ScalingStacks

Proof. The set $S_{00}$ consists of the union of four subsets of maps, corresponding to the four families of fundamental pushouts of types (a), (b), (c), and (d) in Axiom (C.3). The second and last subsets corresponding to the (b) and (d) families pullback to morphisms which are contained in the generating set of $T_{\Theta_{n}}$. Thus it remains to prove the that the same holds for the remaining families (a) and (c). In particular, we wish to show that for each $0\leq i\leq n$, each $0\leq j,k\leq n-i$, and every nondegenerate morphism $C_{i+j}\to C_{i}$ and $C_{i+k}\to C_{i}$, the natural morphism

is contained in $T_{\Theta_{n}}$ where the pushout is formed as in Notation 6.5.

In fact a stronger statement holds (cf. [34, Proposition 4.9]). For each object $o\in\Theta_{n}$ we have a natural bijection of sets

Thus Eq. 13.2.1 is in fact an equivalence in the presheaf category $\pre(\Theta_{n})$. In particular the family (c) pulls back to a family of equivalences, which are hence contained in $T_{\Theta_{n}}$. An virtually identical argument applies the family (a), which also consists of morphisms pulling back to equivalences of presheaves. ∎

Proof. The functor $\sigma_{!}$ is a left adjoint, hence it preserves all colimits in $\pre(\Theta_{n})$, in particular pushouts. Moreover, $\sigma_{!}$ sends the generators of $T_{\Theta_{n-1}}$ to generators of $T_{\Theta_{n}}$. Together these imply the containment $\sigma_{!}(T_{\Theta_{n-1}})\subseteq T_{\Theta_{n}}$. Direct computations, which we leave to the reader, show that $\sigma_{!}$ sends $T_{\Theta_{n-1}}$-local objects to $T_{\Theta_{n}}$-local objects and that the following formula holds,

where $i_{k}:[n]\to[1]$ maps $i\in[n]$ to $0\in[1]$ if $i<k$ and to $1\in[1]$ otherwise. In the above formula when $k=0$ or $n+1$ the space $\map(o_{k},X\times_{Y}Z)$ is interpreted as a singleton space. From this it follows that $\sigma$ preserves fiber products. ∎

Proof. First note that if each of the $b_{i}$ were a representable presheaf, then the map $A\to B$ may be written as a pushout of the generating morphism $\mathcal{T}_{n,\infty}$, hence is manifestly an element of $T_{\Theta_{n}}$ (we leave this as an exercise). The general case, however, reduces to this case as every presheaf is (canonically) a colimit of representables, the functors $\sigma^{[\ell]}_{!}$ commute with these colimits separately in each variable, and $T_{\Theta_{n}}$, being a saturated class, is closed under colimits. ∎

Proof. As colimits are universal in $\pre(\Theta_{n})$, the functors $(-)\times_{C_{i}}\nu H$ preserves all small colimits. Thus the class $[(-)\times_{C_{i}}\nu H]^{-1}(T_{\Theta_{n}})$ is a saturated class in $\pre(\Theta_{n})$. Taking appropriate intersections of these classes and $T_{\Theta_{n}}$ yields the three classes in question. ∎

Proof. By Lemma 13.7, it is enough to check that the generators of $\mathrm{Segal}_{\Theta_{n}}$ and $\mathrm{Comp}_{\Theta_{n}}$ of $T_{\Theta_{n}}$ are contained in $T_{\Theta_{n}}^{(c)}$. For that assume that $[U\to V]$ is a generator of $T_{\Theta_{n}}$ and let $C_{j}\hookrightarrow C_{i}$ be an inclusion. We wish to demonstrate that for all $V\to C_{i}$ we have that

is in $T_{\Theta_{n}}$. Recall that

There are several cases:

1. (a)

   $i=j$. In this trivial case (13.8.1) reduces to $[U\to V]\in T_{\Theta_{n}}$.

2. (b)

   The morphism $V\to C_{i}$ factors as $V\to C_{0}\to C_{i}$. In this case the fiber product $C_{j}\times_{C_{i}}C_{0}$ is either $C_{0}$ or empty. In the latter case (13.8.1) is an isomorphism, and in the former case it is $[U\to V]\in T_{\Theta_{n}}$. Notice that this case covers $\iota_{!}\mathrm{Comp}_{\Delta}$.

3. (c)

   $j=0<i$ and the map $V\to C_{i}$ does not factor through $C_{0}$. In this case it follows that $[U\to V]$ is not in $\iota_{!}\mathrm{Comp}_{\Delta}$, and hence

   $$V=([m];o_{1},...,o_{m})$$

   is representable with $m\neq 0$. Moreover the map

   $$V\to C_{i}=([1];C_{i-1})$$

   consists of a surjective map $[m]\to[1]$ (which specifies a unique $0<k\leq m$; the inverse image of $0\in[1]$ consists of all elements strictly less than $k$) together with map $o_{k}\to C_{i-1}$. A direct calculation shows that in this situation (13.8.1) is either an isomorphism or a generator of $T_{\Theta_{n}}$.

4. (d)

   $0<j\leq i$, the map

   $$[U\to V]\in\sigma_{!}\mathrm{Comp}_{\Theta_{n-1}}\cup\sigma_{!}\mathrm{Segal}_{\Theta_{n-1}}$$

   is a suspension, and the map $V\to C_{i}$ does not factor through $C_{0}$. It follows that $V\to C_{i}$ is the suspension of a map. This case then follows by induction and Lemma 13.3.

5. (e)

   The final case is when $0<j<i$, the map $[U\to V]\in Se_{\Theta_{n}}$ is not a suspension, nor in $\mathrm{Comp}_{\Theta_{n}}$, and the map $V\to C_{i}$ does not factor through $C_{0}$. In this case (13.8.1) is a map of the form described in Lemma 13.5. ∎

Proof. First note that as $\Theta_{n}$ is dense in $\pre(\Theta_{n})$, and $T_{\Theta_{n}}$ is strongly saturated, it is enough to consider $H\in\Theta_{n}$ representable. Let $f\colon U\to V$ be a morphism in $T_{\Theta_{n}}$, let $V\to C_{i}$ be given, and let $H\to C_{i}$ be arbitrary. There exists a unique factorization $H\to C_{k}\hookrightarrow C_{i}$, with $H\to C_{k}$ nondegenerate. Consider the following diagram of pullbacks in $\pre(\Theta_{n})$:

Since $T^{(c)}_{\Theta_{n}}=T_{\Theta_{n}}$, we have $[U^{\prime}\to V^{\prime}]\in T_{\Theta_{n}}$, and if $T^{(b)}_{\Theta_{n}}=T_{\Theta_{n}}$, then we also have $[U^{\prime\prime}\to V^{\prime\prime}]\in T_{\Theta_{n}}$, as desired. ∎

Proof. Let us focus on the case $x=a$. Let $f\colon U\to V$ be in class given on the right-hand side of the asserted identity. We wish to show that $f\in T_{\Theta_{n}}^{(a)}$, that is for any pair of morphism $H\to C_{i}$ and $V\to C_{i}$ we have

is in $T_{\Theta_{n}}$. This follows as there exists a factorization $V\to C_{k}\to C_{i}$ and a diagram of pullbacks:

such that $V\to C_{k}$ is nondegenerate. The analogous result for $T_{\Theta_{n}}^{(b)}$ follows by the same argument and the observation that $H^{\prime}\to C_{k}$ is nondegenerate if $H\to C_{i}$ is such. ∎

Proof. For the special case $i=0$, a more general version of this statement was proven by Rezk [34, Proposition 6.6] and forms one of the cornerstone results of that work. Our current proof builds on Rezk’s ideas.

The fundamental argument is to construct a category $\mathcal{Q}$ along with a functorial assignment of commuting squares

for each $\alpha\in\mathcal{Q}$. This assignment is required to satisfy a host of conditions.

First, each of the functors $A,B:\mathcal{Q}\to\pre(\Theta_{n})$ is required to factor through $\tau_{\leq 0}\pre(\Theta_{n})$, the category of 0-truncated objects. The 0-truncated objects of $\pre(\Theta_{n})$ consist precisely of those presheaves of spaces taking values in the homotopically discrete spaces. There is no harm regarding such objects simply as ordinary set-valued presheaves, and we will do so freely.

Second, we require that for each $\alpha\in\mathcal{Q}$ the natural morphism $A_{\alpha}\to B_{\alpha}$ is in the class $T_{\Theta_{n}}$. As $T_{\Theta_{n}}$ is saturated, this second condition implies that the natural map $\colim_{\mathcal{Q}}A\to\colim_{\mathcal{Q}}B$ is also in $T_{\Theta_{n}}$, where these colimits are taken in the $\infty$-category $\pre(\Theta_{n})$ (hence are equivalently homotopy colimits for a levelwise model structure on simplical preseheaves, see Rk. 13.1).

Third and last, we require that the natural maps $\colim_{\mathcal{Q}}A\to U\times_{C_{i}}H$ and $\colim_{\mathcal{Q}}B\to V\times_{C_{i}}H$ are equivalences in $\pre(\Theta_{n})$ (i.e., levelwise weak equivalences of space-valued presheaves). If all of the above properties hold, then we obtain a natural commuting square

in which the indicated morphisms are in the class $T_{\Theta_{n}}$. As this class is saturated it follows that $U\times_{C_{i}}H\to V\times_{C_{i}}H$ is also in this class. Thus if such a $\mathcal{Q}$ and associated functors can be produced, we will have completed the proof.

At this point we deviate from Rezk’s treatment. Specifically our category $\mathcal{Q}$ and associated functors will differ from his. We will focus on the more complicated case $i>0$, and leave the necessary simplifications in the case $i=0$ to the reader (or simply refer the reader to [34, Proposition 6.6] ).

Under the assumptions of the statement of the lemma we have the following identifications of presheaves:

where $0\leq k\leq m$ and $o_{\alpha},u_{\beta}\in\Theta_{n-1}$ are given. If $i>0$, then the $i$-cell is the representable presheaf $j([1];C_{i-1})$. A nondegenerate map $V\to C_{i}$ includes a nondegenerate map $f:[m]\to[1]$, and likewise a nondegenerate map $H\to C_{i}$ includes a nondegenerate map $g:[n]\to[1]$. Let $m^{\prime}$ be the fiber over $0\in[1]$, and let $m^{\prime\prime}$ be the fiber over $1$. Then $[m]=[m^{\prime}]\cdot[m^{\prime\prime}]$ is the ordered concatenation of $[m^{\prime}]$ and $[m^{\prime\prime}]$. Similarly $[n]=[n^{\prime}]\cdot[n^{\prime\prime}]$ is the ordered concatenation of the preimages of $0$ and $1$ under $g$.

Let $\delta=(\delta^{\prime},\delta^{\prime\prime}):[p]\to[m]\times_{[1]}[n]$ be a map which is an inclusion. There is a unique $-1\leq r\leq p$ such that under the composite $[p]\to[m]\times_{[1]}[n]\to[1]$, an element $s$ maps to $0$ if and only if $s\leq r$ (hence maps to $1$ if and only if $s>r$). Associated to $\delta$ we have a subobject $C_{\delta}$ of $V\times_{C_{i}}H$, of the form $C_{\delta}=\sigma_{!}^{[p]}(c_{1},\dots,c_{p})$, where $c_{\ell}$ is given by the following formula:

if $\ell-1\neq r$, and if $\ell-1=r$ by

where the indices range over all $\delta^{\prime}(\ell-1)<\alpha<m^{\prime}$, $\delta^{\prime\prime}(\ell-1)<\beta<n^{\prime}$, $m^{\prime}<\lambda\leq\delta^{\prime}(\ell)$, and $m^{\prime}<\beta\leq\epsilon^{\prime\prime}(\ell-1)$. We have found the graphical image in Figure 1 to be especially useful in understanding the combinatorics of these subobjects.

As subobjects of $V\times_{C_{i}}H$, the $C_{\delta}$ are naturally arranged into a poset. Let $W$ denote the disjoint union of all the maximal elements of this poset. Let $B_{\bullet}$ denote the simplicial Čech nerve associated to the morphism $W\to V\times_{C_{i}}H$. Each layer of $B_{\bullet}$ consists of a disjoint union of certain $C_{\delta}$. The map $W\to V\times_{C_{i}}H$ is a surjective map of set-valued presheaves. It follows that it is also an effective epimorphism in the $\infty$-topos $\pre(\Theta_{n})$, and hence [28, Corollary 6.2.3.5] the (homotopy) colimit of the simplcial diagram $B_{\bullet}$ is equivalent to $V\times_{C_{i}}H$. We set $\mathcal{Q}=\Delta$ and $B=B_{\bullet}$.

We define $A_{\bullet}$ to be the fiber product of $B_{\bullet}$ with $U\times_{C_{i}}H$ over $V\times_{C_{i}}H$. Because colimits in $\infty$-topoi are universal, we have

Thus all that remains is to show that the natural transformation $A_{\bullet}\to B_{\bullet}$ is levelwise in $T_{\Theta_{n}}$.

As each layer of $B_{\bullet}$ is a disjoint union of certain $C_{\delta}$, it is sufficient to show that the map

is in $T_{\Theta}$ for each $C_{\delta}$. As in the previous construction, there exist unique $0\leq r\leq s\leq p$ such that $\delta^{\prime}(t)<k$ if and only if $t<r$, and $k<\delta^{\prime}(t)$ if and only if $s<t$. The interval $\{r,\dots,s\}\subset[p]$ is precisely the preimage of $\{k\}$ under $\delta^{\prime}$. We then have

and so the desired result follows from Lemma 13.5. ∎

Proof. By Lemma 13.9, it is enough to show that the strongly saturated classes $T_{\Theta_{n}}^{(b)}$ contains the generating sets $\mathrm{Segal}_{\Theta_{n}}$ and $\mathrm{Comp}_{\Theta_{n}}$. By Lemma 13.10, it suffices to show that for any $[U\to V]\in\mathrm{Segal}_{\Theta_{n}}\cup\mathrm{Comp}_{\Theta_{n}}$, any nondegenerate morphism $[H\to C_{i}]\in J_{b}$, and any nondegenerate morphism $V\to C_{i}$ of $\pre(\Theta_{n})$, we must show that

is contained in $T_{\Theta_{n}}$. Observe the following:

• •

  If $[U\to V]\in\mathrm{Segal}_{\Theta_{n}}$ is not in the image of $\sigma_{!}$, then $U\to V$ is contained in $T_{\Theta_{n}}^{(b)}$ by Lemma 13.11.

• •

  If $[U\to V]\in\mathrm{Comp}_{\Theta_{n}}$ is not in the image of $\sigma_{!}$, then $V=C_{0}$, and the only nondegenerate map $V\to C_{i}$ occurs when $i=0$. In this case $U\to V$ is in $T_{\Theta_{n}}^{(b)}$ by [34, Proposition 6.1].

Thus we may restrict our attention to those generators $U\to V$ that lie in the image of $\sigma_{!}$. We proceed by induction. When $n=1$, the set of generators in the image of $\sigma_{!}$ is empty.

Assume that

and let $U\to V$ be an element of $\mathrm{Segal}_{\Theta_{n}}\cup\mathrm{Comp}_{\Theta_{n}}$ that lies in the image of $\sigma_{!}$. Now note that if $C_{i}=C_{0}$, then $U^{\prime}\to V^{\prime}$ lies in $T_{\Theta_{n}}$, again by [34, Proposition 6.1]. If $i\neq 0$, then by Lemma 13.4, the map $V\to C_{i}$ is also in the image of $\sigma_{!}$. In this case, if we have a factorization $H\to C_{0}\to C_{i}$ (which, since $H\to C_{i}$ is nondegenerate, can only happen if $H=C_{0}$), then $U^{\prime}\to V^{\prime}$ is an equivalence (as both are empty). Hence it $U\to V$ lies in $T_{\Theta_{n}}$.

This leaves the final case, where both $[U\to V]$ and $[V\to C_{i}]$ lie in the image of $\sigma_{!}$, and $[H\to C_{i}]$ is nondegenerate with $H=j({[m]};o_{1},\dots,o_{m})\neq C_{0}$, for some $m\geq 1$, $o_{i}\in\Theta_{n-1}$. The nondegenerate map $H\to C_{i}=({[1]};C_{i-1})$ is given explicitly by the following data (see also the proof of Lemma 13.3): a map $i_{k}:{[m]}\to{[1]}$ for some $1\leq k\leq m$ such that $i_{k}(i)=0$ if $i<k$ and $i_{k}(i)=1$ otherwise, together with a single (nondegenerate) map $o_{k}\to C_{i-1}$. In this case we may explicitly compute the pullback

and deduce that it is contained in the class $T_{\Theta_{n}}$.

As $[U\to V]$ is in the image of $\sigma_{!}^{[1]}$, it is of the form $({[1]};U^{\prime\prime})\to({[1]};V^{\prime\prime})$ for some $[U^{\prime\prime}\to V^{\prime\prime}]$ in $\mathrm{Segal}_{\Theta_{n-1}}$. The pullback is then given explicitly as:

This map arises as the right-most vertical map in the following (oddly drawn) commuting square:

The left-most vertical map is a pushout of identities and (by induction) a map in $\sigma_{!}(T_{\Theta_{n-1}})$. Thus by Lemma 13.3 it is contained in $T_{\Theta_{n}}$. Both horizontal maps are contained in $T_{\Theta_{n}}$ by [34, Proposition 6.4], whence the right-most vertical map $[U^{\prime}\to V^{\prime}]$ is also contained in $T_{\Theta_{n}}$, as desired. ∎

Proof. As $i_{!}$ commutes with colimits, to show that $i_{!}(T_{\Theta_{n}})\subseteq S$ it is sufficient to show this property for a subset that generates $T_{\Theta_{n}}$ under colimits. The maps in $\mathrm{Comp}_{\Theta_{n}}$ are clearly mapped into $S$. This leaves the maps $\mathrm{Segal}_{\Theta_{n}}$. We now write $S=S_{n}$ and induct on $n$. When $n=0$, one has $\Upsilon_{0}=\Theta_{0}=\mathrm{pt}$.

Assume that $i_{!}(T_{\Theta_{n-1}})\subseteq S_{n-1}$. The suspension functor $\sigma_{!}\colon\pre(\Upsilon_{n-1})\to\pre(\Upsilon_{n})$ preserves colimits and sends the generators $S_{n-1}$ into $S_{n}$. Hence the suspensions of maps in $i_{!}(T_{\Theta_{n-1}})$ are in $S_{n}$. Moreover, by construction the image under $i_{!}$ of the following map