal morphism $C_{k+1} \to (f_k \times f_k)^* D_{k+1}$ from the commuting square $\array{ C_{k+1} &\stackrel{f_{k+1}}{\to}& D_{k+1} \\ \downarrow^{s \times t} && \downarrow^{s \times t} \\ C_{k}\times C_k &\stackrel{f_k \times f_k}{\to}& D_k \times D_k }$ to the pullback square $\array{ (f_k \times f_k)^* D_{k+1} &\to& D_{k+1} \\ \downarrow && \downarrow^{s \times t} \\ C_{k}\times C_k &\stackrel{f_k \times f_k}{\to}& D_k \times D_k }$ is epi.

The epis in $Sh(S)$ are precsily the stalkwise epis in $Sets$, so alternatively this says that $C_{k+1} \to (f_k \times f_k)^* D_{k+1}$ is stalkwise epi. (Right?)

Call an $\omega$-functor which is is $k$-surjective for all $k\in \mathbb{N}$ an $I$-fibration in the language of Smith’s theorem as used by Lafont et. al.

(Other good names seem to be surjective equivalence or hypercover, I guess.)

2) cofibrations

Take the cofibrations to be those that have the left lifting property with respect to all $I$-fibrations.

This seems to be what Beke’s theorem says we can do, since this is what happens in the model structure for $\omega Categories (Sets)$ and his theorem says that we obtain the model structure internal to the topos of sheaves in entire analogy for structures such as $\omega$-categories which are defineable in terms of “coherent logic”.

While this does not seem to be implied by Dan’s comment

[…] weak equivalences and fibrations defined stalkwise. The cofibrations are determined by the left lifting property […]

it seems to be consistent with it.

Now maybe the trickiest part:

3) weak equivalences

For $C$ an $\omega$-category in $Sets$, let $Core(C)$ be the $\omega$-category obtained by restricting $C$ to all morphisms which have weak inverses (i.e. for which there exists a reverse morphism which is an inverse up to higher morphisms for which there exist reverese morphism… and so on.)

This is functorial $Core : \omega Cat(Sets) \to \omega Cat(Sets) \,.$ So then define the core of an $\omega$-category $C$ internal to sheaves plot-wise: $Core(C) : S^{op} \stackrel{C}{\to} \omega Cat \stackrel{Core}{\to} \omega Cat \,.$

Then say that an $\omega$-functor $f : C \to D$ is essentially $k$-surjective if not necessarily the universal morphism $C_{k+1} \to (f_k \times f_k)^* D_{k+1}$ from before is epi, but its composition with the projection to the quotient by $\omega$-equivalence is $f essentially\, k-surjective \;\Leftrightarrow\; C_{k+1} \to (f_k \times f_k)^* D_{k+1} \to ((f_k \times f_k)^* D_{k+1})/_{\sim} \; is epi \,,$ where the quotient is the pushout $\array{ Q_k &\to& (f_k \times f_k)^* D_{k+1} \\ \downarrow && \downarrow \\ (f_k \times f_k)^* D_{k+1} &\to & ((f_k \times f_k)^* D_{k+1})/_{\sim} }$ with $Q_k$ the pullback $\array{ Q_{k} &\stackrel{f_{k+1}}{\to}& Core(D)_{k+2} \\ \downarrow && \downarrow^{s\circ s \times t \circ t} \\ C_{k}\times C_k &\stackrel{f_k \times f_k}{\to}& D_k \times D_k } \,.$

(All this is just supposed to be the diagrammtic version of what Lafont et al. say in $Sets$.)

Then say that $f$ is a weak equivalence if it is essentially $k$-surjective for all $k$.

Again, using the fact that epis in sheaves are the stalkwise epis in $Sets$, this should say that the weak equivalences in $\omega Cat(Spaces)$ are the stalkwise weak equivalences in $\omega Cat$.

4) fibrations

And now of course take the fibrations to be those $\omega$-functors with right lifting property wrt cofibrations.

It seems to me that it follows that these are the stalkwise fibrations in $\omega Cat$, which would then nicely harmonize with Dan’s comment.

Is that right? Does the system of w-equivalences/fibrations/cofibrations defined this way yield a model structure on $Sh(S,\omega Cat)$?

Posted by: Urs Schreiber on September 28, 2008 3:59 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

I need to get back to this question about the model structure on strict $n$-categories internal to sheaves on some site. Because I realize that I don’t fully understand it yet and that I had mistakes and misunderstandings in the above. And I really need sheaves, not just presheaves.

The crucial subtlety is that business about “stalkwise”. Dan Christensen in his comment was thinking of “objectwise” where I was reading “stalkwise”:

the Dwyer-Hirschhorn-Kan theorem says that for $C$ any cofibrantly generated model category and $D$ any small category, the presheaf category $M^D$ inbherits a model structure whose weak-equivalences and fibrations are the $D$-objectwise such maps in $C$.

Now, what if $D$ has the structure of a site and we want to look at sheaves?

Where do the pointwise/stalkwise weak equivalences come in which Beke has in his list of examples? I gather this is supposed to be clarified by Beke’s article, but I may not follow that yet.

Posted by: Urs Schreiber on October 13, 2008 6:40 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Here is another question I have, on a different but related aspect:

consider two strict 2-categories $C$, $D$ and the model structure on $2 Cat$ induced from that on $\omega Cat$. Then we have the following cute fact:

Weak 2-functors $C \to D$ are in bijection with strict 2-functors $\hat C \to D$ out of a cofibrant replacement $\hat C$ of $C$.

(Here by a weak 2-functor I mean, as usual, one which respects composition and identities only up to coherent invertible 2-cells.)

A description of this is in section 4.1 of Steve Lack’s A Quillen model structure for 2-categories.

So weak 2-functors are “left derived” strict 2-functors. That’s neat.

I think I have a generalization of the construction of the cofibrant replacements $\hat C$ used here from 2-categories to $\omega$-categories. I am inclined to define weak functors $C \to D$ between $\omega$-categories as $\omega$-functors $\hat C \to D$.

My question: has this or anything related been considered elsewhere?

In particular, consider the situation where we restrict to $\omega$-groupoids. Then by Brown-Higgins we know that these are equivalent to crossed complexes of groups.

It seems that under this identitfication the weak $\omega$-functors, now interpreted as maps between crossed complexes, would yield something like a “nonabelian derived Hom-functor” (in that crossed complexes may be nonabelian (and even -oidal) in the lowest two degrees).

Has anyone thought or heard about such?

I am being told that Rosenberg is thinking about nonabelian derived categories, but know very little about it. Maybe there is a relation?

Posted by: Urs Schreiber on September 28, 2008 7:11 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

I am having some discussion about this by private email. Maybe I should say which definition of that special cofibrant replacement $\hat C$ of an $\omega$-category $C$ I have in mind, such that $\omega$-functors $\hat C \to D$ should be called pseudo $\omega$-functors (weak $\omega$-functors) $C \to D$.

So:

Given $C$, define $\hat C$ and an $\omega$-functor $comp : \hat C \to C$ inductively as follows:

- $\hat C$ has the same objects as $C$ and $comp$ is the identity on these.

- assume $\hat C$ and $comp$ have been defined up to and including $k$-morphisms for $k \in \mathbb{N}$. Then let $\hat C_{k+1}$ be freely generated under $(k+1)$-dimensional pasting from

a) the $(k+1)$-morphsims of $C$

b) one new $(k+1)$-morphism $f \to g$ for all pairs $f,g \in \hat C_k$ such that $comp(f) = comp(g)$.

And let $comp_{k+1}$ send all pasting diagrams of generators of type a) to their respective composite in $C$ and send all generators of type b) to the identity $(k+1)$-morphism in $C$ on the image of their source and target.

I think one can easily prove that with respect to the “folk” model structure $\hat C$ is cofibrant and $comp$ is a surjective equivalence (an $I$-fibration). Hence $\hat C$ is a cofibrant replacement of $C$.

Of course $\hat C$ is not just any cofibrant replacement, but a very special one. Somehow it is universal. I still need to formulate that properly.

Posted by: Urs Schreiber on September 29, 2008 12:08 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

cofibrant replacemnet $\hat C$ of an $\omega$-category $C$ […] such that $\omega$-functors $\hat C \to D$ should be called pseudo $\omega$-functors

[…]

Of course $\hat C$ is not just any cofibrant replacement, but a very special one. Somehow it is universal. I still need to formulate that properly.

Zoran Škoda kindly points out to me the relevant article

Francois Métayer: Cofibrant complexes are free

which addresses aspects of this question by showing that

all cofibrant $\omega$-categories are free

(freely generated by polygraphs).

Posted by: Urs Schreiber on September 30, 2008 3:30 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Hi Urs,

The “universality” of the cofibrant replacement you seek is described in a paper of mine

Understanding the small object argument

It shows that cofibrant replacement is a comonad, which is “freely generated” by the generating cofibrations. The idea of defining weak homomorphisms A –> B to be strict homomorphisms A’ –> B appears in Section 7 of that paper. The nice thing is that these weak homomorphisms compose in a strictly associative manner. There are also slides from a talk I gave on this at the Nice PSSL.

I am currently writing up a paper fleshing out the details of this in the cases of tricategories, and of weak omega-categories. It should appear on the arxiv in the next couple of weeks.

Richard

Posted by: Richard Garner on October 20, 2008 11:57 AM | Permalink | Reply to this

### Re: Some Model ω-Questions

The “universality” of the cofibrant replacement you seek is described in a paper of mine

Understanding the small object argument

Thanks! That looks interesting. I’ll have a look.

While I am reading, let me mention that we ran into the following question (in discussion with Zoran Škoda):

it is a not-so-obvious theorem (by Nielsen-Schreier, apparently) that every subgroup of a free group is free.

What is the generalization of this to free categories? free groupoids? free n-categories? free n-groupoids?

I’d think it should be easy to see that every subcategory of a free category is free, but maybe I am mixed up (I am thinking: every morphisms in the subcategory of the free category is either the composite of two nontrivial others or not. If not, it is one of the generators of the subcategory and the subcategory is free on the generators arising this way.) So supposedly the subtlety comes in when passing to free groupoids?

Posted by: Urs Schreiber on October 21, 2008 1:28 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

The notion of free crossed complexes is discussed extensively in Ronnie’s work. They will be handled in the book he is preparing with Higgins and Sivera but the version on the web at the moment is just part 1 and they are in part 2.

I think they are discussed in

(with M. GOLASINSKI), “A model structure for the homotopy theory of crossed complexes”, Cah. Top. Géom. Diff. Cat. 30 (1989) 61-82.

which is now available from NumDam as well as from Ronnie’s publication list.

As to the Neilsen Schreier theorem a beautiful way of viewing this, at least for finitely generated groups, is to consider the free group as the fundamental group of a bouquet of circles. The subgroup then corresponds to a covering space which will be a graph. That graph has a free fundamental group which is isomorphic to the subgroup. END OF PROOF.

Posted by: Tim Porter on October 28, 2008 5:56 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Note that free groups are very different than free categories because they’ve got those inverses thrown in.

Is every submonoid of a free monoid free? Someone must know. Maybe the answer is obvious. Given a set of words in some alphabet, closed under concatenation, can we think of this as freely generated by some subset of those words?

Posted by: John Baez on October 21, 2008 5:35 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

can we think of this as freely generated by some subset of those words?

What about the argument which I gave #:

every morphisms in the subcategory of the free category is either the composite of two nontrivial others or not. If not, it is one of the generators of the subcategory and the subcategory is free on the generators arising this way

I just realized that Ronnie Brown discusses the notion of a free crossed complex, hence of free $\omega$-groupoids.

(I see this in a file which I am not sure I may link to here. Don’t know at the moment which published article this comes from.)

Posted by: Urs Schreiber on October 21, 2008 5:58 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Is every submonoid of a free monoid free? Someone must know. Maybe the answer is obvious. Given a set of words in some alphabet, closed under concatenation, can we think of this as freely generated by some subset of those words?

Ah. I can answer that question… The answer is no, since it isn’t even true for the free monoid on one generator. There are submonoids of the natural numbers under addition that certainly aren’t free. Take the one generated by 2 and 3 for example.

This gives me an excuse to ask a question of my own that I think is at least loosely connected to the topic under discussion:

I’ll ask it in full concrete non-generality because I have a specific application in mind. I am interested in the category of chain complexes of quasi-coherent sheaves over a projective variety (over the complex numbers, say) whose homology sheaves are also quasi-coherent sheaves and I want to put a model category structure on it.

Work of Mark Hovey and James Gillespie (see here for example) explains how to do this, except there are only straightforwardly applicable theorems in their work when the weak equivalences are quasi-isomorphisms of chain complexes. I want the weak equivalences to include slightly more general maps that this: they should be chain maps that induce maps on homology that are not necessarily isomorphisms of sheaves but do induce isomorphisms of sheaf cohomology in all degrees.

I feel that if this is possible it should be part of a more general framework: I have two homotopy theories (that on chain complexes and that inducing sheaf cohomology) that I wan’t to work together nicely to give a single homotopy theory incorporating them both in some way.

Does anyone have any ideas?

### Re: Some Model ω-Questions

Take the one generated by 2 and 3 for example.

Oh, right. I wasn’t thinking straight. Thanks.

Posted by: Urs Schreiber on October 28, 2008 8:08 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Noohi got a head start on this with his butterflies: weak maps of strict 2-groups are essentially built from a span of crossed modules.

Posted by: David Roberts on September 30, 2008 2:13 AM | Permalink | Reply to this

### Re: Some Model ω-Questions

Noohi got a head start on this with his butterflies: weak maps of strict 2-groups are essentially built from a span of crossed modules.

Thanks for the reminder. Still haven’t looked into this. Do his weak maps of 2-groups $G \to H$ coincide with the pseudo-2-functors $\mathbf{B}G \to \mathbf{B}H$ betweent he corresponding strict one-object 2-groupoids? If not, are they more general or more restrictive?

Posted by: Urs Schreiber on September 30, 2008 3:04 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

Okay, on page 16 Noohi states the equivalence of butterfly morphisms with the “right derived hom”. The latter seems to be precisely what we were talking about here, using maps out of cofibrant replacements of 2-categories, only that Noohi restricts attention to one-object 2-groupoids.

Posted by: Urs Schreiber on September 30, 2008 5:06 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

To some extent this is related to doing simplicial derived functors. The interpretation in terms of homotopy coherence is more or less in work by myself and Cordier, and there is related work by Batanin. We took a very constructive approach and hit big combinatorial problems at one point. Perhaps someone could comment also on the complicial set approach of Verity, where our ideas surfaced again.
Posted by: Tim Porter on September 30, 2008 10:07 AM | Permalink | Reply to this

### Re: Some Model ω-Questions

To some extent this is related to doing simplicial derived functors. The interpretation in terms of homotopy coherence is more or less in work by myself and Cordier, and there is related work by Batanin.

Posted by: Urs Schreiber on October 1, 2008 10:09 AM | Permalink | Reply to this

### Re: Some Model ω-Questions

I missed this query sorry.

Our work was in

(with Jean-Marc Cordier) Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.

Since then several others have done similar things, but I am not certain of the references.

Posted by: Tim Porter on October 28, 2008 6:07 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

I missed this query sorry.

Our work was in

(with Jean-Marc Cordier) Homotopy Coherent Category Theory, Trans. Amer. Math. Soc. 349 (1997) 1-54.

Thanks. I had found that meanwhile and started looking at it. Made the appearance to me of being very elegant, though I didn’t quite feel up to it. May have to have another look.

The thing is that I don’t quite feel I have the leisure to learn homotopy coherent theory in its totality right now, but I needed one particular statement for one particular concrete question, this here.

Well, this comes from a general context, which maybe you can help me with:

in essence, I am thinking about the lifting problem and the extension problem in nonabelian cohomology.

So for $C$ and $\omega$-categories and $\hat C$ a replacement for $C$ I am looking at the situation

$\array{ && E \\ && \downarrow \\ \hat C &\to^g& D \\ \downarrow \\ F }$ and am asking for the obstructions to lift $g$ to $E$ and/or to extend it to $F$.

For the case that $E \to D$ is a “shifted central extension” of $\omega$-groups I could reduce this to the question about homotopy limits asked here.

For the case of extensions $C \to F$ I have some pedestrian results, but not really satisfactory.

But there ought to be a much more general theory of lifts and extensions in nonabelian cohomology.

Posted by: Urs Schreiber on October 28, 2008 8:36 PM | Permalink | Reply to this

### Re: Some Model ω-Questions

I do not know the answer as such.
Do you know of a Puppe sequence type result for $\omega$-cats? I have found that such extension questions can often be rephrased in such a context. As an example look at the bitorsors paper of Breen, where, of course, the situation is much simpler. Also Ronnie has some crossed complex fibration results that might help.

Posted by: Tim Porter on October 28, 2008 9:46 PM | Permalink | Reply to this