The n-Category Café

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.

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.

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 $Sh(S,\omega Cat)$?

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?