ScalingStacks

Proof. Since the cells are contained in $\Upsilon_{n}$, it follows that $j_{!}j^{*}C_{i}\simeq C_{i}$. Suppose that $Y\in\precat_{(\infty,n)}$; then the unit $Y\to j_{\ast}j^{\ast}Y$ induces an equivalence

for any cell $C_{i}$.

Now consider the smallest subcategory of $\cat_{(\infty,n)}$ consisting of objects $X$ such that the unit map induces an equivalence $\map(X,Y)\simeq\map(X,j_{\ast}j^{\ast}Y)$ for all $Y\in\precat_{(\infty,n)}$. As we have seen this subcategory contains the cells. It is also closed under colimits since if we write $X\simeq\colim_{\alpha}X_{\alpha}$, where all the $X_{\alpha}$ are in this subcategory, then

It follows that this subcategory is all of $\cat_{(\infty,n)}$, and thus $j^{\ast}$ induces an equivalence $\map(X,Y)\simeq\map(j^{\ast}X,j^{\ast}Y)$. ∎

Proof. For any $K$-local object $X$ of $\pre(\mathcal{R})$, a morphism $Y\to X$ represents an object of $(K^{-1}\pre(\mathcal{R}))_{/X}$ if and only if, for any morphism $U\to V$ of $K_{0}$, the square

is homotopy cartesian, since the horizontal map at the bottom is an equivalence. For this, it suffices to show that the induced map on homotopy fibers over any vertex of $\map(V,X)$ is an equivalence. Unpacking this, we obtain the condition that for any morphism $V\to X$, the map

is a weak equivalence. We therefore deduce that $(K^{-1}\pre(\mathcal{R}))_{/X}$ may be exhibited as a localization $K_{X}^{-1}(\pre(\mathcal{R})_{/X})$, where $K_{X}$ is the strongly saturated class generated by the set of diagrams of the form

in which $\phi\in K_{0}$.

Now suppose $\eta\colon Z\to C_{k}$ a morphism of $K^{-1}\pre(\mathcal{R})$. Since colimits are universal in $\pre(\mathcal{R})$ [28, § 6.1.1], the functor

given by pullback along $\eta$ preserves all colimits, and the universal property of localizations guarantees that the composite

descends to a colimit-preserving functor

(which then must also be given by the pullback along $\eta$) if and only if, for any diagram

in which $0\leq k\leq n$ and $\phi\in K_{0}$, the induced morphism $U\times_{C_{k}}Z\to V\times_{C_{k}}Z$ lies in $K$.

It is clear that it suffices to check this only for nondegenerate morphisms $V\to C_{k}$. It now remains only to show that it suffices to check this for objects $Z$ among the essential image of $\mathcal{R}$. This follows from the fact that the class $K$ is strongly saturated and the fact that $\mathcal{R}$ generates $\pre(\mathcal{R})$ under colimits. ∎

Proof. The 0-truncated objects of $\pre(\Upsilon_{n})$ are precisely those presheaves of spaces taking values in the 0-truncated spaces, i.e., functors $\Upsilon_{n}^{\mathrm{op}}\to\set$. The 0-truncated objects of $\cat_{(\infty,n)}=S^{-1}\pre(\Upsilon_{n})$ consist of precisely those 0-truncated objects of $\pre(\Upsilon_{n})$ which are $S$-local. By Lemma 6.6, the nerve of every gaunt $n$-category is $S$-local, and so the result follows. ∎

Proof. Lemma 8.4 shows that the restriction of $j^{\ast}$ to $\cat_{(\infty,n)}$ is fully faithful. Now to prove that $j^{\ast}$ is essentially surjective when restricted to $\cat_{(\infty,n)}$, it suffices to prove that $S^{-1}\pre(\Upsilon_{n})$ is generated under colimits by the cells. Since every object is a colimit of representables, it suffices to prove that $\Upsilon_{n}$ itself is generated under colimits in $S^{-1}\pre(\Upsilon_{n})$ by the cells. To prove this, we filter $\Upsilon_{n}$ in the following manner.

Let $\Upsilon_{n}^{(0)}=\mathbb{G}_{n}$ be the globular category of cells. For any $k\geq 1$, define $\Upsilon_{n}^{(k)}$ to be the full subcategory of $\Upsilon_{n}$ spanned by the set

That is, $\Upsilon_{n}^{(k)}\subset\Upsilon_{n}$ consists of colimits, formed in $S^{-1}\pre(\Upsilon_{n})$, of diagrams of objects of $\Upsilon_{n}^{(k-1)}$.

We claim that the collection $\{\Upsilon_{n}^{(k)}\}$ forms an exhaustive filtration of $\Upsilon_{n}$, so that we have $\cup_{k}\Upsilon_{n}^{(k)}=\Upsilon_{n}$. First we observe that the strongly saturated class $S$ contains the map

and thus, by induction, the union $\cup_{k}\Upsilon_{n}^{(k)}$ contains $\Theta_{n}$.

It now suffices to show that this union is closed under fiber products over cells. Since colimits commute with fiber products over cells (both in $\gaunt_{n}$ and $S^{-1}\pre(\Upsilon_{n})$) it is sufficient to show that $C_{j}\times_{C_{i}}C_{k}$ is contained in the union for all $i,j,k\leq n$. The fiber products of cells were analyzed in detail in Remark 6.3 and the proof of Lemma 6.7, where it was shown that they can all be obtained from the cells by a finite number of the colimits provided by $S_{00}$. These are colimits in $\cat_{(\infty,n)}$, whence the result follows. ∎

Proof. Let $X\in\precat_{(\infty,n)}$ be an object, and consider the map $RX\to X$. The claim is that $j^{*}RX\to j^{*}X$ is an equivalence. Since the cells generate $S^{-1}\pre(\Upsilon_{n})$ under colimits, it’s enough to observe that $\map(C_{i},RX)\simeq(RX)(C_{i})\to X(C_{i})\simeq Map(C_{i},X)$ is an equivalence, for any cell $C_{i}$. ∎

Proof. By Axiom (C.1), the left Kan extension of $f$ along the Yoneda embedding is a localization $F\colon\pre(\gaunt_{n}^{\omega})\to\mathcal{C}$; the right adjoint is a fully faithful functor $G\colon\mathcal{C}\hookrightarrow\pre(\gaunt_{n}^{\omega})$. Write $W$ for the class of morphisms of $\pre(\gaunt_{n}^{\omega})$ that are carried to equivalences of $\mathcal{C}$ by $F$, so that $\mathcal{C}\simeq W^{-1}\pre(\gaunt_{n}^{\omega})$. The class $W$ is strongly saturated, and by [28, Proposition 5.5.4.16], it is of small generation.

By Axiom (C.4), the class $W$ contains the morphisms of Notation 6.5. We claim further that $T_{0}\subseteq W$. To prove this, it suffices to show that $W$ is stable under the operation $H\times_{C_{i}}(-)$ for any $H\in\gaunt_{n}^{\omega}$.

So let $W^{\prime}\subseteq W$ be the subset consisting of those morphisms $\phi\colon X\to Y$ of $W$ such that for any morphism $Y\to C_{i}$ and any morphism $H\to C_{i}$ of $\gaunt_{n}^{\omega}$, the pullback $H\times_{C_{i}}\phi\colon H\times_{C_{i}}X\to H\times_{C_{i}}Y$ also lies in $W$. By the universality of colimits in $\pre(\gaunt_{n}^{\omega})$, it follows that $W^{\prime}$ is closed under colimits. From Proposition 8.5 for $\mathcal{R}=\gaunt_{n}^{\omega}$, $i=\id$, and $U=W$, we deduce that since $(\mathcal{C},f)$ satisfies Axiom (C.3), there is a subset $W_{0}\subseteq W$ that generates $W$ under colimits and is stable under the operation $H\times_{C_{i}}(-)$ for any $H\in\gaunt_{n}^{\omega}$. Hence $W_{0}\subseteq W^{\prime}$, and so $W^{\prime}=W$.

Since $T_{0}\subseteq W$ (and thus $T\subseteq W$), it follows that $F$ factors through a left adjoint $\precat_{(\infty,n)}\to\mathcal{C}$, which by a small abuse we will also call $F$. Composing this left adjoint with the fully faithful left adjoint $j_{!}\colon\cat_{(\infty,n)}\hookrightarrow\precat_{(\infty,n)}$, we obtain our desired left adjoint $K\mathrel{\mathop{:}}=Fj_{!}$.

To construct the desired natural transformation $\eta$, compose the counit $j_{!}j^{*}\to\id$ with $F$ to obtain $Kj^{*}\to F$, and then restrict along Yoneda to obtain $\eta\colon Kg\to f$. By definition, $\eta$ is an equivalence when restricted to $\Upsilon_{n}$, and thus a fortiori when restricted to $\mathbb{G}_{n}$. ∎