Kerodon

Proof of Theorem 6.2.5.10. Let $\operatorname{\mathcal{C}}$ be an $\infty $-category and let $S$ be a class of short morphisms for $\operatorname{\mathcal{C}}$. For every nonnegative integer $p$, let $\operatorname{\mathcal{C}}^{\leq p}$ denote the smallest simplicial subset of $\operatorname{\mathcal{C}}$ which contains all simplices of priority $\leq p$ (so that a nondegenerate simplex of $\operatorname{\mathcal{C}}$ belongs to $\operatorname{\mathcal{C}}^{\leq p}$ if and only if it has priority $\leq p$). Then $\operatorname{\mathcal{C}}$ is the colimit of the sequence of inclusion maps

We will complete the proof by showing that each of these inclusion maps is inner anodyne. For the remainder of the proof, we fix a positive integer $p$; our goal is to show that the inclusion map $\operatorname{\mathcal{C}}^{\leq p-1} \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}$ is inner anodyne.

Choose a function $\sigma \mapsto \sigma ^{+}$ satisfying the requirements of Lemma 6.2.5.15. For each integer $n \geq 0$, let $\operatorname{\mathcal{C}}^{\leq p}(n)$ denote the simplicial subset of $\operatorname{\mathcal{C}}^{\leq p}$ generated by $\operatorname{\mathcal{C}}^{\leq p-1}$ together with all simplices of the form $\sigma ^{+}$, where $\sigma $ is a simplex of $\operatorname{\mathcal{C}}$ having priority $p$ and dimension $\leq n$. By virtue of requirement $(2)$ of Lemma 6.2.5.15, it suffices to allow $\sigma $ to range over nondegenerate simplices which satisfy these conditions. Note that each $\operatorname{\mathcal{C}}^{\leq p}(n)$ contains the $n$-skeleton of $\operatorname{\mathcal{C}}^{\leq p}$, so that $\operatorname{\mathcal{C}}^{\leq p}$ can be realized as the colimit of the sequence

It will therefore suffice to show that each of these inclusions is inner anodyne. For the remainder of the proof, we fix an integer $n > 0$; our goal is to show that the inclusion map $\operatorname{\mathcal{C}}^{\leq p}(n-1) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}(n)$ is inner anodyne.

Let $\{ \sigma _{t} \} _{t \in T}$ denote the collection of $n$-simplices of $\operatorname{\mathcal{C}}$ which have priority $p$ but are not contained in $\operatorname{\mathcal{C}}^{\leq p}(n-1)$. We first claim that $\operatorname{\mathcal{C}}^{\leq p}(n)$ is generated by $\operatorname{\mathcal{C}}^{\leq p}(n-1)$ together with the collection of $(n+1)$-simplices $\{ \sigma _{t}^{+} \} _{t \in T}$. To prove this, it suffices to show that if $\sigma $ is an $n$-simplex of $\operatorname{\mathcal{C}}$ which belongs to $\operatorname{\mathcal{C}}^{\leq p}(n-1)$, then $\sigma ^{+}$ also belongs to $\operatorname{\mathcal{C}}^{\leq p}(n-1)$. If $\sigma $ is degenerate, then we can write $\sigma ^{+} = s^{n}_ i( \sigma _0^{+} )$ when $\sigma _0$ is an $(n-1)$-simplex of $\operatorname{\mathcal{C}}$ having priority $\leq p$ (Lemma 6.2.5.15), and the desired conclusion follows from the observation that $\sigma _0^{+}$ is contained in $\operatorname{\mathcal{C}}^{\leq p}(n-1)$. We may therefore assume that $\sigma $ is a nondegenerate $n$-simplex of $\operatorname{\mathcal{C}}$. If $\sigma $ has priority $< p$, then $\sigma ^{+}$ also has priority $< p$ and is therefore contained in $\operatorname{\mathcal{C}}^{\leq p-1}$. We may therefore assume that $\sigma $ has priority $p$, and must therefore be of the form $\tau ^{+}$ where $\tau $ is an $(n-1)$-simplex of $\operatorname{\mathcal{C}}$ having priority $p$. Condition $(3)$ of Lemma 6.2.5.15 then guarantees that $\sigma ^{+}$ can be obtained from $\sigma $ by applying a degeneracy operator, and is therefore contained in $\operatorname{\mathcal{C}}^{\leq p}(n-1)$ as desired.

For each $t \in T$, we define the complexity of $t$ to be the integer $c(t) = \sum _{ q = p}^{n} \ell ( \sigma _{t}(p-1) \rightarrow \sigma _{t}(q) )$. Using Proposition 4.7.1.35, we can choose a well-ordering on $T$ for which the complexity function

is nondecreasing. For each $t \in T$, let $\operatorname{\mathcal{C}}^{\leq p}_{\leq t}(n)$ denote the simplicial subset of $\operatorname{\mathcal{C}}^{\leq p}(n)$ generated by $\operatorname{\mathcal{C}}^{\leq p}(n-1)$ together with the simplices $\sigma _{s}^{+}$ for $s \leq t$, and define $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$ similarly. Then the inclusion map $\operatorname{\mathcal{C}}^{\leq p}(n-1) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}(n)$ can be realized as a transfinite composition of inclusion maps $\{ \operatorname{\mathcal{C}}^{\leq p}_{< t}(n) \hookrightarrow \operatorname{\mathcal{C}}^{\leq p}_{\leq t}(n) \} _{t \in T}$. It will therefore suffice to show that each of these inclusion maps is inner anodyne.

Fix an element $t \in T$ and let $L_{t} \subseteq \Delta ^{n+1}$ be the inverse image of $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$ under the map $\sigma _{t}^{+}: \Delta ^{n+1} \rightarrow \operatorname{\mathcal{C}}$, so that we have a pullback diagram of simplicial sets

We will complete the proof by showing that $L_{t}$ coincides with the inner horn $\Lambda ^{n+1}_{p} \subseteq \Delta ^{n+1}$, so that the diagram (6.10) is also a pushout square (Lemma 3.1.2.11). This is equivalent to the following more concrete assertion:

$(\ast _{t})$

For $0 \leq i \leq n+1$, the $n$-simplex $d^{n+1}_{i}( \sigma _{t}^{+} )$ belongs to $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$ if and only if $i \neq p$.

Our proof proceeds by induction on $t$. We consider several cases:

• For $0 \leq i < p$, the $n$-simplex $d^{n+1}_ i( \sigma _{t}^{+} )$ has priority $< p$, and is therefore contained in $\operatorname{\mathcal{C}}^{< p} \subseteq \operatorname{\mathcal{C}}^{\leq p}(n-1) \subseteq \operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$.

• For $i = p$, the $n$-simplex $d^{n+1}_ i( \sigma _{t}^{+} )$ coincides with $\sigma _{t}$ (Lemma 6.2.5.15), which is not contained in $\operatorname{\mathcal{C}}^{\leq p}(n-1)$. Consequently, if $\sigma _{t}$ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n-1)$, then there exists some $t' < t$ such that $\sigma _{t}$ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{\leq t'}(n-1)$ but not in $\operatorname{\mathcal{C}}^{\leq p}_{< t'}(n-1)$. Applying our inductive hypothesis, we deduce that $\sigma _{t} = \sigma _{t'}$, which contradicts the inequality $t' < t$.

• For $p < i \leq n$, condition $(1)$ of Lemma 6.2.5.15 implies that $d^{n+1}_ i( \sigma _{t}^{+} )$ coincides with $d^{n}_{i-1}( \sigma _{t} )^{+}$, and therefore belongs to $\operatorname{\mathcal{C}}^{\leq p}(n-1) \subseteq \operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$.

• Suppose that $i = n+1$ and set $\tau = d^{n+1}_{i}( \sigma _{t}^{+} )$; we wish to show that $\tau $ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$. Note that $\tau $ has priority $\leq p$. If $\tau $ is contained in $\operatorname{\mathcal{C}}^{\leq p}(n-1)$, there is nothing to prove. We may therefore assume without loss of generality that $\tau $ is contained in $T$: that is, we have $\tau = \sigma _{t'}$ for some $t' \in T$. Set $X = \sigma _{t}(p-1) = \sigma _{t'}(p-1)$. For $p \leq q \leq n$, let $f_{q}: X \rightarrow \sigma _{t}(q)$ be the morphism determined by $\sigma _{t}$, and define $f'_{q}: X \rightarrow \sigma _{t'}(q)$ similarly. By construction, the morphism $f_{q}$ coincides with $f'_{q+1}$ for $p \leq q < n$. Moreover, the restriction of $\sigma _{t}^{+}$ to the $2$-simplex $\operatorname{N}_{\bullet }( \{ p-1 < p < n+1 \} $ determines a diagram

  \[ \xymatrix { & \sigma _{t'}(p) \ar [dr] & \\ X \ar [ur]^{ f'_{p} } \ar [rr]^{ f_{n} } & & \sigma _{t}(n) } \]

  which is an $S$-optimal factorization of $f_ n$, so that $\ell ( f'_{p} ) = \ell ( f_ n ) - 1$ by virtue of Lemma 6.2.5.13. It follows that the complexity $c(\sigma _{t'} )$ is given by

  \begin{eqnarray*} c(\sigma _{t'} ) & = & \sum _{q = p}^{n} \ell ( f'_{q} ) \\ & = & \ell (f'_ p) + \sum _{ q = p+1}^{n} \ell (f'_ q) \\ & = & \ell ( f_ n ) - 1 + \sum _{ q = p}^{n-1} \ell ( f_ q ) \\ & = & ( \sum _{q = p}^{n} \ell (f_ q) ) - 1 \\ & = & c(\sigma _{t} ) - 1. \end{eqnarray*}

  We therefore have $t' < t$, so that $\tau = \sigma _{t'} = d^{n+1}_ p( \sigma _{t'}^{+} )$ is contained in $\operatorname{\mathcal{C}}^{\leq p}_{\leq t'}(n) \subseteq \operatorname{\mathcal{C}}^{\leq p}_{< t}(n)$.