### Some ω-Questions

#### Posted by Urs Schreiber

I have some questions on $\omega$-categorical issues in the context of descent and cohomology to the experts. Some of them are accompanied by the figures collected here.

**Question 1) Weak equivalences of orientals.**

With respect to the model category structure on $\omega\mathrm{Cat}$ an $\omega$-functor $F : C \stackrel{\simeq_w}{\to} D$ between $\omega$-categories is a *weak equivalence* if it is *essentially $k$-surjective* for all $k \in \mathbb{N}$ in the sense of definition 4 of Baez-Shulman.

Let $O(\Delta^n)$ be Street’s $n$th oriental, i.e. the free $\omega$-category on the parity $n$-simplex. It seems to be true that $O(\Delta^n)$ is weakly equivalent to the point

$\forall n \in \mathbb{N} : O(\Delta^n) \stackrel{\simeq_w}{\to} pt \,.$

Is that right? Is there a statement and a proof of this in the literature?

**Question 2) $\omega\mathrm{Cat}$-enrichment of cosimplicial $\omega$-categories**

Consider the standard Gray-like closed monoidal structure on $\omega\mathrm{Cat}$ and consider the category $\omega\mathrm{CoSimp} := Funct(\Delta, \omega\mathrm{Cat})$ of cosimplicial $\omega$-categories. It seems to me that using the inner hom in $\omega\mathrm{Cat}$ degreewise cosimplicial $\omega$-categories become naturally enriched over $\omega\mathrm{Cat}$. I am not sure yet about the best formal way to say this, but the simple idea is indicated in figure 2.

Am I right about the $\omega\mathrm{Cat}$-enrichment of $\omega\mathrm{CoSimp}$? Has this been stated anywhere in the literature? What’s the best formal way to characterize it?

**Question 3) descent $\omega$-categories**

The above worries me slightly because it seems obviously right to me, but then the question is why Ross Street defines the descent $\omega$-category with coefficients in the cosimplicial $\omega$-category $E$ not simply as

$hom_{\omega\mathrm{CoSimp}}(O(\Delta^{(-)}), E) \,.$

This seems to be clearly the right answer. And it seems to me that the definition he actually gives (p. 339 here, p. 32 here) is related to that by using the hom-adjunction in $\omega\mathrm{Cat}$ to move the $\omega$-glob around.

So I am thinking the following: Street’s formula for descent $\omega$-categories is actually to be thought of as being the very formalization of the $\omega\mathrm{Cat}$-enrichment of $\omega\mathrm{CoSimp}$ that I was asking for above.

Does that sound right?

**Question 4) codescent and homotopy category of $\omega\mathrm{Cat}$**

For $Y \to X$ a regular epimorphism in $\mathbf{Spaces}$ (your favorite choice), and $Y^\bullet : \Delta^{op} \to \mathbf{Spaces}$ the corresponding simplicial space, finally for
$\mathbf{A}_C : \mathbf{Spaces}^{op} \to \omega\mathrm{Cat}(\mathbf{Spaces})$ an $\omega$-category valued presheaf of the form
$\mathbf{A}_C : X \mapsto hom(\Pi_0(X), C)$
for $C \in \omega\mathrm{Cat}(\mathbf{Spaces})$ a fixed $\omega$-category internal to $\mathbf{Spaces}$ and where $\Pi_0(X)$ denotes the discrete $\omega$-category over $X$, then
define, following Street, the *codescent* $\omega$-groupoid $\Pi_0^Y(X)$ by the property that

$Desc(Y^\bullet,\mathbf{A}_C) \simeq hom(\Pi_0^Y(X), C)$

for all $C$.

I know the codescent $\omega$-category $\Pi_0^Y(X)$ for the case that $C$ is restricted to be an $n$-category, for low $n$. For instance, for $n=1$ we have that $\Pi_0^Y(X)$ is the familiar *Čech groupoid* of the cover $Y$.For higher $n$ it is the Čech groupoid with composition equalities replaced by higher cells.

For the cases (low $n$) that I understand explicitly, we have weak equivalences

$\Pi_0^Y(X) \stackrel{\simeq_w}{\to} \Pi_0(X) \,.$

If the answer to question 1) is positive, then this should be true in general. Anything in the literature on that?

If I replace in the definition of $C$-valued nonabelian cohomology

$H(-, \mathbf{A}_C) := colim_Y hom(\Pi_0^Y(X), C)$

the colimit on the right with something going out of all $\omega$-categories weakly equivalent to $\Pi_0(X)$ I get something very close to the formulation of nonabelian cohomology in terms of homotopy categories of simplicial presheaves as reviewed by Toën. Is anything known about this?

## Re: Some ω-Questions

Question 6) homotopy hypothesis for generalized spacesForgot this one: this concerns figure 5.

In the context of the homotopy hypothesis we pick a notion of $\infty\mathbf{Groupoid}$ and pick a notion of $\mathbf{Spaces}$ such that each space has an fundamental $\infty$-groupoid and each $\infty$-groupoid has a geometric realization as a space – and ask if these two operations yield some kind of equivalence $\infty\mathbf{Groupoids} \simeq \mathbf{Spaces}$.

Consider this for the choice $\mathbf{Spaces} := \mathrm{Sheaves}(\mathbf{Euclid})$ where $\mathbf{Euclid}$ is the full subcategory of $\mathbf{Manifolds}$ on vector spaces and the choice $\infty\mathbf{Groupoids} := \omega\mathbf{Groupoid}(\mathbf{Spaces})$ ($\omega$-groupoids internal to $\mathbf{Spaces}$).

Let $\Pi_\omega : \mathbf{Spaces} \to \omega\mathbf{Groupoids}(\mathbf{Spaces})$ be the functor which sends each space $X$ to its fundamental $\omega$-groupoid $\Pi_\omega(X)$ whose $k$-morphisms are thin homotopy classes of smooth images of the standard $k$-disk in $X$, (constant in a neighbourhood of the boundary of the disk).

Let $|-| : \omega\mathbf{Groupoids}(\mathbf{Spaces}) \to \mathbf{Spaces}$ be the functor which sends each $\omega$-groupoid $C$ to the sheaf $|C| : U \mapsto hom(\Pi_\omega(U),C) \,.$

First sub-question: this $|-|$ is defined without mentioning the $\omega$-nerve of $C$. It should nevertheless yield the right notion of geometric realization in $\mathbf{Spaces}$. Is that right?

Second subquestion: we have Quillen model category structures on both sides. Do $|-|$ and $\Pi_\omega$ induce a Quillen model equivalence? If not, what else do we get? At least an adjunction one way or the other?