### 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.

## 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.